JP2003128041A - Cylindrical folding structure - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a cylindrical folding structure capable of folding and unfolding until parts surrounded by folding lines become an adhered state. SOLUTION: The cylindrical folding structure is provided with a cylindrical wall (1) having a plurality of foldable linear mountain fold lines (M) and valley fold lines (V), a bottom wall (2) for closing one end part in the axial direction of the cylindrical wall (1), and the plurality of folding lines (M and V) satisfying a continuous condition which is a condition for continuing the folding lines (M and V) at the both ends of a developed figure of the side wall, satisfying a folding condition which is a condition capable of folding each parts (P) containing the plurality of folding lines (M and V) crossing at one nodal point in the adhered state and satisfying a closing condition which is a condition for adhering the parts (P) and folding the cylindrical wall (1) in the axial direction when they are folded along the folding lines (M and V).

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a folding structure which can be folded / expanded so as to be deformed between a folded state in which the outer shape is small and a deployed state in which the outer shape is large, and more particularly, it is a cylindrical shape or a rectangular tube. -Shaped, conical or pyramidal folded structures. INDUSTRIAL APPLICABILITY The present invention is applicable to, for example, foldable plastic bottles, paper cups, metal cans such as coffee cans, and instant food containers.

[0002]

2. Description of the Related Art Research on the development of folding and unfolding structures is technically carried out by constructing structures for antennas and solar cells for unfolding in outer space, or by studying plastic buckling using the folding method. Developed in relation to. Moreover, these studies have also been applied to studies aimed at elucidating growth and motor functions of living things such as the mechanism of insect wings and folding of leaves. Through these researches,
Various foldable tubular (cylindrical, prismatic, conical and pyramidal) folding structures are known, for example, Japanese Utility Model Publication No. 59-188837 and Japanese Utility Model Publication 57-6441.
The folding structure disclosed in Japanese Patent Laid-Open No. 5 and Japanese Patent Laid-Open No. 11-223299 is conventionally known.

[0003]

Previously, the folding condition which is a condition for folding each part surrounded by the fold line in a close contact state, and the parts are closely adhered when folded along the fold line, The closing condition, which is a condition for the tubular folding structure to be folded in the axial direction, has not been known. Therefore, conventionally, it is possible to manufacture only a folding structure composed of only parts with a very limited shape, and a folding structure in which the parts cannot be folded in close contact with each other (they can be folded only halfway). However, it was not possible to manufacture a folding structure having an arbitrary shape according to the application.

In view of the above circumstances, the present inventor has conducted repeated research on a foldable / unfoldable tubular structure, and as a result, when the tubular structure is folded along a plurality of fold lines formed on its sidewall, folding is performed. We have derived the conditions that multiple folding lines must satisfy in order for parts surrounded by lines to be folded in close contact. The research conducted by the present inventor will be described below.

(Explanation of Research by the Inventor) As a result of research on the tubular folding / unfolding structure, the inventor found the following. (A) Regarding a pseudo cylindrical wall and a conical wall (cylindrical wall) whose side wall is formed by a large number of divided plane walls (parts), a fold line formed on the outer side of the divided plane wall has a predetermined folding condition and When the closing condition is satisfied, the cylindrical wall is foldable with the divided plane walls in close contact with each other. (B) The present inventor has clarified all the folding conditions and closing conditions of the cylindrical wall, the prismatic wall, the conical wall, and the pyramidal wall.
According to the folding condition and the closing condition, the shape of the divided plane wall (part) may be various shapes other than the conventionally known shape and the shape that has been conventionally studied (isosceles triangle, right triangle, isosceles trapezoid, etc.). It is possible to configure. (C) The present inventor has clarified a general drawing method of a development view of a cylindrical wall, a prismatic wall, a conical wall, and a pyramidal wall that satisfy the folding condition, the closing condition, and the continuous condition.

Next, the details of the research results will be described. Here, we focus on clarifying the possibility of folding from a geometrical point of view, make a general discussion of the folding method using the origami model, and then develop the foldable cylindrical structural model. It will be clarified analytically by using.

1. Folding Method (Explanation of Folding Conditions) First, the folding condition that is a condition for the parts surrounded by the folding lines to be folded in close contact with each other is defined by the number of folding lines and the mountain fold lines and valleys gathering at one node. The fold lines will be explained separately for each pattern. The following description regarding the folding conditions is described in detail in the present inventor's paper, The Japan Society of Mechanical Engineers, Volume 66, No. 643. (1) 1-node 4-fold line method FIG. 1 is a fold line explanatory diagram showing a typical example of a fold line that is a straight line on which a folded structure is folded and a node that is an intersection of a plurality of fold lines. In FIG. 1, the fold lines formed by mountain folds are represented by solid lines (M1, M2, M3) and the valley fold lines are represented by broken lines (V1).
M and NV. It is well known that the following equation holds between NM and NV at the node. ｜ NM-NV ｜ = 2 …………………………………………………… (1) If the total number of folding lines is NT, then NT = NM + NV. If equation (1) is, for example, NM-NV, then NT = 2 (1 + NV)
Therefore, it can be seen that the number of folding lines forming the node is "even". Therefore, it is found that the total number of folding lines NT ≧ 4, and NT = 4 is the minimum total number of folding lines for forming a node. As shown in Fig. 1, the X-axis is the mountain fold line (M
When it is aligned with 3) and folded along the mountain fold lines (M1) and (M2), a valley fold (V1) occurs. At this time, the angles formed by the mountain fold lines (M1) and (M2) and the X axis are α and β, respectively.
And the angle between the valley fold line (V1) and the mountain fold line (M2) is γ, then γ = α …………………………………………………………………… ……… (2) is given. Expression (2) is a relational expression (folding condition expression) of the angle when completely folded along the Y-axis direction along the folding line (M1, M2, M3, V1).

By this operation, the band-shaped paper is folded in half and the axial direction to the right of the node is bent by 2α (when α <β) or 2β (when α> β). When α = β, the axial direction is not bent in the Y-axis direction. At this time, it is easily inferred that the band-shaped paper is bent zigzag when the mountain folds and the valley folds are alternately performed, and becomes cylindrical when only the mountain folds (or the valley folds) are continuously performed. In the present specification, the "planar folding" in which the flat paper is thus folded in a zigzag manner and folded in a new plane is not mentioned, but a folding method for producing a cylindrical structural model that can be folded in the same direction and folded in the Y-axis direction "Cylinder folding" will be described.

(2) 1-node 6-fold line method (type 1) FIG. 2 shows a folding condition of a 1-node 6-fold line of a type in which two valley fold lines are inserted at symmetrical positions of four mountain fold lines. FIG. As shown in FIG. 2, when there are one node and six fold lines where six fold lines intersect at one node, there are combinations in which two valley fold lines are inserted at symmetrical positions of four mountain fold lines. . This is a folding line method frequently used in this specification,
The angular relationship that satisfies the folding condition in this case is shown below. The mountain fold line is (M1), (M2), (M3), (M
4), the valley fold lines are (V1) and (V2), and the overtime of the valley fold line (V1) is the X axis. Fold line (M1) and (V
1), the angles formed by (M2) and (V1) are α and β,
The angles formed by (M3) and (V2) and between (M4) and (V2) are γ and δ. Then, when the angle formed by (V2) and the X axis is set to θ, the folding conditional expression is expressed by the following expression (3). β-α = δ-γ + θ …………………………………………………………………………………………………………………………………………………………………… (3) To be done.

In the case of the 1-node 6-fold line method shown in FIG. 2, the condition for folding at the node O in the Y-axis direction is derived. With the node O as the origin, the XY axes are taken as shown in FIG. Fold lines (M1), (V1), (M2) and vertical line (P) to the X-axis
Let p1, p2, p3 be the angles formed by and, and q be the angle formed by the vertical line (Q) and the fold lines (M4), (V2), (M3).
1, q2, q3, p1 = π / 2-α, p2 = π / 2, p3 = π / 2 + β, q1 = π / 2 + δ + θ, q2 = π / 2 + θ, q3 = π / 2−γ + θ .

By the mountain fold lines (M1), (M2), and the valley fold line (V1), the same direction with respect to the X-axis symmetrical position (point A, point B in FIG. 2) in the region of X <0 is set. The two vectors facing, using the relation p1 to p3, after folding,
An angle of p1−p2 + p3 = −α + β + π / 2 is formed. Further, in the area of X> 0, two vectors that make the X-axis symmetrical position (point C, point D in FIG. 2) your store by the mountain fold lines (M3), (M4) and the valley fold line (V2) are: After folding, an angle of q1−q2 + q3 = δ−γ + θ + π / 2 is formed. After folding, the points A, C and B, D are on the same plane, and the respective vectors are oriented in the same direction. Therefore, these two equations are equally placed to obtain the equation (3). can get. When the valley fold line (V2) coincides with the X axis, θ = 0 is set, and the following expression (3 ′) is established. β-α = δ-γ ………………………………………………………… (3 ′)

(3) 1-node 6-fold line method (type 2) In FIG. 3, valley fold lines (V1) and (V2) are alternately inserted between the mountain fold lines (M1), (M2) and (M3). When it is done 1
It is a figure which shows the folding conditions of the nodal point 6 folding line. As shown in FIG. 3, the XY axis is set with the node O as the origin. Then, in FIG. 3, X which is an extension of the mountain fold line (M4)
The angles formed by the axes and the mountain fold lines (M1) and (M3) are α ^* and β ^* , respectively, and the angles between the fold lines are θ1 to θ4. In the region where X <0, the condition for folding in the Y-axis direction at the node O is derived in the same manner as described above. After folding along the folding line (M4), it rotates by an angle π (= Q1) and faces in the opposite direction.

Next, the symmetrical position of the X axis in the region of X> 0 (see FIG. 3).
Consider the vector in the same direction at points D and E).
A perpendicular line (Q) to the X axis and a fold line (M1), (V1), (M
Assuming that the angles formed by 2), (V2) and (M3) are q1 to q5 as shown in FIG. 3, q1 = π / 2 + α ^* , q2 = π / 2 + α ^* , q3 = π / 2 + α ^* , q4 = π / 2-β ^* , q5 = π / 2−β ^* . For example, by folding at the mountain fold line (M1), the vector is at an angle of 2 with respect to the mountain fold line (M1).
Rotate by × q1. Therefore, the angle Q2 formed by these vectors due to these folds is Q2 = 2 (q1−
q2 + q3−q4 + q5) = π / 2 + α ^* − (θ2 + θ4). After folding, points B and D are on the same plane, and the vectors at this point point in the same direction. Therefore, if this value of Q2 and Q1 = π of the region of X <0 are equal, The folding conditional expression (4) is obtained. α ^* = θ2 + θ4 …………………………………………………………… (4)

Further, in FIG. 3, α ^* + β ^* = θ1 + θ2
Since there is a relation of + θ3 + θ4, if this formula is applied to the formula (4), β ^* = θ1 + θ3 ………………………………………………………… (5) obtain. Therefore, the folding conditional expression of the 1-node 6-fold line method of this type is expressed by α ^* = θ2 + θ4 or β ^* = θ1 + θ3 (6).

From the above description, in the case of the 1-node 6-fold line method, mountain folds (or valley folds) are always made on both sides of the central node. By repeating this folding method, the flat paper is automatically bent into a tubular shape by bending in the same direction. Therefore, using the 1-node 4-fold line method and the 1-node 6-fold line method, it is possible to manufacture a foldable tubular folding structure.

2. Manufacture of tubular folding structure with fold lines (Explanation of closing condition and continuous condition) It is easily inferred that a cylinder that can be folded in the vertical direction can be manufactured by continuously performing equiangular mountain folds. . Hereinafter, it will be considered to manufacture a foldable cylinder by such an operation. Regarding the closing conditions of the cylindrical tubular structure, the above-mentioned Proceedings of the Japan Society of Mechanical Engineers, Volume 66, 64
No. 3, the conditions for closing the conical tubular structure are described in the present inventor's paper, JSME Proceedings 66
Volume 647.

(1) Manufacture of a cylindrical tubular fold structure (1-1) Strip plate in which a cylinder is expanded (explanation of conditions for closing the cylindrical tubular structure) FIG. 4 shows the strip plate along the fold line. FIG. 4B is a diagram for explaining the conditions under which both end portions of the strip plate are joined together to form a cylinder when folded. FIG. It is a figure which shows the change of the direction of a reference axis | shaft when folding along a line. As shown in Fig. 4A, the strips are mountain-folded and valley-folded alternately, or N in the same direction.
Consider the case of diffraction (N: even number). N fold lines (1),
The angles formed by (2), ..., (N) and the X-axis are θ1, θ2, ..., θN, and the axial directions after folding are X1, X2, ..., XN, respectively.
By the first folding operation (fold line (1)), the right side portion of (1) becomes the back surface. This operation creates a new axis (X1)
Forms an angle of 2θ1 = Θ2 with the X0 axis (see FIG. 4B). When the second fold is made at the fold line (2), the X2 axis becomes the reference axis X0.
And the angle Θ2 = 2θ1-2θ2. X3 by Fold (3)
The axis is an angle of X0 and Θ3 = 2 (θ1−θ2 + θ3). By these series of folding operations, the front and back surfaces appear alternately,
The angle ΘN formed by the XN axis and the reference axis by folding operation N times
(When N = even) is expressed by the following equation. ΘN = 2 {θ1−θ2 + θ3−… −θN} …………………………………… (7) When this strip is folded, the left and right edges of the strip are joined together without any gaps. The condition (close condition) is given by the following expression (8) when n is an integer other than 0. ΘN / 2π = n ……………………………………………………………… (8)

(1-2) Cylinder with Folding Line and Pseudo Cylinder Composed of Folding Line Group of One-node Four-Folding Line Method Next, a band folded so as to satisfy the closing condition (equation (8)). A specific example in which the left and right ends of the plate are joined together without a gap will be described. 5A and 5B are explanatory diagrams of an example of folding into a regular quadrangle along a folding line that satisfies the closing condition and the folding direction is the same direction (either mountain fold or valley fold).
FIG. 5B is a diagram showing a fold line of the strip in the unfolded state, FIG. 5B is a diagram showing a state in the middle of folding, and FIG. 5C is a diagram showing a folded state. In FIG. 5A, the folding lines (1), (2), (3), and (4) that are folded in the same direction (either mountain fold or valley fold) of the strip that extends in the X-axis direction that is the reference axis are respectively X
Angles θ1, θ2, θ3, θ4 are formed with respect to the axis, and θ1
= Θ3 = 135 °, θ2 = θ4 = 45 °. That is, the fold lines (1), (2), (3), (4) are 45 ° (=
It is formed in zigzag at π / 4). Further, the left side of the folding line (1) of the X axis is the X0 axis, and n = 1, 2, 3, 4
In this case, the X-axis portion on the right side of each folding line (n) is defined as the Xn-axis (n = 1 to 4). In FIG. 5C, the angle Θ2 formed by the axis X2 and the axis X0 is Θ2 = 2 (θ1−θ2) = 2 (135 ° -4
5 °) = 2 × 90 ° = 180 ° = π. Also, the axis X
The angle Θ4 formed by 4 with the axis X0 is Θ4 = 2 (θ1−θ2 + θ3−θ
4) = 2 (135 ° −45 ° + 135 ° −45 °) = 2
π. Therefore, n in the equation (8) is n = (Θ
4 / 2π) = 1, and the axis X4 overlaps with the axis X0. In this case, both ends of the strip plate are joined together without a gap.

From the above description of FIG. 5, the strip plate is bent in the same direction along the folding line having the same folding direction (either mountain fold or valley fold) to form a regular N-gon (N is an even number).
It is understood that when folding is performed, folding lines (1), (2) ..., (N) having an angle θ = π / N with respect to the reference axis X may be formed in zigzag at equal intervals.

(1-2-1) Specific Examples of Cylinder with Folding Line and Pseudo Cylinder with Main Folding Line Consisted of Horizontal Folding Lines Considering the upper and lower ends of the band-shaped paper in FIG. 4A as horizontal folding lines,
Assume a plane paper in which these are connected in several stages in the Y-axis direction.
The parallel horizontal fold line (group) is named the main fold line. When the two ends of the plane paper thus formed in the vertical direction are joined together, a tubular folding structure with folding lines is formed. Specific examples thereof are shown in FIGS. FIG. 6 is the same as FIG.
When the strip-shaped plate shown in A is bent in the same direction by π · (N−2) / N at equal intervals to form a regular N-gon, and N = 6
It is a figure which shows the typical development view in the case of. In the tubular structure shown in FIG. 6, when the folding angle is θ in the formula (7),
Considering that it can be bent by 2θ, six zigzag mountain fold lines (1) to (6) which are at an angle π / 6 with the horizontal fold line are introduced at equal intervals. Each mountain fold line is bent by π / 3, and finally a cylindrical structure is produced which is folded into a hexagonal sectional shape.

FIG. 7 is a trapezoidal element of unequal sides obtained by decomposing twice the angle (π / 3) between the mountain fold line and the horizontal fold line of FIG. 6 into α = 2π / 9 and β = π / 9. It is a development view of a pseudo cylinder constituted by. In the case of folding into a regular hexagon, the angle division can be arbitrarily selected as long as the total is π / 3. 8 shows the mountain fold line in the Y-axis direction of FIG. 6 by α.
= Π / 3 mountain fold line I and β = π / 6 valley fold line II, which is an explanatory view of a cylinder produced by introducing six sets of disassembled fold lines. FIG. 8B is the same as FIG. 8A.
FIG. 8 is a diagram showing a half-folded state of a folded cylinder produced when the both ends of the developed view of FIG. 8 are joined, and FIG. 8C is a diagram showing a state where the folded cylinder of FIG. 8B is further folded. Figure 8A
In here, as long as α-β = π / 6, the values of α and β can be freely selected. FIG. 9 is a diagram in which the points A and B of FIG. 6 are matched and the mountain folds are removed from the horizontal fold line. The diamond pattern ((1) to (1) to It is a development view of (3)). At this time, the cross-sectional shape at the horizontal fold line portion becomes an equilateral triangle, which corresponds to a model of diamond buckling in plastic buckling of a thin-walled cylinder.

FIG. 10 is a development view of a deformed diamond pattern composed of isosceles triangular elements. FIG. 11 is an explanatory view of a pseudo-cylindrical body having a development view that is symmetrical with respect to a horizontal fold line and that can be folded. FIG. 11A is a development view, and FIG. 11B is both ends of the development view of FIG. 11. The figure which shows the half-folded state of the folding cylinder produced when joining
FIG. 11C is a view of the same thing as FIG. 11B seen from a different direction. The five types of developed views shown in FIGS. 6 to 10 are applicable to all horizontal folding lines, but the developed views shown in FIG. 11 can be folded. In FIG. 11, it is needless to say that the folding condition expression and the expression (3) are satisfied at the point A due to the symmetry, but the expression (3) is also established at the point B. FIG. 12 is a diagram showing an example of a development view of a fold formed by only folding lines similar to the point B in FIG. FIG. 13 is a development view of a foldable cylindrical wall having a plurality of polygonal parts (flat plate walls) formed by folding lines. The cylindrical wall with the exploded view of FIG. 13 can create a foldable cylinder with multiple shaped polygonal parts.

In the development view of the tubular structure, when the parts surrounded by the plurality of folding lines are made of a flexible material such as paper, the tubular structure is formed with the folding lines extended. The thing becomes a cylinder. However, when the parts are made of a material that does not have flexibility, the cylindrical structure is a cylinder having unevenness on the surface because the parts are flat when the fold line is extended. In the present specification, the tubular structure having irregularities on its surface is referred to as a pseudo cylinder.

(1-2-2) Cylinder with a fold line in which the main fold line is inclined with respect to the horizontal line (along the spiral) or pseudo-cylindrical FIG. 14 is a development in which FIG. 6 is inclined by π / 6. FIGS. 14A and 14B are explanatory views of the pseudo cylinder having the drawings, and FIG. 14B is a diagram showing a half-folded state of a folding cylinder manufactured when both ends of the developed view of FIG. 14A are joined. FIG. 14A
Corresponds to a view obtained by cutting FIG. 6 with a straight line GH that is inclined by π / 6 with respect to the horizontal line and using the cutting line as the horizontal lower end. Figure 1
In 4A, the main fold line is inclined with respect to the horizontal line, and extends along the spiral around the central axis of the tubular structure on the surface of the tubular structure. This is called a spiral-shaped cylindrical folded structure. When the left and right ends of the development view of the spiral type structure are joined, generally, the continuity of the fold line is not always satisfied at both ends of the development view. When the development view is given by the trapezoidal element as in the case of FIG. 14, continuity can be maintained by appropriately selecting the upper base length Lu of the trapezoid.

(1-2-3) Conditions for continuous folding lines at both ends of the developed view (continuous condition) FIG. 15 is an explanatory diagram of a method for maintaining continuity when the two ends of the developed view are joined. is there. In FIG. 15, N points of trapezoidal elements are drawn in the direction of the main fold line (angle ψ) with the origin O as a base point to define a point A. When the height of the trapezoid is h, the length is OA = N {(h / tan θ) + Lu} in the case of a regular N-gon. Draw m (even number) elements below the Nth trapezoidal element, and define the point B as shown in the figure. The point B needs to be on the X-axis for the development to be continuous for any ψ. AB =
Since mh, tan ψ = OA / OB, the following equation (9)
To get Lu = {2N−m · tan ψ / tan θ} h / tan ψ (9) That is, if Lu is properly determined by the expression (9), the continuity of the left and right ends of the development diagram in these cases is determined. Is obtained.

(Proof that the closing condition is satisfied) When the cylinder having the above-mentioned folding line is folded, it is generally unknown whether or not the closing condition in the circumferential direction (see the equation (8)) is satisfied. Verify this. In the cylinder given in FIG. 14, consider the folding line of the strip plate portion (micro width D) at the lowermost end of this development view. There are 18 fold lines here, and fold lines with the same slope appear repeatedly every 6 lines from the left side, so they are made up of 3 sets of 6 fold lines. Using Expression (7), the rotation angle of the axis line by these fold lines is ΘN = 2 {(α + ψ) −ψ + ψ−ψ + (α + ψ) −ψ} × 3, where ψ (= π / 6) is the inclination angle. = 12α ... (10). Since α = π / 6, ΘT = 2π in equation (7)
Since the above expression (8) is satisfied, it can be seen that the closing condition is satisfied after folding.

Therefore, as shown in FIG. 14A, one node 4
A cylindrical structure that is obtained by cutting at an arbitrary angle so that the fold lines of the development view after cutting in a tubular structure composed of fold lines whose main fold lines are horizontal The structure is also folded in the axial direction if the development view before cutting satisfies the closing condition.

(1-2-4) Concrete Example of Folding Line Cylinder or Pseudo Cylinder with Main Folding Line Inclining with respect to Horizontal Line (along Spiral) In the cross-sectional view formed as described above 16 to 19 show specific examples of the tubular structure in which the main fold line is inclined with respect to the horizontal line. 16 is an explanatory view of a pseudo-cylindrical body having a development view in which FIG. 7 is inclined by π / 6, FIG. 16A is a development view, and FIG. 16B is manufactured when both ends of the development view of FIG. 16A are joined. It is a figure which shows the half-folded state of a folding cylinder. FIG. 17 is a development view in which FIG. 8 is inclined by π / 6. In the examples shown in FIGS. 16 and 17, when the cylinder is manufactured by joining the left end and the right end of the developed view, the pattern formed by the fold lines is not generally continuous, but a diagram formed so as to satisfy the above-described continuity condition. 16, the development view of FIG. 17 is continuous at both left and right ends. FIG. 18 is a spiral type of FIG. 11, and points A,
It is obtained by cutting along a straight line connecting D. The angle (~ 0.193π) shown in Fig. 18 indicates the angle formed by this cutting line and the horizontal line. In this case, since the shape of the triangular element is given, the angle of the valley fold line is limited. Become. Figure 1
9 is a view showing a portion cut out by two parallel straight lines AB 'and C'D in the development view of the cylindrical body shown in FIG. 12, so that A and B'and D and C'are overlapped. FIG. 20 is a development view of what becomes a foldable cylindrical body by connecting the left and right edges of FIG. 19. The cylindrical wall with the exploded view shown in FIG. 19 can create a foldable cylinder with multiple shaped polygonal parts.

(1-2-5) Others In addition, there are innumerable cylindrical structures formed by a group of folding lines that satisfy the folding condition, the closing condition and the continuous condition. An example thereof is shown in FIG. FIG. 20 is a development view of a collapsible cylinder having quadrangular elements (parts) of arbitrary shapes. In FIG. 20, when the straight line extending AF is AE, the folding condition is ∠BAE = ∠
DAC = α. The value of α can be arbitrarily determined as α = 180 ° / N (N is a positive integer). For example N
= 8, α = 180 ° / 8 = 22.5 °. Therefore, ∠BAE = ∠DAC = α = 22.5 °
As a result, by setting the length of AE to an appropriate arbitrary value, a foldable cylinder having parts of arbitrary shape can be created.

(1-3) Cylinder with fold line constituted by fold line group of 1-node 6 fold line method and pseudo cylinder (1-3-1) Confirmation of closing condition FIG. 21 satisfies the closing condition and the folding direction. 21A is an explanatory view of an example of folding into a regular hexagon by folding lines that are alternately inverted (inverted in the mountain fold direction and the valley fold direction), and FIG. 21A is a fold line (1) to ( 12) and FIGS. 21B to 21F are views showing a state in the middle of folding, FIG.
FIG. 1G is a diagram showing a folded state. In FIG. 21A, folding lines (1), (3) shown by solid lines, which are folded in the same direction (for example, the mountain folding direction) of the strip extending in the X-axis direction which is the reference axis,
, (11) form angles θ1, θ3, ..., θ11 with respect to the X axis, and θ1 = θ3 = ... = θ11 = 60 °. Further, folding lines (2), (4), ..., (1 shown by dotted lines that are folded in the direction opposite to the folding lines (1), (3) ,.
2) form angles θ2, θ4, ..., θ12 with respect to the X axis, and θ2 = θ4 = ... = θ12 = 30 °. The virtual line (13) shown in FIG. 21 is a fold line when the strip is folded.
It is a line that overlaps with (1). In FIG. 21, the angle Θ12 formed by the axis X12 and the axis X0 in the solid line is Θ12 = 2 (θ1−θ2 + θ3 −... +
θ11−θ12) = 2 (60 ° −30 ° + 60 ° −... + 60
° -30 °) = 2 × π. Therefore, n in the equation (8) becomes n = (Θ12 / 2π) = 1, and the axis X12
Overlaps the axis X0. In this case, both ends of the strip plate are joined together without a gap.

From the description of FIG. 21, the strip plate is bent in the same direction by a fold line in which the folding direction is alternately inverted (either mountain fold or valley fold) to form a regular N polygon (N
Is an integer), (θ2m
It is understood that the mountain fold lines and the valley fold lines (1), (2), ... (2N) that satisfy −θ2m + 1) = π / N (m is an integer) may be formed in a zigzag at equal intervals.

(1-3-2) Concrete Example of Cylinder with Folding Line and Pseudo Cylinder with Main Folding Line consisting of Horizontal Folding Line Group 1 Similar to the case of the nodal 4 fold line method, the band-shaped paper of FIG. 21A is used. Consider the upper and lower ends as horizontal folding lines, and assume a plane paper in which these are connected in several stages in the Y-axis direction. Then, the parallel horizontal fold line (group) becomes the main fold line. When the two ends of the plane paper thus formed in the vertical direction are joined together, a tubular folding structure with folding lines is formed. A concrete example thereof is shown in FIG.
~ Shown in FIG. 22A and 22B are explanatory views of a pseudo-cylindrical body having a development view that is symmetrical with respect to the horizontal fold line and that can be folded. FIG. 22A is a development view, and FIG. 22B is both ends of the development view of FIG. 22. It is a figure which shows the half-folded state of the folding cylinder produced when joining. In the example shown in FIG. 22, when the left end L and the right end R of the development view are joined to manufacture a cylinder, the pattern formed by the fold lines is continuous. And α = β = π
Since it is set to / 6, the folding condition is satisfied, and Θ6 = 2 (6 × π / 6) = 2π, and n in the above formula (8) becomes n = (Θ6 / 2π), so it is closed. Satisfies the conditions. Therefore, the cylinder shown in FIG. 22 is folded.

FIG. 23 is an explanatory view of a folding cylindrical body having a fold line which is a generalization of FIG. 22, and FIG. 23A is a developed view.
FIG. 23B is a diagram showing a half-folded state of a folding cylinder produced when the both ends of the developed view of FIG. 23A are joined. Assuming the angles α and β as shown in FIG. 23A and considering the rotation angle by the folding line of the same six parallelogrammic parts at the lowermost end, the following formula is obtained. ΘT = 2 {(α + β) -β} × 6 = 12α (11) If α = π / 6 is used, ΘT = 2π and the condition is closed (see formula (8)). Is satisfied. As can be seen from the equation (11), it can be seen that ΘT does not depend on the β value. That is, in this model of a structure with a tubular fold line that is a 1-node 6-fold line type and the main fold line is a horizontal fold line group, the condition for folding into a regular N polygon is α = π / N, and The angle β can be freely selected. That is, it does not depend on the value of the angle β. Therefore, the fold line of the development view of the tubular structure shown in FIG. 23A satisfies the folding condition, the closing condition, and the continuous condition, and thus can be folded.

FIG. 24 is a developed view of the 6th stage of FIG.
FIG. 6 is a development view in the case where α is set to 30 ° and the value of β is changed for each stage. As shown in FIG. 24, even if the value of β is independently set for each stage, the folding condition and the like can be satisfied. FIG. 25 is a development view of a repetitive spiral type obtained by reversing the spiral mountain fold line and the valley fold line of FIG. 23A for each stage. This developed view can also be obtained by matching points A and B in FIG. The structure with the cylindrical fold line shown in FIG. 25 can also satisfy the folding conditions and the like.

(1-3-3) Cylinder with Folding Line and Pseudo-Cylinder with Folding Line Group with Main Folding Line Inclined (along Spiral) In FIG. 26, the line connecting AB in FIG. 23 is horizontal. It is a figure corresponding to what inclined the whole angle so that it becomes. FIG. 26
Is a development view of a tubular structure studied by Guest, who is a researcher of folded structures, and is made of triangular divided flat plates. The main fold line is spiral and the main fold line is one round. FIG. 3 is a developed view of the cylindrical structure when the spiral (1) rises by one step each time. They analyzed what characteristics the cylinder represented by the expanded view of FIG. 26 exhibits when folded, using the angles (α, β) between the spirals as variables, but it was not possible to show the complete folding condition. There wasn't. In FIG. 26, the main fold line is inclined with respect to the horizontal line, and when cut along a straight line connecting AB of FIG. 23 as in the developed view of FIG. 26, continuity of the fold lines at both ends of the developed view Is satisfied. However, since it is generally unknown whether or not the closing condition is satisfied, this will be verified.

(Verification of Closing Condition) In the above equation (7), Θ6 = 2 {6 × (α + β + ψ) -6 (β + ψ) + ψ- (β + ψ)} = 12α-2β. When this is used in the conditional expression (8) for closing, the following expression is obtained. Θ6 / 2π = (6α−β) / π = n ……………………………… (12) Therefore, when α = π / 6, β = 0, π ...
The closing condition is not satisfied. In this case, α = π / 5, β =
By setting π / 5, it is possible to manufacture a structure with a foldable cylindrical fold line that satisfies the closing condition and that can be folded along a spiral in which the main fold line rises by one step. Also, the formula (1
As can be seen from the process of deriving 2), by increasing the number of parallelograms arranged at the bottom from 6 to 7, Θ7 / 2π = (7α−β) / π = n, and α The closing condition holds even when = β = π / 6. That is, by appropriately setting the number of parallelograms in the lowermost stage, it is possible to manufacture a foldable cylindrical fold structure along a spiral in which the main fold line satisfying the closing condition is raised by one stage. Further, in the case where the spiral is raised by two steps every time the main fold line makes one turn, if the same derivation as in the equation (12) is performed, then Θ6 = (6α−2β) / π = n. Therefore, in general, the expression of the closing condition when the number of parallelograms at the bottom is L and the main folding line is raised by M steps is: ΘL = (L × α-M × β) / π = n ……………………………… (13)

(1-4) Manufacture of Foldable Pseudo Cylinder The present inventor investigated the folding property in the axial direction using a pseudo cylinder made of a polypropylene sheet having a thickness of 0.2 mm according to the above-mentioned development view, and I confirmed that is possible. When the spiral folding model shown in FIGS. 14 and 17 is pushed by the material testing machine, the upper part of the cylinder is folded while rotating while the lower part is stopped. The results of observing the evolution of these folds show that the proposed model is capable of good folding, and
It was shown that the load required for complete folding was an extremely low value of 20 to 40 N (diameter of the cylinder before folding; about 100 mm).

(1-5) Summary of Research on Cylindrical Body with Folding Line In the above description, N = 6 (partial N = 3, 8) is taken as an example,
The method of manufacturing the pseudo cylinder in which the developed view is divided into triangular elements, trapezoidal elements or quadrangular shapes of arbitrary shape and is folded into a regular N-gonal shape has been described. The folding angle at one node is (N-2) / N, except when the main folding line, which is difficult to satisfy the continuity at the left and right ends of the development view, is horizontal and is composed of an odd number of trapezoidal elements.
By setting π, it is possible to manufacture a folded structure for any N value (N ≧ 3, integer). Further, if the angle of the fold line is selected so as to satisfy the expression (8) and the length of the fold line is appropriately selected, it is possible to manufacture a folding structure that is not a regular N-gonal shape. When the cylinder is made of a thin polymer sheet, it seems that it is easy to mold it into a shape as shown in FIGS. 8B and 22B. Therefore, it is considered that a container such as a foldable PET bottle can be manufactured by performing a molding process with such a shape. The valley fold line
In the case of forming the spiral type, it is generally easier to expand and contract in the axial direction than that of the horizontal type. This seems to have to be considered in improving the folding structure.

(2) Manufacture of a conical tubular fold structure (2-1) 1-node 4-fold line method, and a cone composed of a fold line group whose main fold line is formed along the circumference and Pseudo-cone (examination of re-examination of folding conditions and closing conditions) Regarding the cones with fold lines, the folding conditions at each node are the same as those of the cylindrical tubular fold line structure described above (2), (3). , (6), but it is rare in the case of a cylindrical tubular folding structure that is clearly established, and verification is required for each fold line arrangement pattern. Further, it is necessary to derive the closing condition similarly to the structure with the cylindrical tubular folding line. Regarding the conical tubular folding structure, when a plurality of parts surrounded by fold lines are arranged along the circumferential direction, the fold line connecting the plurality of parts along the circumferential direction Called the main fold line.

(2-1-1) Folding Condition FIG. 27 shows a main part of a development view of a conical tubular fold structure in which a main fold line is formed along the circumference and parts are formed by trapezoidal elements. FIG. First, the folding conditions are examined. In FIG. 27, two straight lines (AC, BD) which form an angle α with AB from points A, B on the outer edge having an apex angle 2Θ with respect to the center O connect the midpoint I and the center O of AB. The points C and D are determined so as to be symmetrical with respect to the straight line OI and have the apex angle φ ^* . Hereafter, points are arranged in the central direction with the apex angle between 2Θ and φ ^* . In this way, the trapezoidal parts formed by the folding line groups all have similar shapes, and the fixed point C,
The line connecting D is parallel to AB. That is, if the point H is set on the extension of the straight line DC, AB and CD are parallel and ∠ACH = α
Since the parts are similar, ∠ECF = α. Therefore, the folding conditional expression (2) is satisfied.

(2-1-2) Closing Conditions The closing conditions for the conical tubular folding structure can be derived in the same way as the closing conditions for the cylindrical tubular folding structure described above. However, in the case of a cone, the development view has a central angle, which must be taken into account. Similar to the above-mentioned closing condition of the cylindrical folded structure, consider a strip plate formed by the trapezoidal ABCD in FIG. 27 and the trapezoidal ACFG formed on the right side thereof. The acute angle formed by each fold line of this strip and the main fold line along the circumference is α. By folding along this fold line,
AB and AG are bent by 2 × ∠CAB = 2α. Similarly, when N trapezoids are arranged along the circumference, they are bent by 2α × N by folding. The sum of the angle 2αN that changes due to this bending and the central angle {(2Θ + φ ^* ) N / 2} that the development view has from the beginning is 3 for one rotation.
When it becomes 60 ° (= 2π), both ends of the strip plate are joined (closed) without a gap. Therefore, the closing condition in the case of FIG. 27 is the following expression (14) 2αN + (2Θ + φ ^* ) N / 2 = 2π .... Let θ1, θ2, ... be the angles formed by
Using equation (7) and using Θ N, the central angle is φ (= (2Θ
+ Φ ^* ) N / 2), we have ΘN + φ = 2π ……………………………………………………………… (15) It was derived.

(2-1-3) Concrete Example of Cone and Pseudo Cone Consisting of 1-node 4 Folding Line Method and Folding Line Group in which Main Folding Line is Formed along Circumference FIG. 28 and FIG. Similarly, a conical wall with a fold line that is divided into isosceles trapezoids by a fold line and is folded into a regular N pyramid,
N = 6, φ ^* = π / 36, 2θ = π / 12 in FIG.
28 is an explanatory view of the pseudo-conical wall having a development view in the case of FIG.
28A is a developed view, and FIG. 28B is a perspective view of a state in which the conical wall with folding line having the developed view of FIG. 28A is half-folded. FIG. 29 is an explanatory view of how to draw the development view of FIG. 28. In FIG. 29, line segments (1) and (2) are drawn from the points A and G at the same angle φ, and line segments OB and OH are taken symmetrically with respect to the perpendicular line drawn from the point O to the base AG of ΔOAG. Let B and H be the intersections of. Φ is taken in the opposite direction from the points B and H, and the intersections with OA and OG are C and I. With such an operation, zigzag folding lines ABCDE ... And GHLJ. The folding condition at each node has been clarified in the above description. Also,
Since N = 6, φ ^* = π / 36, 2Θ = π / 12, the central angle is π / 3, and α = 5 from the formula (14) of the closing condition.
If π / 36, it is derived that the conical tubular folding structure can be folded.

(2-2) Concrete Example of Cone and Pseudo Cone Consisting of 1-node 4 Folding Line Method and Folding Line Group with Main Folding Line Inclined to Circumference FIG. FIG. 30A is an explanatory view of a pseudo cone having a development view in which 28 is made into a spiral shape. FIG. 30A is a development view, and FIG. is there. The tubular structure of FIG. 30 is a structure in which the folding line group is configured such that the main folding line is formed along the equiangular spiral in FIG. 28. FIG. 30A,
The exploded view of the trapezoid in which the fold lines are arranged along the spiral as shown in FIG. 30B can also form a foldable conical tubular structure. Since each fold line is formed by an equiangular spiral, the angular relationship established in FIG. 28 is also established in FIG. 30 (property of conformal spiral transformation). Therefore, the folding condition is satisfied because the angular relationship between the folding lines at each node is preserved. Regarding the closing condition, it is necessary to confirm whether it is individually established from the relationship between the angle between each folding line and the central angle. The continuous conditions will be described later.

(2-3) A cone and a pseudo-cone (2-3 -1) Folding Condition FIG. 31 is an enlarged view of a main part of a development view in the case where a development view of a cone whose main fold line is formed along the circumference is composed of N isosceles triangles having an apex angle 2Θ. . The valley fold line (broken line) in FIG. 31 is called a main fold line. 0 for the vertex, A for the outer edge,
B, C, and D are drawn, and a straight line forming an angle α with the perimeter is drawn from these points, and the respective intersections are set to E, F, and G. Points E, F,
Similarly to the above from G, a line forming an angle α with the line segments EF and FG is drawn, and the intersections thereof are defined as H and I. With this drawing, the development view is divided by two types of isosceles triangular elements.
Due to the symmetry, O, H, B and O, I, C form a straight line, and a diamond pattern symmetrical to the left and right of the straight line OF is obtained. Straight line O
F forms a right angle with the perimeter BC. The fold line forming the node F corresponds to that in FIG. ∠CFG = ∠BFE = β, and in consideration of the symmetry at the node F, δ in the folding conditional expression (3) is set as α, and γ is set as β. To study the folding condition at the node F, the angle between the valley fold lines EF and FG is required. Assuming the extension line FJ of the valley fold line EF, ΔOEF is ∠OFE = π / 2− from the isosceles triangle.
Θ, and ∠OFJ = π−∠OFE = π / 2 + Θ. △ OFG is also an isosceles triangle, so ∠OFG = π / 2-
From Θ, the relationship of ∠GFJ = ∠OFJ−∠OFG = 2Θ ……………………………… (16) is obtained. This ∠GFJ is the valley fold line EF and FG
Corresponds to the angle between and.

Regarding ΔEFB, ∠EBF = π−
From 2β, ∠OBF = π / 2−β. △ OBC is an isosceles triangle, so ∠OBC = π / 2-Θ = ∠OBF +
∠ From FBC, π / 2-Θ = π / 2-β + α That is, β-α = Θ ……………………………………………………………… (17) Relationship is obtained. When the relations of Expressions (16) and (17) are applied to the folding conditional expression (3), β-α- (δ-γ +
θ) = β−α− (α−β + 2θ) = 0. Therefore, the folding condition is satisfied, and the conical tubular folding structure shown in FIG. 31 can be folded. At this time, if ∠HFI = γ ^* is set, when folded at the node F,
The angle formed by the valley fold lines EF and FG is γ ^* -2α.

The folding condition will be verified for a more general case where each part as shown in FIG. 31 is not composed of an isosceles triangle. FIG. 32 is an enlarged view of a main part of a development view of a conical wall with a fold line when the element is divided into isosceles triangular elements by the fold line. In FIG. 32, the points on the outer side are A,
B, C, D ... At each point, the line segment forming the angle α with the outer edge is drawn in the upper right direction, and the line segment forming the angle δ is drawn in the upper left direction.
Let F and G (∠BOF = θ ^* ). A straight line is drawn from these points to the upper left with the line segments EF and FG at the angle α and to the upper right at the angle δ, and the intersections thereof are designated as H and I. Points O, H, B
And O, I and C form a straight line. Asymmetric diamond patterns are obtained on the left and right of the straight line OF. ∠BFE = β, ∠CFG = γ
Let J be the intersection of EF and BC. △ OBC and △ OC
D, ΔOEF and ΔOFG are isosceles triangles each having an apex angle 2θ, and ∠DCJ = ∠GFJ = 2θ, and ∠OFJ = ∠O
Get CJ. That is, the points O, F, C, and J are on the same circle, and ∠CJF = ∠FOC = 2θ−θ ^* . △ BF
Paying attention to J, the following equation is obtained. β-α = 2Θ-θ ^* ……………………………………………… (18) ∠CFJ = γ-2Θ obtained from the angular relationship around the point F Δ = ∠CFJ + (2Θ obtained from the angular relationship
When used for −θ ^* ), the following equation (19) is obtained. δ−γ = −θ ^* …………………………………………………… (19)

Substituting θ ^* in equation (19) into equation (18) and considering that the angle formed by the valley fold lines EF and FG is 2Θ, the following folding conditional expression (20 ') is established. One. β-α = δ-γ + 2Θ ……………………………………………… (20 ') If ∠HFI = γ ^* is set in the same way as before, when folded at the node F, The angle formed by the valley fold lines EF and FG is γ ^* -(α +
δ). Considering the folding of a regular N-sided polygon, equating this value with (N-2) / N · π and using γ ^* + (α + δ) = π-2Θ obtained from the geometrical relation The folding conditional expression (20) is obtained. (Α + δ) = π / N-Θ ……………………………………………… (20) If α and δ satisfying the equation (20) are selected, it is possible to fold it with unequal scale triangle elements. An unfolded view of a regular N-gon fold structure is obtained.

The folding condition in the case of the developed view in which the folding lines are formed along the equiangular spiral will be verified. 33 is an explanatory view of a development view of a conical wall with a foldable fold line having a fold line along an equiangular spiral, and FIG. 33A is an overall explanatory view, FIG.
3B is an enlarged view of a main part of FIG. 33A. Similar to FIG. 31, the angle formed by one pattern with respect to the center O is 2Θ,
Generally, it is expressed as shown in FIG. 33A. This FIG. 33A
Is drawn as follows: First, line segments (1) and (2) are drawn in the upper right direction so as to form an angle ψ with the radiation OA and OI from the center O starting from the points A and I. Next, draw line segments (4) and (5) from the points A and M in the upper left direction to form an angle φ with the radiation (ψ and φ
The values are α and δ in FIG. 32, and ψ = π / 2−Θ−α, φ = π /
2-Θ−δ). Assuming that the intersections of (1) and (5) and (2) and (4) are F and B, the points B and F come on concentric circles. Similarly, when the above operation is performed at the points B and F, the points C, J and G are determined, and the points D, K and H are sequentially determined. That is, the row of points F, G, H ... Taken in the upper right direction from the point A always forms an angle ψ with the radial direction, and the row of points A, B, C, D, E forms an angle φ with the radial direction. Is drawn as. Point A. If a line connecting F, G, and H is a new curve (1) and a line connecting points A, B, C, and D is a new curve (4), these two curves are equiangular with the radial direction. While becoming a line toward the center. That is,
Each of these points lies on a conformal helix exiting from the center O.
In FIG. 33A, (1), (2) and (3) are counterclockwise spirals, (4) and
(5) and (6) are clockwise spirals.

As shown in FIG. 33A, if the angle formed by the line segments AB, BC, ...
The angle formed by G and GH is 2 (Θ-Θ ′). The folding condition is examined using an enlarged view of two rectangles on the left and right of the point F (FIG. 33B). These rectangles are congruent and the line segment BF,
FG makes an angle 2Θ. The angle relationship between ψ, φ and α to δ is as shown in the figure. Since ΔOBF in FIG. 33A is an isosceles triangle with an apex angle 2Θ, α + φ = π / 2−Θ and δ + ψ = π
/ 2-Θ, and α + δ = π- (φ + ψ) -2Θ ……………………………… (21) is obtained. From the internal angle relation of ΔABF or ΔMFN, β + γ = π− (φ + ψ) ……………………………… (22) is obtained. The following equation is established from the equations (21) and (22). [beta]-[alpha] = [delta]-[gamma] +2 [Theta] (23) Considering that the line segments BF and FG form an angle 2 [Theta], the folding conditional expression (3) is established. That is, it is understood that the folding condition is automatically established when the folding line is drawn with the equiangular spiral. (Continuity condition) Further, the radius R ₁ of the points B and F is given by the following equation using the sine law with the radius of the developed view being R ₀ . R ₁ / R ₀ = sin {2 (Θ−Θ ′) + ψ} = p …………………… (24) The second stage point (C, J, G…) and the third stage from the outer circumference. The radii of the points (D, K, H ...) Are sequentially given by p ² , p ³ ... That is, if the values of p, p ² , p ³ ... Are the same at both ends of the development view, it means that the fold lines are continuous at both ends of the development view. Therefore, these p, p ² , p ³ ... Correspond to the continuous condition in the conical structure.

(2-3-2) Closing condition Since the closing condition is the same as the closing condition described in (2-1-2) above, a description thereof will be omitted.

(2-3-3) Concrete of a cone and a pseudo-cone constituted by a folding line group in which the main folding line formed by the 1-node 6-folding line method and the valley folding line is formed along the circumference Example FIG. 34 is an explanatory view of a conical tubular folding structure having a development view formed by folding line groups having the same angular relationship as FIG. 31, FIG. 34A is a development view, and FIG. 34B is the development of FIG. 34A. It is a perspective view of a conical wall with a fold line which has a figure in a half-folded state. The conical tubular fold structure of FIG. 34 is a development view when N = 3, 2Θ = π / 6 and α = π / 8, and is formed by a fold line group having the same angular relationship as FIG. 31. It is foldable because it is designed. Fig. 35
Is a development view of the conical tubular fold line structure when divided into isosceles triangular elements by fold lines, where N = 3, 2Θ = π
/ 9, α = π / 9 , δ = π / 6 and expand view of the (theta ^*
= About 0.0688π). Since the folding line of the tubular folding structure in FIG. 35 is formed by the folding line group having the same angular relationship as in FIG. 32, the tubular folding structure in FIG. 35 can be folded. FIG. 36 is a development view of a conical wall with a fold line in the case of being divided into inequilateral triangular elements by a fold line having an angle α at the upper right and an angle δ at the upper left at the point F in FIG. 32. 36 is a development view when the δ values are the same as those in FIG. 35. The tubular folding structure of FIG. 36 can also be folded. When it is applied to the folding lines of the equiangular spiral type, when φ = φ is the above-mentioned FIG. 34A, and when φ ≠ φ, FIG. 36 corresponds.

(2-4) 1-node 6-fold line method, and the main fold line formed by the mountain fold line is a cone and a pseudo-cone (2-4 -1) Folding condition The main fold line includes a fold line group formed along a cone and a pseudo-cone, and a fold line group (type 1) corresponding to the fold condition expression (3) There are two types of folding line groups (type 2) corresponding to conditional expression (6). (Type 1) FIG. 37 shows 1 for explaining the folding condition.
It is a principal part enlarged view of a 2nd step strip plate. As shown in FIG. 37, points A, B, and G are set on the outer circumference at intervals of a central angle 2θ with respect to the central point O, and angles ψ are formed with respect to OA, OB, and OG.
^* Forms a straight line and the center angle 2θ ^* each point of the intersection of the straight line that forms E, F, take the H. Similarly, points I, J and K are taken, and angles α to δ, p and q are taken as shown in FIG. The strip thus formed has a similar shape to the shape of the first stage and the second stage.

In FIG. 37, ∠OAB = ∠OBA = π
Considering / 2-Θ, the following equation is obtained. p = π / 2−Θ + ψ ^* , β = π / 2 + Θ−γ−ψ ^* , δ = π / 2−Θ− (γ + ψ ^* ) ……………………………… (25-1) OA If we take point C so that ΔOCE becomes an isosceles triangle, then ∠OCE = π / 2−θ ^* , so paying attention to ΔAEC, ∠AEC = π / 2−θ ^* −ψ ^* To get △ OCE
In consideration of the fact that and OEF are isosceles triangles, ∠AEF = q = π− (∠OE
Using C + ∠OEF + ∠AEC), the following equation (25-1)
To get q = π / 2 + 2θ ^* + ψ ^* + θ, α = γ-2θ ^* (25-2) In FIG. 37, the folding condition is examined by taking the point F as an example. From equations (25-1) and (25-2), β-α = π /
2 + Θ−ψ ^* + 2θ ^* −2γ and δ−γ = π / 2−Θ−
ψ ^* −2γ is obtained. That is, the following expression (27) is obtained. β−α = δ−γ + (p−q) = δ−γ + 2 (Θ + θ ^* ) ………… (26) The angle formed by the valley fold lines (1) and (2) in FIG. 37 is 2Θ from the periodicity,
Similarly, the angle formed by the folding lines (2) and (3) is 2θ ^* . That is, considering that (1) and (3) form an angle of 2 (Θ + θ ^* ), Expression (26) shows that the conditional expression for folding at node F is established.

(Type 2) FIG. 38 is the second stage in FIG. 37.
When the valley fold line of is taken in the opposite direction to that of the first step at an angle γ
FIG. The intersection of this valley fold line and OA (O; center)
Is K, △ KEO is ψ^*And was newly obtained
The rectangle EFIK is similar to that of the first stage. Fold at point F
The appearance of the line corresponds to the folding line group of the folding conditional expression (6).
It Fold this mountain fold line (1) to the mountain fold line FH
Θ of the folding condition expression (6)₁~ Θ_FourIs θ₁= Δ, θ₂= Γ, θ
₃= Α, θ_Four= Β. The line segments FH and FE in FIG. 38 are angles
Since 2Θ is formed, α in the folding conditional expression (6) is^*And β^*Is a figure
On 38,   α^*= Δ + γ + 2Θ, β^*= Α + β + 2Θ ………………… (27-1) Becomes The first strip in FIG. 38 is the first strip in FIG. 37.
Since it is the same as, the above equations (25-1) and (25-2) are used.
It can be used. Therefore, the equation (25−
Using 1) and (25-2), α in the above equation^*, Β^*Is   α^*= Π / 2-ψ^*−Θ = β + γ   β^*= Π / 2-ψ^*−θ−2θ^*= Α + δ                                   …………………………………… (27-2) Becomes Θ in these equations₁= Δ, θ₂= Γ, θ₃= Α, θ_Four
= Β is used, equation (3) is obtained and the folding condition is satisfied.
One. In this way, if you draw a valley fold line in the opposite direction for each step,
A folding structure is created by a group of folding lines.

(2-4-2) Closing Condition FIG. 39 is a development view of a conical wall with folding lines consisting of N isosceles triangular elements (vertical angle 2Θ), and only one step is written out as a curved strip portion. It is a figure. Similar to the case of the cylindrical tubular folding structure described above, the conditions for closing the conical tubular folding structure composed of folding lines of the 1-node 6-folding method are considered. Here, assuming that mountain folds and valley folds are periodically introduced, the angles formed by the fold lines with the outer sides AB, ... Are ζ and η (0 ≦ (ζ, η) ≦ π / 2). When this strip is bent along these folding lines, it is bent by φ = 2 (ζ−η) N in the circumferential direction. Originally, this strip had an angle ψ = 2N
Θ Since it was bent, in order to join both ends of this strip without gap after folding, the angle φ that changes due to bending and the angle (center angle) ψ that was bent from the beginning
And the sum must be 360 ° (= 2π) for one rotation.
That is, φ + ψ = 2π must be satisfied in order for the conical tubular folding structure to close. This corresponds to the conical closing condition, which is expressed by the following equation (21). φ + ψ = 2 (ζ−η + Θ) N = 2π ……………………………… (28)

(2-4-3) 1-node 6-fold line method, and a specific example of a cone and a pseudo-cone constituted by a fold line group in which a main fold line formed by mountain fold lines is formed along the circumference Example FIG. 40 shows N = 6, γ + ψ ^* = π / in the conical tubular fold structure having the fold line shown in FIG. 37.
3, development view when ψ ^* = π / 6 and γ = π / 6 (2
40 is an explanatory view of a pseudo-cone wall having Θ = π / 18), and FIG.
40A is a development view, and FIG. 40B is a perspective view of the conical tubular folding structure having the development view of FIG. FIG. 41 shows a conical tubular fold structure having fold lines shown in FIG. 37, where N = 6, γ + ψ ^* = π /
3 is a development view (2Θ = π / 6) when ψ ^* = π / 4 and γ = π / 12. FIG. 42 is a development view when the number of stages in the development view of FIG. 40A is reduced and the value of ψ ^* is increased for each stage. In FIG. 42, in the case of a conical structure, ψ ^* + γ = 60 °. Under ψ ^* + γ = 60 °, ψ ^* and γ are divided. Each stage can be arbitrarily divided into ψ ^* and γ values. With an equiangular spiral, the pattern becomes smaller toward the center, so to avoid this, ψ ^*
Is small. FIG. 43 is a development view forming the same conical wall as the folding structure having the development view of FIG. 42. FIG. 43 is a development view of a conical wall having the same shape as FIG. 42. In the developed view of FIG. 43, joining of both side edges is easier than in FIG. 42.

FIG. 44 is an explanatory view of the pseudo-conical structure having the fold line shown in FIG. 38, FIG. 44A is a development view, and FIG.
FIG. 44B is a perspective view showing a state in which the pseudo-conical structure having the development view of FIG. 44A is half-folded. FIG. 45 shows 2Θ = π /
It is a development view (N = 6) of the repeating spiral type fold lined conical structure obtained as 6, ψ ^* = π / 6, γ = π / 6.

(2-4) The 1-node 6-fold line method, and the main fold line formed by the mountain fold line is formed by a fold line group formed by being inclined (along the spiral) with respect to the circumference. Cone and Pseudo-cone FIG. 46 is a development view of a folded conical wall with a fold line in which the spiral in the circumferential direction of FIG. 40A is moved up one step at the right end. When the development view of FIG. 46 is a conical wall, the right side edge and the left side edge are connected so that the right end points A, B, C, ... And the left end points D, E, F, D overlap. To do. As described above, when equiangular spirals or inverted equiangular spirals are combined, the folding condition at the node is automatically established. So that it becomes 2π, the above-mentioned conditional expression (1
5) or (28). Further, for the nodes on these developed views, the p-value used in the equation (24) used in the explanation of the continuous condition is obtained, and the radius p,
It can be determined from the intersection of the concentric circles of p ² , p ³ ... And the radius. (2-5) Using the produced folding conical shell and its characteristic polypropylene sheet having a thickness of 0.2 mm, FIG.
The folding state of the conical shell of FIG. 51B manufactured in the developed view shown by A and the conical shell of FIG. 56B manufactured in the developed view shown in FIG. 56A was observed. As a result, it was found that good folding was possible as predicted by the origami model.

3. Further Study on Cylindrical Cylindrical Folding Structure 1. Section and 2. In the section, a study on a tubular folding structure based on a paper previously published by the present inventor was described, and the results of further study of this study will be described in detail below. (3-1) Purpose A development view in which a foldable / developable cylinder, a conical shell, a disc, or the like is patterned with rectangular or hexagonal elements having a congruent or similar shape has already been described above. If these development drawings can be created by combining two or more different elements (pseudo)
It is considered that it becomes possible to manufacture a rectangular tube, a pyramid, or the like having an atypical cross section different from the cylindrical cross section (for example, a rectangular cross section). In the following, we will create foldable prismatic cylinders and pyramid shells by combining different shaped element groups, increase their formability and functionality, and this technique will be used not only for space structures but also for industrial products and commercial products. The purpose of this paper is to develop a basic model for the purpose of increasing the versatility of this technique.

(3-2) Basic development view of foldable tubular structure FIG. 47 is a diagram for explaining a folding method of a foldable / unfoldable plane paper, and FIG. 47A is a view of a conventionally known foldable plane paper. FIG. 47B is an explanatory view of an essential part of a developed paper, and FIG. 47B is an explanatory view of an essential part of a developed paper of a flat paper which is cylindrical after being folded. FIG. 48
FIG. 48A is a development view of a foldable / expandable tubular structure, FIG. 48A is a development view of a tubular foldable structure composed of the same triangular elements (parts), and FIG. 48B is the same isosceles trapezoidal element (parts). FIG. 4 is a development view of the tubular folding structure configured in FIG. FIG. 49 is an exploded view of a foldable / expandable tubular structure, FIG. 49A is an exploded view of a tubular foldable structure composed of different trapezoidal elements (parts), and FIG. 49B is a different triangular element (parts). It is a development view of the configured cylindrical folding structure. FIG. 47 shows an example of a basic development view for folding a flat paper that can be simply developed, or for manufacturing a foldable pseudo cylinder. Figure 47A is 1
The most basic form of the nodal four-fold method is that all vertical fold lines are isomorphic and equiangular in a zigzag shape, which gives a planar fold. FIG. 47B shows two types of folding lines of the equiangular zigzag of FIG. 47A, which are alternately introduced. When the exploded view of FIG. 47B is folded, a plurality of fold lines (that is, main fold lines) connected along the horizontal form a curved line. In other words, it becomes the basic form of a foldable cylinder (square tube).

FIG. 48A is another basic model for forming a cylinder, and FIG. 48B is a modification thereof. FIG. 49A is a further diversification of the shape of the trapezoidal element of FIG. 48B. FIG. 49B is a basic model of a variant model in which the basic model of FIG. 47A has one node and six fold lines. The developed views of FIGS. 47A, 48A, and 48B consist of parallelograms, trapezoidal elements, or triangular elements of the same shape. Meanwhile, FIG. 48B and FIG.
The developed views of FIGS. 9A and 49B consist of two types of trapezoidal elements or triangular elements. When the number of elements in the horizontal direction is N, in the case of FIGS. 48A and 48B, which is composed of one type of element, a regular N-gonal cylinder (each cylinder) type tubular folding structure can be regarded as a pseudo cylinder. However, in other models except FIG. 47A, the cross-section is not a regular N-gonal shape but an odd-shaped tubular folded structure.

(3-3) Model of foldable deformed rectangular cylinder (3-3-1) Modeling of deformed rectangular cylinder by 1-node 4-fold line method FIG. 50 shows a cylindrical folded structure having a cross-shaped cross section. 50A is an explanatory view, and FIG. 50A is a plan view when a structure formed by joining both ends of the developed view of FIG. 50A is folded. FIG. 51 is an explanatory view of a tubular folding structure having a rhombic cross section, FIG. 51A is a development view, and FIG. 51B is FIG.
It is a top view at the time of folding the structure formed by joining both ends of the development view of 1A. 52 is an explanatory view of a cylindrical folding structure having a pseudo-elliptical cross section, and FIG. 52A is a development view,
FIG. 52B is a plan view of a structure formed by joining both ends of the developed view of FIG. 52A when the structure is folded. 53 is an explanatory view of a tubular folding structure having a rectangular cross section.
53A is a developed view, and FIG. 53B is a plan view when a structure formed by joining both ends of the developed view of FIG. 53A is folded.

Typical developed views considered as atypical models are shown in FIGS. 50A to 53A, and shapes after folding of each developed view are shown in FIGS. 50B to 53B. Here, a shape that was completely folded into a cross shape, a rhombus shape, a pseudo-elliptical shape, and a rectangular shape was selected. 50A to 5
3A was obtained based on the original models shown in FIGS. 47B and 48B. FIG. 50A corresponds to FIG. 47B, and two types of zigzag peak and valley fold lines in FIG. 47B (angles formed with horizontal fold lines; 75
There are 4 sets of (30 °). The fold line in the development view of FIG. 50A is 2 × (75-30) × 4 = 2 × 360, and the conditional expression (13) for closing the cylindrical tubular folding structure
Are satisfied. 51B to 53B are derived from FIG. 48B and have the shapes of FIGS. 51B to 53B after being folded, and the angle with the horizontal line is set in the same manner as in FIG. 50A so as to satisfy the condition of closing as a cylindrical tubular folding structure. Has been done.

(Cylindrical structure whose main fold line is spiral) FIG. 54 is a development view of a cylindrical tubular fold structure whose main fold line is spiral, and FIG. 54A corresponds to the development view of FIG. 50A. 54B is a view corresponding to the development view of FIG. 51A. FIG. 55 is a development view of a cylindrical tubular folding structure having a main fold line along a spiral, and FIG. 55A is the same as FIG. 52A.
55B is a view corresponding to the development view of FIG. 53A. FIG. 56 is a development view of a cylindrical tubular folding structure having a main fold line along a spiral, and FIG.
It is a figure corresponding to A.

When the developed views of FIGS. 50A to 53A are cut at an arbitrary angle so that the broken lines of the developed view after cutting are continuous left and right, the main fold lines of FIGS. 54A, 54B, 55A, and 55B are cylindrical. A development view is obtained that forms a spiral pattern centered on the central axis. For example, FIG. 54A is obtained by cutting the sloping line AB of FIG. The deformed cylinders obtained from the developed views by these cutting methods are also folded in the axial direction. Also, FIGS.
If the basic pattern of 3A satisfies the condition of closing in the circumferential direction, regardless of the angle of the inclined line to be cut, the pattern shown in FIG.
The spiral development view shown in FIG. 5 satisfies the condition of closing in the circumferential direction after folding. The expandability (the contraction rate of the outer shape before and after folding: folding / expansion rate) of these spiral folding structures is generally superior to that of the basic patterns of FIGS. It has the characteristics of. When the basic pattern of FIG. 49A is applied, a development view of a pseudo pseudo-cylindrical cylinder can be obtained if it is composed of trapezoidal elements of various shapes. Based on this, after obtaining a development view having horizontal fold lines as shown in FIGS. 50A to 53A, cutting this with an inclined line to create a spiral development view, a pseudo-cylindrical structure as shown in FIG. 56 is obtained. A development drawing is given.

(3-3-2) Modeling of odd-shaped cylinder by 1-node 6-fold line method FIG. 57 is composed of a fold line formed in zigzag in the horizontal direction and a fold line formed in zigzag in the vertical direction. It is a figure explaining the angular relationship of the folding line group. (Regarding conditional expressions such as folding condition and closing condition)
Horizontal mountain fold line of zigzag that forms angle Θ1 and Θ2 with x-axis (D
BAC) and y-axis alternate angles Θ3, Θ4 and x-axis and a fold line (EAF) almost perpendicular to each other. The fold diagram by a line is shown. Each node and angle are determined as shown in FIG. ∠BAG =
If α, ∠GAE = β, ∠CAH = γ, ∠FAH = δ, then from FIG. 57, α + β = π / 2 + Θ1-Θ3, γ + δ = π / 2-Θ2-Θ4 ……… I get (29). The following expression (30) is obtained by combining the expression (29) with the folding conditional expression (3). β-δ = γ-α + Θ1 + Θ2 + Θ4-Θ3 ………………………………………… (30)

In FIG. 57, the extension line of the line segment GA and the line segment AH
If the angle formed by is θ1, then θ1 = β + Θ3 + π / 2 + Θ2 + γ-π, so if β in equation (29) is used in this relational equation, θ1 is given by the following equation (31) . θ1 = γ-α + Θ1 + Θ2 ……………………………………………… (31) The folding conditional expression at node A is β-from the folding conditional expression (3). Since α = δ-γ + θ1, substituting θ1 in equation (31) into this folding conditional expression, β-δ = Θ1 + Θ2 …………………………………………………… …………… (32) is obtained. Further, the following expression (33) is established from the expressions (32) and (30). γ-α = Θ3-Θ4 …………………………………………………… (33) Next, the angle between the extension line of the valley fold line BI and the valley fold line BF. Θ2
Then, similarly, θ2 = γ ^* -α ^* -(Θ1 + Θ2). Therefore, the folding conditional expression at the point B is expressed by the following expression (34). γ ^* -α ^* = Θ3-Θ4, β ^* -δ ^* =-(Θ1 + Θ2) ……………… (34) Summarizing these, the folding conditional expressions (32) to (3
4) is γ-α = γ ^* -α ^* = Θ3-Θ4 ………………………………………… (35) β-δ = δ ^* -β ^* = Θ1 + Θ2…. …………………………………………… (36).

In FIG. 57, the closing condition (closing condition in the circumferential direction after folding) is expressed by the following equations (37) and (38) (N; even number). (N / 2) ・ 2 (β + β ^* ) = 2π ……………………………………………… (37) (N / 2) ・ 2 (δ + δ ^* ) = 2π ……………………………………………… (38) Here, if the sides AB≡a, AC≡b, AF≡c, AE≡d are expressed, the sine theorem is △ ABF , ΔAFH, a / c = sin δ ^* / sin
Obtain γ ^* , b / c = sin δ / sin γ. From these relationships,
The following expression (39) is obtained. a / b = sinβsinα ^* / (sinα · sinβ ^* ) (39) The following relational expressions (39 ') and (40) are obtained by the same procedure. a / b = sinγ sinδ ^* / (sinδ ・ sinγ ^* ) ……………………………… (39 ′) c / d = sinβ ^* sinγ / (sinα ^*・ sinδ) = sinγ ^* sinβ / (sinδ ^*・ Sinα) ……………………………… (40) The following relational expression is given as a further geometric constraint condition. α-α ^* + β ^* -β = γ ^* -γ + δ ^* -δ = Θ1 + Θ2 (41) Eight angles (α to δ ^* ) are unknowns, and Length ratio a /
Given b and c / d, there are nine constraint equations of equations (35) to (41), and it is generally difficult to solve.

Therefore, here, Θ1 + Θ2 = 0 and Θ3-
Consider two simple cases where Θ4 = 0. The former is when Θ1 = Θ2 = 0 and the horizontal fold line is straight, and the latter is when Θ3 = Θ4 and the vertical fold line is straight. (I) When Θ1 = Θ2 = 0 When the horizontal fold line is a straight line, the vertical fold line is zi
Since gzag is parallel, the angular relationship is α = α ^* , β = β ^* , γ = γ ^* , δ = δ ^* ……………………………… (42). If Θ1 + Θ2 = 0, then equations (34) and (35)
Is represented by β = δ, β ^* = δ ^* , γ-α = γ ^* -α ^* = Θ3-Θ4 (43).

(Specific Example in the Case of Θ1 = Θ2 = 0) FIG. 58 is an explanatory view of a specific example of the tubular folding structure having a straight fold line connected in the horizontal direction. FIG. Development view of a cylindrical structure whose fold line in the direction becomes zigzag shape,
FIG. 58B is a development view of a tubular structure in which a vertical folding line is curved. FIG. 59 is an explanatory view of a specific example of a tubular folding structure in which the fold lines connecting in the horizontal direction have straight fold lines, and a development view in which the fold lines are inverted step by step in the development view of FIG. 58A. 58 and 59 show folding diagrams when the shapes and dimensions of the parallelogram elements arranged side by side in the horizontal direction are the same. Here, the value of the angle δ is given by the equation (37).
Is determined by the horizontal closing condition of (δ = π / N). Equation (37) is satisfied by equating β in the second stage with δ. If Θ1 + Θ2 = 0 and β = δ in equation (30), then γ-α = Θ3-Θ4
Then, the equation (38) is automatically satisfied. That is, as shown in FIG. 57B, β = δ = π / N is selected for each stage, and α and γ can be folded even in a development view in which arbitrary values are selected. As a result, the vertical mountain fold line is zigzag (Fig. 58).
A) or curved (FIG. 58B).

FIG. 59 is a repetitive fold line of FIG. 58A with one jump in the opposite direction.
become that way. The folding condition at the node O in the 1-node 6-fold line method is the horizontal mountain fold line (line segment A in FIG. 58).
Since the extended line of (O) coincides with the line segment OB, α + β = α + δ,
It is given by simultaneously satisfying γ + δ = β + γ. Since both equations satisfy β = δ, the developed views of FIGS. 58A and 58B automatically satisfy the folding condition and the closing condition. This shows that if the development view is composed of parallelograms, and if the angle β = δ is selected so that only the closing condition is satisfied, it is possible to design a development view of a pseudo-cylinder that can be folded in any pattern at each stage. If equations (42) and (43) are used in equation (39), a / b = 1. This shows that when the horizontal folding line is a straight line, it is not possible to make a development view of a collapsible irregular shape having different horizontal parallelogram dimensions.

Using the repetitive folding line in FIG.
When the dimensional lengths of the sides are different in the horizontal direction, that is, an unfolded view folded in a rectangular shape is obtained. Figure 60
(N = 4). FIG. 60 is a development view of a tubular folding structure configured by repetitive folding lines, and is a development view of a structure having a rectangular cross section after being folded. The folding condition at each contact in FIG. 60 is satisfied exactly as in FIG. Δ and δ ′ in FIG. 60
Is the sum of π / 2, and they are constant for vertical triangular elements (parts) (angles other than δ and δ'are arbitrary).
From this, the tubular structure having repetitive fold lines,
It can be seen that the degree of constraint regarding the folding condition is milder than that of the structure of the type shown in FIGS. 58A and 58B.

(Ii) Θ3 = Θ4, that is, a mountain fold in the vertical direction
If the wire is straight In FIG. 61, the folding lines connected in the vertical direction form a straight line.
Explain the angular relationship of a tubular fold structure with fold lines
FIG. If Θ3 = Θ4, the model of the tubular structure is
Since it becomes like FIG. 61, α^* = γ, β ^* = δ, γ^* = α, δ
^* = β. Therefore, equation (35) is γ- when Θ3 = Θ4
α = 0. Further, from FIG. 57, α + β-Θ1 + Θ3 = γ + δ + Θ2 +
Since Θ4, if we set Θ3 = Θ4, γ = α, this relational expression
Becomes β-δ = Θ1 + Θ2, and the above equation (32) is also satisfied at the same time.
Be done. That is, the folding condition of the equations (32) to (36)
Is expressed by a single equation (37).     α = γ, β-δ = Θ1 + Θ2 ……………………………………………… (37) In equation (36), α^* = γ, β^* = δ and α = γ
When,     a / b = sinβ / sinδ …………………………………………………… (38) To get a / b value is given and the closing condition (β + δ = π / N)
When used in (38), β and δ can be obtained by numerical calculation, and equation (3
Θ1 + Θ2 is determined from 7). Therefore, from FIG.
Since a / b = sin Θ2 / sin Θ1, the obtained Θ1 + Θ2 is used.
Θ1 and Θ2 will be decided when there is. Figure 62 is connected vertically
Cylindrical fold having folding lines forming straight lines
FIG. 62A is a diagram of a specific example of a structure having a / b = 2.
5, a development view in the case of N = 4, FIG. 62B shows a / b = 2.
5 is a development view in the case of 5, N = 6. FIG. 62A tubular folding
The folded structure has a / b = 2.5, N = 4 (β + δ = π /
4), the tubular folding structure of FIG. 62B has a / b = 2.
5, N = 6 (β + δ = π / 6), so
Folding conditions and closing conditions are satisfied, so fold
Can be removed. Here, α = γ can be freely selected.

(3-4) Manufacture of a pseudo-conical shell by combining different element shapes and creation of an atypical pyramid shell model by this (3-4-1) Model of a conical shell with different patterns The method of dividing the development view into similar rectangular and hexagonal elements by using folding lines to manufacture a foldable conical tubular structure has been described above. At this time, the element dimension becomes geometrically smaller as it gets closer to the center, and a complicated process is required when manufacturing and processing the element. Therefore, here, we consider a modified model in which the size gets larger than the dimension given by the conformal helix as it approaches the center. 63 is a diagram of a modified model of an equiangular spiral, FIG. 63A is an explanatory diagram of spiral folding lines, and FIG. 63B is a diagram showing a specific example when N = 6 in FIG. 63A. As shown in FIG. 63A, points A, B, C, ... Are on the outer circumference, and points D, E, F, ...
Take H, I ... The lower left corner of the upper triangle obtained by dividing the rectangular element drawn by connecting each point (∠DAE, ∠EBF, ∠GDH
Etc.) are all set as .delta.0 as used in the description of FIG.

Now, consider the folding condition at the point E. The angle θ1 formed by the line segment AE that divides the element ABED and that (line segment EI) of the element on the upper right is θ1 = ∠ADE = ∠BEF≡p
β0 + p + γ1-π. From the sum of the interior angles of △ ADE, p + α0 + δ0
get = π. If p is deleted from these equations, β0-α0 = δ0-γ1 + θ1 becomes (39), and the folding condition is satisfied. This shows that a foldable pseudo-conical tubular structure can be manufactured by keeping the angle δ0 constant for each step. This δ0 value is determined by the closing condition, and is determined by the following equation, where NΘ is the initial bend of the development view (Θ; the angle formed by the lower side of the rectangular element with respect to the center). NΘ + 2δ0 × N = 2π …………………………………………………… (40) Figure 63B shows an example where N = 6 (Θ = 20 °, δ0 = 20 °).

FIG. 64 is a diagram of a modified model of the repetitive conformal helix, and FIG. 64A is an explanatory diagram of the spiral fold line, and FIG. 64B.
FIG. 64B is a diagram showing a specific example when N = 6 in FIG. 64A. FIG. 64A shows a repetitive model of the model of FIG. 63A in which even-numbered-stage rectangles are taken in the opposite direction from the outer circumference.
In addition, each point and angle are determined in exactly the same manner as in FIG. 63A. Point E
Consider folding in. Point on extension of line segment EF
Take J The folding condition at point E is ∠HEJ = ∠GED + ∠AEB
And ∠BEJ = ∠HEG + ∠DEA are established at the same time. ∠DEJ
= Θ, so ∠HEJ = δ0 + γ1 + Θ, and ∠JEB = π-p = α0
Since + δ0, the above condition holds. Therefore, FIG.
In the modified model of the iterative conformal spiral of 4A, the folding condition is satisfied and the closing condition is the same as the case of FIG. 63A. An example of this repeating tubular structure is shown in FIG. 64B (N =
6, Θ = 20 °, δ0 = 20 °).

(3-4-2) Manufacture of atypical pyramidal cylinder FIG. 65 is an explanatory view of an atypical pyramidal structure, and FIG.
Is a diagram for explaining the angular relationship, and FIG. 65B is a specific example of an atypical pyramidal structure. As shown in FIG. 65A, the zigzag mountain fold line ACDE from the point A on the outer circumference to the center O (see FIG. 6).
5A fold line), and consider the fold line of ACDE
Be jΘ (j: positive value, Θ: fan-shaped minute division angle), and fold line AC,
It is assumed that CD, DE ... Form angles φ0, φ1, φ2 ... With the radial direction. Similarly, a mountain fold line (a fold line in FIG. 65A) extending from the point B in the direction opposite to the fold line AC is drawn first, and this forms the angles ψ0, ψ1, ψ2 ... With the radial direction (swing angle; kΘ ).
These folding lines (,) are taken alternately, and a development view 65
Draw A and divide the sector into trapezoidal elements. Figure 65A
The angle formed by the line segments AB, DG, etc. in the center of the
Let θ be the angle between the left and right line segments IC, JD, FK, and GL. Place points M and N on the extension of line segment IC and KF. The folding condition at point C is ∠ACM = π / 2-[(n / 2 + j) Θ + φ0], ∠DCF = π / 2-
Putting [(j + m / 2) Θ + φ1] equally, φ0-φ1 = (mn) Θ / 2 …………………………………………………… (41) Becomes Similarly, the folding condition at the point F is expressed by the following equation (42). ψ0-ψ1 = (mn) Θ / 2 …………………………………………………… (42)

In FIG. 65A, ∠DCF = π / 2- (j + m / 2) Θ
Since -φ1, ∠ICD = π / 2-nΘ / 2 + φ1, the angle Φ≡∠ICD-∠DCF after folding at point C and the angle Ψ after folding at point F are given by ), (44). Φ = 2φ1 + {j + (mn) / 2} Θ ………………………………………… (43) Ψ = 2ψ1 + {k + (mn) / 2} Θ ………………………… ……………………… (44) Consider the case where the shape after folding is a rectangle, and set j = k = 1, and assume that each vertex is folded at 90 °. If m and n, which give the aspect ratio of the rectangle, are set to mn = 2, then equations (43) and (4
4) becomes 2 (φ1 + Θ) = 2 (ψ1 + Θ) = 90 ° ………………………………………… (45). Also, equations (41) and (42) become the following equation (46). φ0-φ1 = ψ0-ψ1 = (mn) Θ / 2 = Θ …………………………………… (46) If we set Θ = 5 ° in equation (45), φ1 = ψ1 = 40 °, and φ0 = ψ0 = 45 ° from equation (46). Φ2, φ3 ... Are obtained by the same procedure, and finally the change of φ (= ψ) is 45 ° → 40 ° → 45 ° → 40
Given in °. FIG. 65 shows a development view when m = 3 and n = 1.
Shown in B.

4. Relationship between a cylindrical tubular folding structure and a conical tubular folding structure In the following, first, the equiangular spiral folding line is reversed at an arbitrary angle,
Alternatively, a method for drawing a developed view of a conical (trapezoidal) cylindrical structure that can be folded by triangular elements configured by intersecting in the same direction will be described. Next, a method for producing a developed view composed of rectangular elements having a greater developability, which is devised based on this, will be described. (4-1) Basic Relationship of Conversion from Foldable Cylinder to Conical Shell FIG. 66 is a diagram illustrating a development view of a foldable cylindrical tubular structure, and FIG. 66A is a development view of a buckling pattern of the cylinder. FIG. 66 is a diagram corresponding to FIG. 9, FIG. 66B is an explanatory diagram of angles of each element, and FIG. 66C is a diagram inclined so that AB in FIG. 66A is horizontal. 67A and 67B are explanatory views of a foldable cylindrical tubular structure, FIG. 67A is a drawing in which FIG. 66B is redrawn, and is a drawing corresponding to FIG. 22A, and FIG. It is a figure corresponding to FIG. 66B. Figure 6
8 is an explanatory view of a foldable cylindrical tubular structure,
68A is a diagram in which the GF in FIG. 67B is inclined so as to be horizontal, and FIG. 68B is a diagram in which FIG. 68A is redrawn.

66 to 68 are developed views of the folding of the cylinder. FIG. 66A is a folding basic form obtained from a buckling pattern of a cylinder when axially compressed, and α = β = π / 6 (see FIG. 66).
(See B)). When this is cut along the slanted line segment AB in FIG. 66A and AC is joined to the lower side DE, FIG. 66C is obtained, which can be transformed as shown in FIG. 67A. Figure 6
When cut with FG in 7B, it goes up 1 step (up 1 step),
Alternatively, when cut by FH, a two-step raised main fold line is a development view of a tubular structure along a spiral. The one step up is shown in FIG. 68A, which is equivalent to FIG. 68B.
When the both ends of the development view of FIG. 68 are joined to form a cylinder, the right end of the inclined fold line extending from the lower left corner (point F in FIG. 68A)
Is continuous with the left end of the fold line above this mountain fold line. This is equivalent to a cylinder model in which Guest et al. Investigated folding characteristics by numerical calculation. In FIG. 67A, six parallelogram elements are arranged along the circumferential direction of the tubular structure.
7 pieces, 2 steps up to 8 pieces. The folding condition of these models is β = π / 6, and when α = π / 6, it is folded in a regular hexagonal shape, but unless α is π / 6, it is not folded in a regular polygonal shape. . Also,
When the number of steps increases, the contraction in the radial direction becomes large when folded.

69 to 71 are developed views of a conical tubular folding structure corresponding to the cylindrical tubular folding structure shown in FIGS. 66 to 68. 69A is a development view of a conical tubular folding structure that can be folded, and FIG.
66A is a development view of a conical structure corresponding to the cylindrical structure of FIG. 66A, and FIG. 69B is a development view of a conical structure corresponding to the cylindrical structure of FIG. 67A. 70 is a development view of a foldable conical tubular folding structure, which is a development view of a conical structure corresponding to the cylindrical structure of FIG. 68B. FIG. 71 is a development view of a conical tubular folding structure that can be folded, FIG. 71A is a development view of a conical structure that corresponds to the two-step-up cylindrical structure in FIG. 67B, and FIG. 71B is a diagram. It is a repainted version of 71A. Note that FIG.
FIG. 1B is a redrawing of FIG. 71A, showing that it is composed of six elements in the direction of the upward gradual gentle mountain fold. In the following, the fold line in the form of FIG. 69A will be referred to as reverse spiral, and the fold line in the form of FIG. 69B will be referred to as same direction spiral.

(4-2) Construction method of folding conical shell by spiral fold line (4-2-1) Consists of triangular elements composed of spiral fold lines of mountain folds intersecting in opposite directions FIG. 72 is an exploded view showing the drawing and the angle regarding the folding line group of the backward spiral, and is a view corresponding to FIG. 33. As shown in Fig. 72, fold the line so that point A is placed on the outer circumference and form a corner with the radius
Draw AB and define point B (angle that line segment AB forms with the center:
m Θ). Similarly, the point C is further developed from the point B at the angle φ in the upper right direction, and the point C is determined, and then the point D is determined. Let this line be.
A line segment AE is drawn in the upper left direction from the point A so as to form an angle ψ with the radial direction (angle splayed by the line segment AE: nΘ), and the same angle ψ, nΘ is drawn to determine points E, F, G. Draw a line rising from point E to the right in the same way as. In addition, a polygonal line (points I, B, H ...) Is drawn from the point B in the same procedure as that, and a development view is constructed by equiangular spiral folding lines. Using the sine theorem for △ 0AB and △ 0AE, if the radius of B is R1 and the radius of point E is R1 ^* , p≡R1 / R0 and q≡R1 ^* / R0
Are respectively given by p = sin φ / sin (φ + mΘ), q = sin ψ / sin (ψ + nΘ) ……………………… (46). The dimensionless radii of points C and D are p ² , p ³ ..., Points F, G ...
The dimensionless radii of are given by q ² , q ³ ...
Becomes an equiangular spiral. Also, the dimensionless radii of points E, B, J ... are the geometric progressions of p / q (≡r) such as q, p, p ² / q, p ³ / q ² This sequence of points also comes on a conformal spiral. The rectangular elements composed of fold lines are similar figures.

Here, the folding condition at each node is confirmed. Figure 72 shows the angles of the triangular elements that make up this rectangle.
The angles are represented by the angles α to δ as shown in FIG.
From the relationship of the sum of the internal angles of △ OEB and △ ABE, α + δ + φ + ψ + (m + n) Θ = π, β + γ + φ + ψ = π ……………………… (47 ) Get The following equation (48) is derived from the above two equations (47). β-α = δ-γ + (m + n) Θ ……………………………………
……………… (48) The angle formed by the line segments EB and BJ is (m + n) Θ
Therefore, the expression (48) shows that the folding condition is satisfied at the point B. Due to the similarity of rectangular elements, all nodes satisfy the folding condition. From ∠0EH = ∠0AB = φ, the line segment EH
And the intersection of AB and the four points 0, E, A are on the same circle. Therefore, δ = β + mΘ holds, and from equation (48), δ-β = mΘ, γ-α = nΘ …………………………………………………… (49) obtain. Further, the following expression (50) is obtained from the expression (47). α = π-δ- (φ + ψ)-(m + n) Θ …………………………………… (50) r≡p / q is the sine theorem in ΔOAB, It is given by the following equation (51). r = sin (φ + δ) / sin [φ + δ + (m + n) Θ] ……………………………………… (51)

Next, a concrete example will be considered. When the point K in FIG. 72 is on the outer circumference, p / q ² = 1. Generally expressed by p / q ^k = 1 (k; integer). The following expression (52) is obtained from q = sin ψ / sin (ψ + nΘ) = p ^{(1 / k)} . cosψ = sinψ ・ f (p) ……………………………………………… (52) where f (p) ≡ (1-p ^{(1 / k)} cosmΘ) / (p ^{(1 / k)} sinm Θ) Therefore ψ is a function of p or φ. cos ² ψ + sin ² ψ
= With 1, Taking sinψ> 0 sinψ = (1 + f2) -0.5 i.e. [psi is ψ = arcsin (1 + f2) -0.5 ................................................... (53) Also, from r = p / q = p ^{(1-1 / k)} , as above, δ + φ
Is δ + φ = arcsin (1 + g2) ^-0.5 ………………………………………… (54) where g (p) ≡ [1-p ^{(1-1 / k )} cos (m + n) Θ] / [p ^{(1-1 / k)} sin
(m + n) Θ].

After folding, the condition for closing in the circumferential direction is N = (number of elements in the circumferential direction), which is the number of mountain fold lines in the upward direction on the development view, and the bending angle Φ due to folding is Φ = 2 [δ + [π -φ-ψ- (m + n) Θ]] ・ N …………………………………… (55). The initial bend (angle between both ends) of the developed view is Ψ
Since = (m + kn) NΘ, the conditional expression (28) (Φ + Ψ = 2π) that closes in the circumferential direction after folding is [δ + π-φ-ψ- (m + n) Θ] + (m + kn) Θ / 2N = π / N ………………………… (56). 73 is shown together with the constants using an example of a developed view obtained by calculating φ satisfying the formulas (53) and (54) in the formula (56) and performing numerical calculation. FIG. 73 is a development view of a specific example of the conical tubular folding structure having a reverse spiral folding line group. The cylindrical structure of FIG. 73 is similar to that of FIG.
Corresponding to the spiral mountain fold line of A, p = q ^k
From the condition of (k = 2), the continuity of the fold lines at the left and right edges of the development view is established. In FIG. 73, N = 3, Θ = 5 °, m = 3, n = 2
The calculation result of p = 0.8873, q = 0.9419 was obtained.

(4-2-2) Exploded view consisting of triangular elements composed of spiral fold lines of mountain folds intersecting in the same direction. FIG. 74 is a drawing and explanation of angles regarding a fold line group of spirals in the same direction. 74A is an overall view, and FIG. 74B is FIG. 74A.
FIG. As shown in Fig. 74A, a point A (radius R0) is taken on the outer circumference of the circle, and a line segment AB that forms an angle φ with the radius is drawn (line A
(An angle formed by B with respect to the center: mΘ), and point C and point D are sequentially determined from point B using the same φ and mΘ values. This fold line is Next, define point E from point A at an angle ψ (<φ) in the same way (line segment
AE angle: Θ), follow the same procedure to determine points F, G, H and draw a polygonal line. p≡R1 / R0 (radius of R1; B) is given by p≡sin φ / sin (φ + mΘ) ……………………………………………… (57). Similarly, the dimensionless radius q of the point E is given by the following equation (58). q≡sin ψ / sin (ψ + Θ) ……………………………………………… （58） Therefore, the dimensionless radii of points B, C, D are p ² , p ³ , p ⁴ at point F,
Those of G and H are given by q ² , q ³ and q ⁴ , respectively. Then points B, C, D
Draw the angle φ in the same way as you obtained the polygonal line starting from. With such a procedure, a rectangular pattern as shown in FIG. 74A is obtained, and the radius of each point is as shown in FIG. 74A. Columns of points A, I, J, and K on the diagonal line (dotted line) from point A to the rectangle (white circles)
The fold line connecting The dimensionless radius of these points is
^They are represented by 1, (pq), (pq) ² and (pq) ³ , respectively. Therefore, the fold lines are in the form of an equiangular spiral.

Specifically, the dimensionless radius is calculated. r≡pq is expressed by the following equation (59) using the sine theorem for △ OAI (∠
EAI = δ). r = sin (ψ + δ) / sin [ψ + δ + (m + 1) Θ] ……………………………… (59) For node I, verify the folding condition. All the similar rectangular elements are diagonally divided, and angles α to δ are determined as shown in FIG. 74B. From FIG. 74B, φ = γ + δ + ψ. If we take the point L on the extension of the radius OE, then ∠AEL = ψ + Θ and ∠OEI = φ, so ∠AEI = ∠AEL + ∠LEI = ψ + Θ + π-φ. △ A
From the relationship of the sum of the internal angles of EI, α is α = π-∠AEI-δ = φ-ψ-
δ-Θ, and γ becomes γ = φ- (ψ + δ) from the relation of ∠OAB. From the internal angle of △ OAB ∠ABI = (π-φ-mΘ) + ψ, β =
π-γ-∠ ABI = φ-ψ-γ + mΘ = δ + mΘ Line AI
Is an angle of (m + 1) Θ with respect to the center, and the angles made by the line segments AI and IJ are
Considering (m + 1) Θ, the following equation (60), β-α = δ-γ + (m + 1) Θ ……………………………………………………… (60 ) That is, the folding condition (Equation (3)) at node I holds.

Next, the continuous condition is verified. Assuming that the number of rectangular elements in the circumferential direction is N, the continuous condition on the left and right of the development diagram in the case of one step rise is q = p ^N. This value is calculated as follows. From q≡sin ψ / sin (ψ + Θ) = p ^N, we perform the same calculation as we obtained equation (53), and ψ = arcsin (1 + f2) ^-0.5 ……………………………… …………………… (61) However, we get f (p) ≡ (1-p ^N cos Θ) / (p ^N sin Θ). Therefore, ψ is a function of p or φ. r
= p ^{N + 1} and r = sin (ψ + δ) / sin [(ψ + δ) + (m + 1) Θ],
We obtain r = p ^{N + 1} sin [(ψ + δ) + (m + 1) Θ] = sin (ψ + δ). By the same procedure, the following equation is obtained as ψ + δ. ψ + δ = arcsin [(1 + g2)] ………………………………………… (62) However, g≡ [1-p ^{N + 1} cos [(m + 1) Θ] ] / (p ^{N + 1} sin [(m + 1) Θ])

After folding, the conditions for closing in the circumferential direction are introduced. Bending angle Φ due to folding in case of one step rise
Is given by Φ = [N [(ψ + δ) -ψ]-(φ-ψ-δ-Θ)] × 2 ……………………………… (63). The initial bend Ψ of the development view of the conical shell is Ψ = [(N-1) m + (m-1)] Θ = (Nm-1) Θ …………………………………… (64 ), The closing condition (Φ + Ψ = 2π) is the following expression (65). (N + 1) δ + (ψ-φ) + (Nm + 1) Θ / 2 = π …………………………………… (65) Formulas (61) and (62) By using φ in (65) and calculating φ satisfying the equation (65) by a numerical meter calculation, the developed view of FIG. 70 is obtained. 2N [G-2φ-F] -NmΘ + 2 (N-1) π = 0 ………… when the mountain fold line (main fold line) along the spiral goes up by 2 steps per revolution ……………………………… (66) The condition for closing is given. Expressions (61) and (62) are converted into Expression (6)
FIG. 75 shows the result obtained by the numerical calculation used for 6). Figure 7
5 is a development view of a specific example of a conical tubular folding structure having folding lines in the same direction. In addition, in FIG. 75, calculation is performed with N = 8, Θ = 5 °, m = 2.4, and p = 0.98547, q
= 0.868859 was obtained.

(4-2-3) Exploded view composed of rectangular elements In FIG. 72, a group of two equiangular spiral mountain fold lines (,) and a valley fold line connecting their intersections forms a 3-shape. An exploded view of the elements was constructed. Based on this, here the development view is divided by similar trapezoidal elements. 76 is an explanatory view of a conical structure composed of trapezoidal elements, FIG. 76A is an explanatory view of a drawing method and an angular relationship, and FIG. 76B is a development view of a specific example of the structure formed by the drawing method of FIG. 76A. Is. In FIG. 76A, the fold line is extended from the point A (radius R0) on the outer circumference to the upper left direction at the angle nΘ extending from the center first, and then to the upper right direction by kΘ, and this is alternately repeated to fold the zigzag fold line (ADEFG). Draw (alternate angles φ and ψ between line segments and radial direction). Draw the same zigzag mountain fold lines BHIJK, CLMNP ... from points B and C on the outer circumference that are arranged at equal intervals at the central angle (m + n) Θ.
Next, connect the points C, D, A, H, B ... Or L, M, F, E, J, K ..., and use this as the mountain fold line in the circumferential direction. Etc. to the valley fold line. Assuming that the radius of point D is R1 and the radius of point E is R2, p≡R1 / R0 and q≡R2 / R1 are p≡sin φ / sin (φ + nΘ) q≡sin ψ / sin (ψ + kΘ)… ………………………………………………… (67) The dimensionless radii of points L, D, H on the same diameter are p, points M, E, I,
And the points N, F, and J are given by pq, p ² q ,. If the angle formed by the bases AH and CD of the trapezoid with the radial direction is ξ (mΘ; the angle formed by the line segment AH with respect to the center) and the angle formed by the valley fold lines DI and LE with the radial direction is χ, the p and q values are p ≡ sin ξ / sin (ξ + Θ) q≡sin χ / sin [χ + (k + m + n) Θ] …………………………………………… (68)

Next, the folding condition at the point D is derived.
Place point Q on the extension of line segment CD. Set point S on the extension of radius OD. The folding condition at the point D is represented by ∠ADQ = ∠EDI. From ∠ODQ = ∠CDS = ξ + mΘ, ∠ADS = φ + nΘ, ∠ADQ
= π- [ξ + φ + (m + n) Θ]. Since ∠EDI = χ-ψ,
By arranging them equally, χ-ψ + ξ + φ + (m + n) Θ = π ……………………………………………… (69) is obtained as a folding condition. At point E, ∠JED = ∠JET + ∠DET = (π-ξ) + (ψ + kΘ)
DEL = ξ- (ψ + kΘ) and ∠FEL = [π-χ- (k + m + n) Θ] -φ
When is placed equally, the above equation (69) is obtained. That is, the folding conditions at all points are expressed by equation (69).

Next, considering the curved strip plate portion in the circumferential direction, the conditions for closing in the circumferential direction after folding are derived. If the number of trapezoidal elements in the circumferential direction is N, the number of
Bend only by (χ-ψ) N. The initial bending (angle between both ends) of this strip is (m + n) ΘN. Putting the sum of these as 2π, the closing condition is (χ-ψ) + (m + n) Θ / 2 = π / N ……………………………………………… (70 ) Get Folding conditional expression (69) and closing conditional expression (70)
The following expression (71) is obtained. ξ + φ = (N-1) / N ・ π- (mn) Θ / 2 ………………………………………… (71) Eq. (67), (68) is equal to Then, if m, n, k are given and Eq. (71) is used, φ and ξ can be calculated.
Is calculated. An example of the development view thus obtained is shown in FIG.
Shown in B. Incidentally, FIG. 76B shows a development view in the case of p = 0.9573 and q = 0.8746.

FIG. 77 shows the configuration of FIG. 7 formed by trapezoidal elements.
77A is an explanatory view of a conical structure different from FIG. 6, FIG. 77A is an explanatory view of a drawing method and an angular relationship, and FIG. 77B is a developed view of a structure of a specific example formed by the drawing method of FIG. 77A. The developed view of FIG. 76A is zigzag drawn from points A, B and C.
The mountain fold line (for example, CDAHB) is formed by connecting the vertices of the mountain fold line by shifting by half a wavelength. FIG. 77A shows a development view in which this is further shifted by one wavelength and drawn. The angles are defined as in Figure 76A. The folding conditions of the representative points G and F are ξ + φ + χ-ψ + (m + nk) Θ = 0 ………………………………………… (72) ξ + φ + χ-ψ + (m + n) Θ = 0 ………………………………………………… (73). Since the above two equations (72) and (73) must be satisfied at the same time, k = 0 must be satisfied in the development view of such a pattern. That is, with ψ = 0, the equation (6)
8) is represented by the following formula. p = sin φ / sin (φ + nΘ), q = constant ………………………………………… (74) The dimensionless radius of points K, D, H ・ ・ is p, point L, Those of E ···, points M, F ·· are respectively given by pq, p ² q ··. The radius of point E is p ² q,
This is also expressed as sinχ / sin [χ + (m + n) Θ], so pq ² = sinχ / sin [χ + (m + n) Θ] …………………………………… ……… (75) is obtained. Further, the following expression (76) is also established. p ² q = sin ξ / sin [ξ + mΘ] …………………………………………… (76)

The closing condition is that the bending angle Φ = 2 (χ-ξ) N,
As the initial turning angle Ψ = (m + n) ΘN (N; number of elements), the following equation (7
Given in 7). Φ + Ψ = 2 (2χ-ξ) N + (m + n) Θ) N = 2π …………………………………… (77) Let χ and ψ be unknowns and formulas (74)-( FIG. 77B shows a development view obtained by solving (77). In FIG. 77B, p =
The developed view for 0.8990 and q = 0.9806 is shown.

FIG. 78 is composed of rectangular elements different from those in FIG.
Drawing method and angle relation of conical tubular fold structure
FIG. Based on the model of FIG. 74,
Consider an expanded view composed of rectangular elements. As in Figure 78
Starting from point A (radius R0) on the outer edge, make an angle φ with the radial direction.
Draw a line segment AB (angle between AB and the center; m Θ). next
Draw a line segment BC that makes an angle ξ with the radial direction from point B, and
Let nθ be the angle. In the same procedure, every time m
Take angles φ and ξ and draw zigzag fold line CDEF ・ ・ G. this
It should be a fold line in the circumferential direction. Infinite if the radius of point B is R1
The original radius R1 / R0 is     p≡R1 / R0 = sin [φ / (φ + mΘ)] ………………………………………… (78) Given in. If the radius of the point C is R2, the following equation (79) becomes
It stands.     r≡R2 / R1 = sin [ξ / (ξ + nΘ)] ………………………………………… (79) The dimensionless radius R2 / R0 of the point C is represented by pr, and the points D, E, F
Dimensionless radius is p²r, p²r ², p³r²Represented by ...

Next, a line segment AI forming an angle ψ with the radial direction from the point A is drawn (angle formed by the line segment AI with respect to the center; Θ). The dimensionless radius q of the point I is expressed by q≡sin ψ / sin (ψ + Θ) …………………………………………………… (80). Starting from the point I, conversely to the fold line, first determine the points J and K at the angle ξ (angle to be stretched nΘ) and then the angle ψ (angle to be stretched mΘ).
Repeat this to draw the same folding line. With such a drawing, the fan-shaped film can be divided by similar hexagonal elements. The zigzag line FEMLT ··· obtained by this division is the fold line, and the line dividing the hexagonal element diagonally is the (valley) fold line.

The folding condition at the point E is derived. Place point S on the extension of line segment DE and point R on the extension line of radius 0E. ∠DER = ξ
Since + nΘ and ∠REF = π-φ, ∠FES = φ-ξ-nΘ. Since ∠MEP = χ−ψ, these are equally placed to obtain the folding conditional expression (81). φ + ψ-χ-ξ = nΘ …………………………………………………… （81） The folding conditional expression at point M is also expressed by the above expression (81). .
That is, the folding conditional expressions for all nodes are given by Expression (81).

Next, the continuous condition will be examined. Fold line is angle φ
Is bent N times at an angle ξ at N times and reaches point G. That is, the dimensionless radius of point G is given by p ^N r ^N-1 . When this value is equal to the dimensionless radius q of the point I, the spiral pattern is raised by one step, and the continuous condition of the left and right folding lines of the fan shape is satisfied, which is expressed by the following equation (82). q = p ^N r ^N-1 ……………………………………………………… (82) Due to the continuity of hexagonal elements, the angle formed by the diagonal line AK and the radial direction. Is χ, and the angle formed by this diagonal with respect to the center is (m + n + 1)
Considering that it becomes Θ, the dimensionless radius s of the point K is s = sinχ / sin [χ + (m + n + 1) Θ] …………………………………………………… (83) is given by. Since s = pqr, s is expressed by the following equation (84). s = p ^{N + 1} q ^N …………………………………………………………… (84)

Next, consider the condition for closing in the circumferential direction in the case of one step up. Consider a circumferentially curved strip UV that is divided by two dotted lines. The angle between line segment AI and line segment AK is χ
The angle ζ between -ψ and line segment BC and line segment DW is ζ = ξ-χ-Θ.
In this strip, there are (N-1) formers and one latter. In other words, by this folding operation, Φ≡2 [(N-1) (χ-ψ)
-Just bends. The initial bending angle Ψ of the strip is [mN
Since + n (N-1) -1] Θ, the condition (Φ + Ψ = 2π) that both ends of this strip portion are closed after bending is given by the following equation (85). 2Nχ-2 (N-1) ψ-ξ + [mN + n (N-1) -1] Θ = 2π ………………………… (85)

(Concrete Example) FIG. 79 is an explanatory view of a concrete example of a foldable conical tubular fold structure having fold lines formed based on the drawing method of FIG. 78, and FIG. 79A shows one step up. FIG. 79B is a development view of two stages higher. When φ, ψ, ξ and χ are unknowns and four relational expressions (81), (82), (84), (85) are numerically solved using the notation expressions of p, q, rs, these The value of is determined. FIG. 79A shows an expanded view of one step up obtained by N = 7. Figure 79A
Is calculated with N = 7, Θ = 5 °, m = 2, n = 1, and the developed result is p = 0.8869, q = 1.007, r = 0.9525, s = 0.7551. Show.

In the case of two steps up, the continuous condition of the developed view is q = p ^{N / 2} · r ^{N / 2-1} ………………………………………………………… (86) Substituting equation (86) into s = pqr, s is represented by the following equation (87). s = p2 / (2-N) qN / (N-2) …………………………………………………… (87) After folding, the condition to close in the circumferential direction is Φ = 4 ((N / 2-1) (χ-
ψ) -ζ], Ψ = [mN + n (N-1) -2 (n + 1)] Θ, using 4 [(N / 2-1) (χ-ψ)-(ξ-χ- Θ)] + [(m + n) N-2 (n + 1)] Θ = 2π ............ (88) Fig. 79 shows the two-tiered development view obtained with N = 8.
Shown in B. At this time, calculation was performed with N = 8, Θ = 5 °, m = 2, n = 1, and p = 0.8479, q = 0.9981, R = 0.9490, S = 0.8031 was calculated. In FIGS. 79A and 79B, it can be seen that the developed view is composed of six elements in the upward direction of the circumferential direction, even though N = 7 and 8 respectively (FIGS. 69B, 70, and 7).
1A).

5. Consideration 3. A foldable cylindrical or conical tubular fold structure that combines the folding conditions at each node, the continuity of the left and right ends of the development view, and the condition that both ends of the development view are closed in the circumferential direction after folding. I examined how to make things. It has been found that modeling development drawings with different dimensions results in an extremely large number of constraint equations, and it is often not easy to create developments with the same pattern. Although not described in the text, it has been proved mathematically impossible to make the rectangular cross-section model shown in FIG. 60 or the four-sided pyramid model shown in FIGS. 63B and 64B into a spiral pattern.
From the viewpoint of systematically studying the folding method in the early years, it is also important to show that such folding cannot be done in order to clarify its limit. Although models are included herein that may be difficult to engineer immediately, the development of such models also systematizes folding techniques to make them more sophisticated and useful. It is believed that it will contribute to the development of a folding structure. Also, if we consider the folding lines of these models as truss members,
It can be used as a basic model of a folding truss structure and also useful for creating new mechanical elements.
As seen in the movement to replace plastic products with paper products from the viewpoint of environmental protection, and the demand for folding and storing instant food containers, etc., the culmination of the folding technology mentioned above is the era of answering these. It seems that it is becoming a request.

Also, 4. Based on the result of constructing an exploded view by dividing the same shape such as a foldable cylinder, a conical shell, or similar shape elements in a node, we discuss a method of designing an exploded view by dividing these with different element shapes. , A method for making a foldable irregular-shaped rectangular tube and a pyramidal fold structure by applying this. First, formalize a method of constructing a development diagram with the intersections of equiangular spiral mountain fold lines that intersect in opposite directions at arbitrary angles, and the triangular elements that are divided by connecting those intersections that intersect in the same direction. did. Next, based on these results, we examined a method of dividing the development view with rectangular elements. As a result, generalization of the development view composed of triangular elements and
A general treatment for rectangular elements has been established.
If we consider the fold line as a truss member, we used 6 fold lines per node.
The conical shell consisting of triangular elements has a structure with a strong degree of constraint in which the nodes are connected by six members. On the other hand, since the structure consisting of rectangular elements consists of four fold lines, the degree of constraint is weak and as a result the structure is easy to deploy. In addition, since the structure of this rectangular element is easier to introduce the folding line than that of the triangular shape, it is considered to be advantageous from the point of manufacturing and processing the product. A foldable container made of paper or polymer film can be manufactured with a simple development drawing of only one stage by using the model in Fig. 8 (b) .Therefore, such a foldable model is used for the food container and its processing method. It is considered possible to develop

6. Summary Cylindrical, prismatic, or pyramidal tubular folding structure that is foldable by patterning mountain fold lines that make up rectangular or hexagonal elements and valley fold lines that divide these elements into two, and using different shaped element groups I described how to design. These expand the functionality of the folding technique and enhance the formability, and are considered to be the basic model when assembling a systematic folding storage method that is commonly used for industrial products and consumer products. . In addition, we examined the method of dividing the development view into triangular elements and rectangular elements by equiangular spiral fold lines to obtain a foldable conical shell development view, and extended the method to calculate it analytically. The development diagram is shown. Inflatable structures for space structures that require strict performance still require detailed study of deployment performance and stability, but it seems that it is not difficult to apply them to simple household goods.

In view of the problems of the prior art and the above-mentioned research results, the tubular folding structure of the present invention has the following items (O0
1) and (O02) are considered technical issues. (O01) To provide a foldable / unfoldable tubular fold structure having a shape that has never been seen before. (O02) To provide a tubular folding structure that can be folded / unfolded until the parts surrounded by the fold line come into close contact with each other.

[0106]

The present invention, which has solved the above-mentioned problems, will now be described. Elements of the present invention include elements of the embodiment in order to facilitate correspondence with the elements of the embodiment described later. Add the ones in parentheses to the symbol. The reason why the present invention is described in association with the reference numerals of the embodiments described later is to facilitate the understanding of the present invention and not to limit the scope of the present invention to the embodiments.

(Invention) In order to solve the above-mentioned problems, the tubular folding structure of the present invention has the following constitutional requirements (A01) to (A01).
It is characterized by having (A07). (A01) Having a plurality of polygonal parts (P) and a linear part connecting part that connects the outer sides of the parts (P) to each other, and is foldable along the linear part connecting part Cylindrical wall (1) on which linear folding lines (M, V) are formed
The fold lines (M, V) are a plurality of mountain fold lines (M) whose one side is a mountain fold when viewed from one side of the cylindrical wall (1), and one or more valley fold lines which are a valley fold. (V) with the cylindrical wall (1), (A02) bottom wall (2) that closes one axial end of the cylindrical wall (1), (A03) fold line at both ends of the side wall development view The plurality of fold lines (M, V) having fold lines (M, V) satisfying a continuous condition which is a condition for (M, V) being continuous, (A04) The mountain fold line (M) And a plurality of nodes that are the intersections of the valley fold lines (V) are arranged at a predetermined interval, and the difference between the number of the mountain fold lines (M) and the number of the valley fold lines (V) that intersect at one node is 2. Folding condition which is a condition that allows each part (P) including a plurality of folding lines (M, V) intersecting at the one node to be folded in close contact with each other. The plurality of fold lines having a satisfactory fold line (M, V) (M, V), (A
05) A fold line that satisfies a closing condition, which is a condition for the part (P) to be in close contact and the cylindrical wall (1) to be axially folded when folded along the fold lines (M, V) The plurality of fold lines (M, V) having (M, V),
(A06) The cylindrical wall (1) in which the diameter of the cross section orthogonal to the central axis is constant along the central axis direction, (A07) the cross section having a polygonal or triangular shape of a pentagon or more, or a circular shape. Tube wall (1).

In the tubular folding structure of the present invention having the above-mentioned structural requirements (A01) to (A07), the diameter of the cross section of the tubular wall (1) perpendicular to the central axis is formed to be constant along the central axis direction. Therefore, it is a cylindrical folding structure of a cylindrical or square tube type. One end of the tubular wall (1) in the axial direction is closed by the bottom wall (2), and the tubular folding structure can accommodate an object inside the tubular wall (1) as a container. The cylindrical wall (1) has a plurality of polygonal parts (P) and linear part connecting parts that connect the outer sides of the parts (P) to each other. A linear fold line (M, V) that can be folded along is formed.
Further, the fold line (M, V) of the cylindrical wall (1) is a valley fold with a plurality of mountain fold lines (M) whose one side is a mountain fold when viewed from the one side of the cylindrical wall (1). And one or more valley fold lines (V). The plurality of fold lines (M, V) are continuous conditions that are the conditions for the fold lines (M, V) to be continuous at both ends of the development view of the side wall, and the plurality of fold lines intersect at the one node. Folding conditions under which each part (P) including (M, V) can be folded in a close contact state, and when the parts (P) are folded along the folding line (M, V) The cylindrical wall (1) has folding lines (M, V) that satisfy the closing condition, which is a condition for axially folding the cylindrical wall (1). Therefore, a plurality of folding lines (M,
V) when the tubular folding structure is folded, unlike the conventional case where the tubular folding structure becomes unfoldable before the parts (P) come into close contact with each other, the plurality of polygonal parts ( P) Fold each other until they are in close contact. Then, the parts (P) can be expanded from the state where they are in close contact with each other to the state before being folded. Also, the cylinder wall (1)
Has a polygonal or triangular shape having a pentagonal shape or more, or a circular cross section, it is possible to provide a tubular folding structure having various shapes of cross sections, which has not been available in the past. Further, the tubular folding structure can be used for applications such as being transported by being folded and having a small outer shape, and being unfolded at the time of use.

In order to solve the above-mentioned problems, the tubular folding structure of the present invention has the following constitutional requirements (A01) to (A01).
05), (A06 ') can be provided. (A01) Having a plurality of polygonal parts (P) and a linear part connecting part that connects the outer sides of the parts (P) to each other, and is foldable along the linear part connecting part Cylindrical wall (1) on which linear folding lines (M, V) are formed
The fold lines (M, V) are a plurality of mountain fold lines (M) whose one side is a mountain fold when viewed from one side of the cylindrical wall (1), and one or more valley fold lines which are a valley fold. (V) with the cylindrical wall (1), (A02) bottom wall (2) that closes one axial end of the cylindrical wall (1), (A03) fold line at both ends of the side wall development view The plurality of fold lines (M, V) having fold lines (M, V) satisfying a continuous condition which is a condition for (M, V) being continuous, (A04) The mountain fold line (M) And a plurality of nodes that are the intersections of the valley fold lines (V) are arranged at a predetermined interval, and the difference between the number of the mountain fold lines (M) and the number of the valley fold lines (V) that intersect at one node is 2. Folding condition which is a condition that allows each part (P) including a plurality of folding lines (M, V) intersecting at the one node to be folded in close contact with each other. The plurality of fold lines having a satisfactory fold line (M, V) (M, V), (A
05) A fold line that satisfies a closing condition, which is a condition for the part (P) to be in close contact and the cylindrical wall (1) to be axially folded when folded along the fold lines (M, V) The plurality of fold lines (M, V) having (M, V),
(A06 ') Cylindrical wall (1) whose cross-sectional diameter decreases toward one end in the central axis direction

In the tubular folding structure of the present invention having the above-mentioned structural requirements (A01) to (A06 '), the diameter of the cross section becomes smaller as the tubular wall (1) goes to one end in the central axis direction. Therefore, it is a conical or pyramid-shaped tubular folding structure. Since one end portion of the cylindrical wall (1) in the axial direction is closed by the bottom wall (2), the cylindrical folding structure can store an object inside as a container. The cylindrical wall (1) has a plurality of polygonal parts (P) and linear part connecting parts that connect the outer sides of the parts (P) to each other. A linear fold line (M, V) that can be folded along is formed. Further, the fold line (M, V) of the cylindrical wall (1) is a valley fold with a plurality of mountain fold lines (M) whose one side is a mountain fold when viewed from the one side of the cylindrical wall (1). And one or more valley fold lines (V). The plurality of fold lines (M, V) are continuous conditions that are the conditions for the fold lines (M, V) to be continuous at both ends of the development view of the side wall, and the plurality of fold lines intersect at the one node. Folding conditions that are conditions under which each part (P) including (M, V) can be folded in a close contact state, and when the parts (P) are folded along the folding line (M, V), the part (P) is in close contact Then, a fold line (M, M that satisfies the closing condition, which is a condition for axially folding the cylindrical wall (1)).
V). Therefore, when the tubular folding structure is folded along a plurality of folding lines (M, V), unlike the conventional case where the tubular folding structure cannot be folded before the parts (P) are brought into close contact with each other. , The plurality of polygonal parts (P) are folded to a close contact state. Then, the parts (P) can be expanded from the state where they are in close contact with each other to the state before being folded. Also, there is no conventional one,
It is possible to provide a conical or pyramidal tubular folding structure with various shaped cross sections. Further, the tubular folding structure can be used for applications such as being transported by being folded and having a small outer shape, and being unfolded at the time of use.

Further, the cylindrical folding structure having the above-mentioned structural requirements (A01) to (A07) or (A01) to (A06 ') can also have the following structural requirement (A08). (A08) The plurality of folding lines (M, V) formed only on a part of the cylindrical wall (1) in the central axis direction. The above configuration requirements (A01) to (A08) or (A01) to (A06 '),
In the tubular folding structure provided with (A08), since the plurality of folding lines (M, V) are formed only in a part of the tubular wall (1) in the central axis direction, the folding It is possible to manufacture and provide a deployable tubular folding structure. Further, since the fold line (M, V) is formed only in a part, an object is accommodated in a portion where the fold line (M, V) is not formed, and the fold line (M, V) is formed. By folding, the outer shape can be reduced while accommodating objects. Therefore, it can be used in a wider range of applications than a structure in which folding lines (M, V) are formed on the whole.

The tubular folding structure having the above-mentioned constituents (A01) to (A07) or (A01) to (A06 ') can also have the following constituents (A09). (A09) The plurality of fold lines (M, V) having fold lines (M, V) formed along a spiral about the central axis of the cylindrical wall (1). The above configuration requirements (A01) to (A0
7), (A09) or (A01) to (A06 '), (A09)
In the tubular folding structure provided with, since the plurality of folding lines (M, V) are formed along a spiral centered on the central axis of the tubular wall (1), a folding / unfolding structure which has not been available in the past can be obtained.
It is possible to manufacture and provide a deployable tubular folding structure.

Further, the above structural requirements (A01) to (A07),
(A01) to (A06 '), (A01) to (A08), (A01)
~ (A06 '), (A08), (A01) to (A07), (A0
9) or (A01) to (A06 ') and (A09), the tubular folding structure has the following constitutional requirements (A010) and (A0).
11) can be provided. (A010) The cylindrical wall (1)
Of the plurality of polygons having only the triangular polygonal parts (P) formed by two mountain fold lines (M) and one valley fold line (V) when viewed from the outside. Parts (P), (A011) Of the two mountain fold lines (M), one mountain fold line (M) is arranged along the horizontal plane of the cylindrical wall (1), and the other mountain fold line (M). M) is the cylindrical wall (1)
The two mountain fold lines (M) arranged at a certain angle with respect to the generatrix.

The above constituent elements (A01) to (A07), (A01)
0), (A011), (A01) to (A06 '), (A010),
(A011), (A01) to (A08), (A010), (A01
1), (A01) to (A06 '), (A08), (A010),
(A011), (A01) to (A07), (A09) to (A011)
Alternatively, in the tubular folding structure including (A01) to (A06 ') and (A09) to (A011), the plurality of polygonal parts (P) are viewed from the outside of the tubular wall (1). And has only the triangular polygonal part (P) formed by two mountain fold lines (M) and one valley fold line (V). The two mountain fold lines (M) are arranged such that one mountain fold line (M) of the two mountain fold lines (M) is arranged along a horizontal plane of the cylindrical wall (1), and The other mountain fold line (M) is arranged at a certain angle with respect to the generatrix of the cylindrical wall (1). Therefore, it is possible to provide a foldable / unfoldable tubular fold structure, which has not been available in the related art, and is folded while rotating around the central axis during folding to reduce the outer shape.

Further, the above-mentioned constituent elements (A01) to (A07),
(A01) to (A06 '), (A01) to (A08), (A01)
~ (A06 '), (A08), (A01) to (A07), (A0
The tubular folding structure provided with 9) or (A01) to (A06 '), (A09) can have the following constituent element (A012). (A012) The plurality of polygonal parts (P) having only parts (P) of an isosceles quadrangle having no pair of parallel sides. The above configuration requirements (A01) to (A07), (A012), (A0
1) to (A06 '), (A012), (A01) to (A08),
(A012), (A01) to (A06 '), (A08), (A01
2), (A01) to (A07), (A09), (A012) or (A01) to (A06 '), (A09), (A012), the tubular folding structure has a plurality of polygons. Since the part (P) includes only the part (P) that is not an equilateral quadrilateral and does not have a pair of parallel sides, it is possible to provide a foldable / unfoldable tubular fold structure that has not existed in the related art.

(Examples) Next, specific examples (examples) of the embodiments of the present invention will be described with reference to the drawings, but the present invention is not limited to the following examples. To facilitate understanding of the following description, in the drawings, the front-rear direction is the X-axis direction, the left-right direction is the Y-axis direction, and the up-down direction is the Z-axis.
In the axial direction, the directions or the sides indicated by arrows X, -X, Y, -Y, Z, -Z are respectively forward, backward, rightward, leftward, upward, downward, or frontward, rearward, rightward. , Left side, upper side, and lower side. In addition, in the figure, "." In "○" means an arrow from the back of the paper to the front,
"X" in "○" means an arrow from the front to the back of the paper.

(Embodiment 1) FIG. 80 is a development view of a foldable / unfoldable conical food container as a cylindrical folding structure according to Embodiment 1 of the present invention. 81 is an explanatory view of the unfolded state (before folding) of the food container of FIG. 80.
FIG. 8A is a plan view of the food container before folding as seen from above, FIG.
1B is a side view of the food container before folding. FIG. 82 is an explanatory view of the food container of FIG. 80 in a half-folded state (during folding), FIG. 82A is a plan view of the half-folded food container seen from above, and FIG. 82B is a half-folded food container. It is a side view. 83 is an explanatory view of the food container of FIG. 80 in a completely folded state, FIG. 83A is a plan view of the food container in a completely folded state seen from above, and FIG. 83B is a food in a completely folded state. It is a side view of a container.

In FIG. 80, a food container A as a conical tubular folding structure capable of folding / expanding has a conical side wall (cylindrical wall) 1 and a disc-shaped bottom wall 2. As shown in FIGS. 80 to 83, the side wall 1 has no fold line formed on the upper portion 1a and the lower portion 1b of the outer side surface, and has a fold line only on the central portion 1c. A large number of mountain fold lines M (see solid lines in FIG. 80) and a large number of valley fold lines V (see dotted lines in FIG. 80) that are concave are formed on the central portion 1c when the outer surface is convex when folded. There is. Among the many mountain fold lines, the plurality of mountain fold lines connected at the upper end and the lower end of the central portion 1c are the upper main fold line M1 and the lower main fold line M2. Since these two main folding lines M1 and M2 are formed endlessly on a cross section perpendicular to the central axis of the food container A and are not formed along a spiral, they are formed between the two main folding lines M1 and M2. Only the open part (1 step) can be folded.

Mountain fold lines M other than the main fold lines M1 and M2
And the valley fold line V are arranged so that a total of four fold lines of three mountain fold lines M and one valley fold line V intersect at a node that is an intersection of the plurality of fold lines M and V. There is. That is,
Number of mountain fold lines M intersecting at nodes = 3, number of valley fold lines V = 1
And the difference is 2 (= 3-1). The mountain fold line M and the valley fold line V other than the main fold lines M1 and M2 are
The food containers A are arranged at equal intervals with respect to the generatrix direction of the food container A so as to satisfy the folding condition and the closing condition at each node. Therefore, it is formed (enclosed) by the folding lines M and V of the food container of the first embodiment.
The part P, which is a portion, is formed in an isosceles triangle (triangle), all triangles having a base on the upper 1a side are the same triangle, and all triangles having a bottom on the lower 1b side are also the same triangle.

(Operation of Embodiment 1) In the food container A of Embodiment 1 having the above-mentioned structure, the folding lines M and V of the central portion 1c are composed of folding line groups satisfying the folding condition, the closing condition and the continuous condition. The main folding lines M1 and M2 are formed endlessly with respect to the central axis. Therefore, by folding the food container A of Example 1 while twisting, the food container A can be axially folded / expanded from the expanded state shown in FIG. 81 to the half-folded state shown in FIG. 82 to the completely folded state shown in FIG. 83. You can

The food container A thus formed can be used, for example, as a container for instant miso soup or a container for instant noodles. In this case, in the process of transportation from the producer to the consumer, the food container A
By accommodating a material or the like having a small volume before being expanded by pouring hot water in the upper portion 1a or the lower portion 1b and folding the central portion 1c, the outer shape of the container can be reduced and shipped. Therefore, since the outer shape of the container becomes smaller, it is possible to transport a larger amount than before, and the transportation cost can be suppressed. When the consumer makes miso soup or ramen, the consumer unfolds the central portion 1c of the container A to provide a food container A that can be poured as it is by pouring hot water. After use, by folding the central portion 1c again, the outer shape becomes smaller and the collection or disposal becomes easier.

Therefore, it is possible to provide a foldable / unfoldable food container having a fold line group formed on the outer wall, which has not existed in the past. Also, the food container A of Example 1
Is the upper part 1a or the lower part 1b in which no fold line is formed.
By accommodating things in the center and folding the central portion 1c,
It has an unprecedented function that can reduce the external shape while accommodating objects. The food container A of Example 1 can be used for applications such as transporting and transporting the food container A in a state of being folded and having a small outer shape and unfolding at the time of use, which has not been possible conventionally.

Generally, the larger the angle formed between the upper and lower main folding lines M1 and M2 and the valley folding line V, the more difficult it is to naturally expand from the completely folded state due to the mechanical relationship. The force required to axially compress when folded from the state increases. Therefore, it is desirable to set the angle formed to be large (45 ° or more), and by setting in this way, the food container A in the completely folded state can be maintained in the folded state without applying external force. However, it can be prevented from developing naturally. At the same time, the unfolded food container A will not be folded even if some force is applied in the axial direction, so that the container can be prevented from being folded during a meal. Furthermore, if the angle formed is set to a large value and the upper opening of the folded food container A is closed with a lid or the like, the air inside the food container A is sealed, so that it can be naturally expanded during transportation and transportation. It can be prevented more effectively.

Further, in the food container A of Example 1, the central portion 1
Although the folding line is formed only in the portion c, the folding line can be formed in the upper portion 1a or the lower portion 1b. Also, a fold line is formed on the upper portion 1a and the central portion 1c, a fold line is formed on the central portion 1c and the lower portion 1b, or between the upper portion 1a and the lower portion 1b. Can be formed. Further, it is possible to form the fold line on the entire surface without forming the fold line partially. When formed in this way, it can be used for folding / unfolding cups, tableware, water bottles, etc. that can be used outdoors. If it is used for PET bottles, coffee cans, etc., it can be folded into small pieces and discarded. Furthermore, instead of forming only one fold line, two or three fold lines are formed.
It is also possible to form a plurality of steps such as steps. By forming a plurality of stages, it becomes possible to fold smaller in the axial direction than when forming one stage.

(Method for Manufacturing Food Container) Next, a method for manufacturing the food container A of Example 1 will be described. As shown in FIG. 80, the food container of Example 1 has side walls 1 having fold lines.
The food container A can be manufactured by bending a sheet of plastic, paper or the like on which the developed view is formed into a tubular shape and adhering both ends thereof, and then adhering the separately formed bottom wall 2 to the side wall 1. However, this method is difficult to mass-produce. A manufacturing method using a vacuum forming method used for plastic molding for mass-producing a plastic foldable / expandable food container is exemplified below. When the container is made of a material other than plastic (styrofoam, iron, aluminum, paper, etc.), it can be manufactured by another molding method depending on the characteristics of the material. When the food container A is manufactured from the development view, it is possible to manufacture it by coating or fusion bonding with a resin or the like instead of adhering it with an adhesive or the like. It is sectional drawing of the manufacturing apparatus of a food container, and is a figure which shows the process of pre-expansion of a vacuum forming method. FIG. 85 is a cross-sectional view of the apparatus for manufacturing a foldable / unfoldable food container according to the first embodiment of the present invention, which is a diagram showing a press-fitting step of the vacuum forming method. FIG. 86 is a cross-sectional view of the apparatus for manufacturing a foldable / unfoldable food container according to the first embodiment of the present invention, which is a diagram showing a forming step of a vacuum forming method.

84 to 86, the food container manufacturing apparatus B has a molding apparatus B1 and a press-fitting apparatus B2. The molding apparatus B1 includes a molding table 21 having an opening 21a at the top.
And the molding material support portion 2 provided on the upper part of the molding table 21.
2 and a molding die 23 supported inside the molding table 21 so as to be movable in the vertical direction. The outer shape of the molding die 23 is formed smaller than the opening 21 a of the molding table 21. The side portion 24 of the inner wall of the mold 23 has a conical shape corresponding to the food container A to be manufactured, and has a concave portion 24a for forming a mountain fold line and a valley fold line for forming a fold line to be formed. The convex portion 24b is formed. Mold 23
A plurality of air holes 25a are formed in the bottom portion 25 of the inner wall, and all the air holes 25a communicate with an air reservoir 26 provided in the lower portion of the bottom portion 25. An air inlet / outlet passage 27 is connected to the air reservoir 26, and the air inlet / outlet passage 27 is connected to an air pump (not shown). The air inlet / outlet passage 27 has an air valve 27a (see FIG. 85) that shuts off the flow of air.
(See) is provided.

84 to 86, the press-fitting device B is used.
No. 2 is arranged so as to be able to approach / separate from the molding apparatus B1, and a nozzle 31 facing the molding apparatus B1 is provided. An intake passage 32 is connected to the nozzle 31, and the intake passage 32 is connected to a pump (not shown). The gas that has flowed into the intake passage 32 by a pump (not shown) is configured to be jetted from the nozzle 31 toward the molding apparatus B1.

(Explanation of Manufacturing Method) In the food container manufacturing apparatus B having the above-described structure, in the pretensioning step shown in FIG. Hold in position. First, a molding material A ′ made of the same material as the food container A is mounted on the molding material support portion 22. The molding material A'covers the opening 21a of the molding table 21 so as to seal it. Then, from the air inlet / outlet 27 to the molding device B
When air is made to flow toward 1, the inside of the molding table 21 is sealed by the molding material A ', so that the molding material A'expands toward the outside by air pressure.

In the press-fitting step shown in FIG. 85, the inflow of air into the air inlet / outlet passage 27 is stopped, and the air valve 27a is closed to shut off the air inlet / outlet passage 27, whereby the molding table 2
Seal 1 air. After that, the molding die 23 is raised and the press-fitting device B2 is brought close to the molding device B1. At this time, the molding material A'is made to approach the molding material A'expanded toward the outside from the nozzle 31 of the press-fitting device B2, whereby the molding material A'is molded.
Is deformed to the inner wall (24, 25) side. However, in this state, the molding material A'is not yet molded into the same shape as the inner wall of the molding die 23.

In the molding step shown in FIG. 86, the molding die 23 is held in a state of being projected from the opening 21a of the molding table 21, and the press-fitting device B2 is held in a state of being inside the molding die 23. . Since the molding die 23 projects beyond the opening 21a, the molding material A'is held by the upper end of the molding die 23, and the molding material A'and the inner wall 2 of the molding die 23.
The space between 4 and 25 is sealed. And the nozzle 31
Stop the expected injection from, open the air valve 27a,
From the air inlet / outlet path 27 to the air hole 25a and the air reservoir 26
The air inside the molding die 23 is discharged via the. At this time, since the molding die 23 and the molding material A ′ are hermetically sealed, when the air inside the molding die 23 is exhausted to a negative pressure, the molding material A ′ comes into close contact with the inner wall of the molding die 23. And is molded into the same shape as the inner walls 24 and 25 of the molding die 23. Therefore, the food container A having a conical wall in which the mountain fold line forming concave portion 24a forms the mountain fold line M and the valley fold line forming convex portion 24b forms the valley fold line V.
Are formed inside the mold 23.

(Embodiment 2) FIG. 87 is a development view of a foldable / unfoldable conical container as a tubular folding structure according to Embodiment 2 of the present invention. 88 is an explanatory view of the unfolded (before folding) state of the container of FIG. 87, FIG. 87A is a plan view of the unfolded container seen from above, and FIG. 87B is a side view of the unfolded container. 89 is an explanatory view of the container of FIG. 87 in a half-folded state (during folding).
89A is a plan view of the container in the half-folded state seen from above, FIG. 89B
[Fig. 3] is a side view of the container in a half-folded state. FIG. 90 corresponds to FIG.
9 is an explanatory view of a state in which the container of No. 7 is completely folded, and FIG.
OA is a plan view of the container in a completely folded state seen from above, and FIG. 90B is a side view of the container in a completely folded state. In the description of the second embodiment, constituent elements corresponding to those of the first embodiment are designated by the same reference numerals, and detailed description thereof will be omitted.

In FIG. 87, the container C as a cylindrical foldable structure of the second embodiment is a foldable / unfoldable cylindrical foldable structure, unlike the first embodiment.
The part P surrounded by is composed of a scalene quadrangle. Further, in the container C of Example 2, three endlessly formed main folding lines M1, M2, M3 are arranged,
It can be folded for two steps. Therefore, by folding the container C of Example 2 while twisting,
It is possible to fold / unfold the unfolded state shown in FIG. 8 through the half-folded state shown in FIG. 89 to the fully folded state shown in FIG.

(Operation of Embodiment 2) Like the container A of Embodiment 1, the container C of Embodiment 2 having the above-described structure stores objects in the upper portion 1a and the lower portion 1b where the fold line is not formed. The outer shape can be reduced by folding the central portion 1c. Therefore, it is possible to provide a foldable / unfoldable container C having a fold line group formed on the cylinder wall 1, which has not existed in the past. In addition, since the container C of the second embodiment has two foldable portions, it can be made smaller in the axial direction than the container A of the first embodiment. When folding, do not fold both tiers. For example, fold only one tier when storing and transporting items inside, expand fold lines when using for meals or exhibitions, and both tiers when discarding. Usage such as folding is also possible. In addition, the container C of Example 2 is similar to the container A of Example 1 in that not only the central portion 1c but also the fold lines M and V are formed on the upper portion 1a and the lower portion 1b, and the fold line M and V are formed on the entire surface. It is also possible to form V. Further, similarly to the container A of the first embodiment, it is desirable that the container C of the second embodiment also sets a large angle between the main fold lines M1, M2, M3 and the valley fold line V (more preferably, 4).
By setting in this way, it is possible to prevent the folded state from being naturally expanded or being folded during use. .

(Embodiment 3) FIG. 91 is a development view of a foldable / unfoldable conical container as a tubular folding structure according to Embodiment 3 of the present invention. 92 is an explanatory view of the unfolded (before folding) state of the container of FIG. 91, FIG. 92A is a plan view of the unfolded container seen from above, and FIG. 92B is a side view of the unfolded container. 93 is an explanatory view of the container of FIG. 91 in a half-folded state (during folding).
93A is a plan view of the half-folded container as seen from above, FIG. 93B
[Fig. 3] is a side view of the container in a half-folded state. FIG. 94 corresponds to FIG.
9 is an explanatory view of a state in which the container of No. 1 is completely folded, and FIG.
4A is a plan view of the container in a completely folded state seen from above, and FIG. 94B is a side view of the container in a completely folded state. In the description of the third embodiment, constituent elements corresponding to those of the first embodiment are designated by the same reference numerals, and detailed description thereof will be omitted.

In FIG. 91, a container D as a foldable / expandable conical folding structure of the third embodiment is different from that of the first embodiment in that a part P surrounded by fold lines M and V is composed of an isosceles square. Has been done. Therefore, by folding the container D of Example 3 while twisting in the axial direction, the container D can be folded / unfolded from the unfolded state shown in FIG. 92 to the half-folded state shown in FIG. 93 to the fully folded state shown in FIG. 94. . Similarly to the container A of the first embodiment, the container D of the third embodiment has folding lines M and V formed not only in the central portion 1c but also in the upper portion 1a and the lower portion 1b, and the folding lines M and V are formed on the entire surface. It is also possible to form V. Further, similarly to the container A of the first embodiment, the container D of the third embodiment desirably has a large angle formed between the main fold lines M1 and M2 and the valley fold line V (more preferably 45 ° or more), By setting in this way, it is possible to prevent the folded state from being naturally expanded or folded during use.

(Embodiment 4) FIG. 95 is a development view of a foldable / developable cylindrical container as a tubular folding structure according to Embodiment 4 of the present invention, and FIG. FIG. 95 is a development view of a cylindrical container that can be folded in two stages.
B is a development view in which both end surfaces of FIG. 95A are formed to be linear. The cylindrical container E formed by joining the left and right ends of the developed view shown in FIG. 95 is a portion between the main fold lines M1 to M3 formed endlessly (the central portion 1C, as in the case of the above-mentioned embodiments). ) Is foldable and folds / unfolds from the unfolded state to the half-folded state to the fully folded state. Note that if unevenness is formed on the left and right ends of the developed view as in FIG. 95A, it is difficult to join the left and right ends.
When the development view is formed as in 5B, both ends are easily joined, and the cylindrical container E can be easily formed from the development view.
In addition, the container E of the fourth embodiment, like the container A of the first embodiment, includes not only the central portion 1c but also the upper portion 1a and the lower portion 1b.
It is also possible to form the folding lines M and V on the entire surface or to form the folding lines M and V on the entire surface. Further, the container E of Example 4
Are the main folding lines M1, M2, similar to the container A of the first embodiment.
It is desirable to set a large angle between M3 and the valley fold line V (more preferably 45 ° or more). By setting in this way, the folded state can be naturally expanded or folded during use. Can be prevented.

(Embodiment 5) FIG. 96 is a development view of a conical container according to a fifth embodiment of the present invention, which is a development view of a conical container having a main fold line formed along a spiral. The conical container F formed by joining the left and right ends of the developed view shown in FIG. 96 can be folded / unfolded from the unfolded state through the half-folded state to the fully folded state, as in the case of each of the embodiments. However, since the main fold line is not formed endlessly but is formed along the spiral, the folding conditions and the like are satisfied at the upper end portion and the lower end portion of the central portion F1c where the fold lines M and V are formed. There is a high possibility that there will be unresolved nodes.
That is, distortion may occur in each part P of the central portion 1c and the cylindrical wall 1 of the upper portion 1a or the lower portion 1b. Further, the container F of Example 5 is similar to the container A of Example 1 in that not only the central portion 1c but also the upper portion 1a and the lower portion 1c
It is also possible to form the folding lines M and V on b and to form the folding lines M and V on the entire surface. Furthermore, the container C of Example 5
Similarly to the container A of the first embodiment, it is desirable to set the angle formed by the main fold lines M1 and M2 and the valley fold line V to be large (more preferably 45 ° or more). It is possible to prevent it from unfolding naturally from its original state or being folded during use.

(Modifications) Although the embodiments of the present invention have been described in detail above, the present invention is not limited to the above-mentioned embodiments, but is within the scope of the gist of the present invention described in the claims. Thus, various changes can be made. Modifications of the present invention are exemplified below. (H01) Instead of the fold line group formed in the central portion 1c of the container of each of the above-described examples, various fold line groups shown in the research results of the present inventor can be used. That is, it is possible to use an arbitrary folding line group that satisfies the folding condition, the closing condition, and the continuous condition. Further, by using these folding line groups, not only those having a circular cross section but also polygonal cross sections such as a polygonal tube or a pyramidal tubular folding structure can be manufactured. (H02) In each of the above embodiments, both ends of the tubular structure may be closed without closing the one end by the bottom wall 2.

[0139]

The packing device of the present invention described above can achieve the following effects (E01) and (E02). (E01) It is possible to provide a foldable / unfoldable tubular fold structure having a shape that has not been achieved in the past. (E02) It is possible to provide a tubular folding structure that can be folded / unfolded until the parts surrounded by the fold lines come into close contact with each other.

[0140]

[Brief description of drawings]

FIG. 1 is a fold line explanatory diagram showing a typical example of a fold line that is a straight line for folding a fold structure and a node that is an intersection of a plurality of fold lines.

FIG. 2 is a diagram showing a folding condition for a 1-node 6-fold line of a type in which two valley fold lines are inserted at symmetrical positions of four mountain fold lines.

FIG. 3 shows mountain fold lines (M1), (M2), (M
It is a figure which shows the folding conditions of 1 nodal point 6 fold line when the valley fold lines (V1) and (V2) are alternately inserted between 3).

FIG. 4 is a view for explaining a condition in which both ends of the strip are joined to form a cylinder when the strip is folded along the fold line, and FIG. 4A is a diagram showing the strip and the fold line and the fold line. FIG. 4B is a diagram showing the angle of FIG. 4A, and FIG. 4B is a diagram showing the change in the orientation of the reference axis when folded along the folding line shown in FIG. 4A.

FIG. 5 is an explanatory diagram of an example in which a folding condition is satisfied and a folding direction is the same direction (either mountain fold or valley fold) into a regular tetragon, and FIG. 5A is an expanded state. 5B is a diagram showing a folding line of the strip plate, FIG. 5B is a diagram showing a state in the middle of folding, and FIG. 5C is a diagram showing a folded state.

FIG. 6 is a plan view of the strip-shaped plate shown in FIG.
FIG. 2 is a diagram showing a typical development view when a regular N-gon is formed by bending in the same direction by −2) / N at equal intervals and when N = 6.

FIG. 7 is a trapezoid of unequal sides obtained by decomposing twice the angle (π / 3) between the mountain fold line and the horizontal fold line of FIG. 6 into α = 2π / 9 and β = π / 9. It is a development view of a pseudo cylinder composed of elements.

FIG. 8 is a graph showing the mountain fold line in the Y-axis direction of FIG. 6 as α =
FIG. 8A is an exploded view and FIG. 8B is an explanatory view of a cylinder produced by introducing six sets of disassembled fold lines into a π / 3 mountain fold line I and β = π / 6 valley fold line II. FIG. 8 is a diagram showing a half-folded state of a folded cylinder produced when the both ends of the developed view of FIG. 8A are joined, and FIG. 8C is a diagram showing a further folded state of the folded cylinder of FIG. 8B.

FIG. 9 is a diagram in which points A and B in FIG. 6 are aligned with each other and a mountain fold portion is removed from a horizontal folding line, and a diamond pattern (isosceles triangle having a base angle π / 6 in the horizontal direction ( It is a development view of (1) to (3).

FIG. 10 is a development view of a deformed diamond pattern composed of isosceles triangular elements.

11 is an explanatory view of a pseudo-cylindrical body having a development view that is symmetrical with respect to a horizontal fold line and that can be folded, FIG. 11A is a development view, and FIG. 11B is the same as FIG.
FIG. 11C is a diagram showing a half-folded state of a folded cylinder produced when the both ends of FIG. 11B are joined, and FIG. 11C is a diagram of the same thing as FIG. 11B seen from a different direction.

12 is a diagram showing an example of a development view of a fold formed only by folding lines similar to the point B in FIG.

FIG. 13 is an exploded view of a foldable cylindrical wall having a plurality of polygonal parts (flat plate walls) formed by folding lines.

14 is an explanatory view of a pseudo-cylindrical body having a development view in which FIG. 6 is inclined by π / 6, and FIG. 14A is a development view,
FIG. 14B is a diagram showing a half-folded state of a folding cylinder produced when the both ends of the developed view of FIG. 14A are joined.

FIG. 15 is an explanatory diagram of a method of maintaining continuity when both ends of the development view are joined.

16 is an explanatory view of a pseudo-cylindrical body having a development view in which FIG. 7 is inclined by π / 6, FIG. 16A is a development view,
FIG. 16B is a diagram showing a half-folded state of a folding cylinder produced when the both ends of the developed view of FIG. 16A are joined.

FIG. 17 is a development view in which FIG. 8 is inclined by π / 6.

FIG. 18 is a spiral type of FIG. 11 and is obtained by cutting along a straight line connecting points A and D in the figure.

FIG. 19 is a view showing a portion cut out by two parallel straight lines AB ′ and C′D in the development view of the cylindrical body shown in FIG. 12, and A and B ′ and D and C. FIG. 20 is a development view of what becomes a foldable cylindrical body by connecting the left and right edges of FIG. 19 so that ′ overlap.

FIG. 20 is a quadrilateral element (part) having an arbitrary shape.
It is a development view of a refoldable cylindrical body having.

FIG. 21 is an explanatory diagram of an example of folding into a regular hexagon along a fold line that satisfies the closing condition and that the folding directions are alternately inverted (inverted between the mountain fold direction and the valley fold direction), and FIG. 21A is an expanded view. Folding lines (1) to (1
2), FIG. 21B to FIG. 21F are views showing a state in the middle of folding, and FIG. 21G is a view showing a folded state.

22 is an explanatory view of a pseudo-cylindrical body having a development view that is symmetrical with respect to a horizontal fold line and that can be folded, and FIG. 22A is a development view and FIG. 22B is the same as FIG.
It is a figure which shows the half-folded state of the folding cylinder produced when joining both ends of the development view.

FIG. 23 is an explanatory view of a folding cylindrical body having a fold line that is a generalization of FIG. 22, and FIG. 23A is a developed view,
FIG. 23B is a diagram showing a half-folded state of a folding cylinder produced when the both ends of the developed view of FIG. 23A are joined.

[FIG. 24] FIG. 24 is a developed view of 6 stages of FIG. 23A.
FIG. 6 is a development view in the case where α is set to 30 ° and the value of β is changed for each stage.

FIG. 25 is a development view of a repetitive spiral type obtained by reversing the spiral mountain fold line and the valley fold line of FIG. 23A for each stage.

FIG. 26 is a researcher of a folded structure.
Corresponding to a development view of a tubular structure examined by Guest et al., It is made of a triangular divided flat plate, and the main fold line has a spiral shape, and the spiral (1) has one step for each revolution of the main fold line. FIG. 3 is a developed view of the cylindrical structure when the present inventor rises.

FIG. 27 is an enlarged view of a main part of the development view of the conical tubular folding structure in which the main fold line is formed along the circumference and the part is formed by the trapezoidal element.

28 is a conical wall with a fold line that is divided into isosceles trapezoids by a fold line and is folded into a regular N pyramid in the same manner as FIG. 27, where N = 6, and φ ^* = π / 36 in FIG. 27. Two
28A is an explanatory view of a pseudo-conical wall having a development view in the case of Θ = π / 12, FIG. 28A is a development view, and FIG. 28B is a perspective view of a state in which the folding conical wall having the development view of FIG. 28A is half-folded. It is a figure.

29 is an explanatory diagram of how to draw the development view of FIG. 28. FIG.

30 is an explanatory view of a pseudo-cone having a development view in which FIG. 28 is made into a spiral shape, FIG. 30A is a development view, and FIG. 30B is a conical wall with folding lines having the development view of FIG. 30A. It is a perspective view in the state half-folded.

FIG. 31 is an enlarged view of a main part of a development view in a case where a development view of a cone in which a main fold line is formed along the circumference is composed of N isosceles triangles having an apex angle 2Θ.

FIG. 32 is an enlarged view of a main part of a development view of a conical wall with a fold line in the case of being divided into isosceles triangular elements by the fold line.

FIG. 33 is an explanatory view of a development view of a foldable conical wall with folding lines having folding lines along an equiangular spiral, FIG. 33A is an overall explanatory view, and FIG. 33B is an enlarged view of a main part of FIG. 33A. It is a figure.

34 is an explanatory view of a conical tubular folding structure having a development view formed by folding line groups having an angular relationship similar to FIG. 31, and FIG. 34A is a development view and FIG.
34B is a perspective view of the conical wall with fold lines in the half-folded state having the development of FIG. 34A.

FIG. 35 is a development view of a conical tubular fold line structure in which fold lines divide into isosceles triangular elements, where N = 3, 2Θ = π / 9, α = π / 9, FIG. 7 is a development view when δ = π / 6 (θ ^* = about 0.0688π).

36 is a development view of a conical wall with a fold line in the case of being divided into inequilateral triangular elements by a fold line having an angle α at the upper right and an angle δ at the upper left at the point F in FIG. 32. ,
FIG. 36 is a development view when the Θ, α, and δ values are the same as those in FIG. 35.

FIG. 37 is a view for explaining a folding condition 1
It is a principal part enlarged view of a 2nd step strip plate.

38 is a diagram showing the case where the valley fold line of the second step in FIG. 37 is taken in the opposite direction to that of the first step at an angle γ.

FIG. 39 is a view in which a development view of a conical wall with folding lines consisting of N isosceles triangle elements (apex angle 2Θ) is considered, and only one step thereof is written out as a curved strip portion.

40 is a conical cylindrical folded structure having the fold line shown in FIG. 37, where N = 6, γ + ψ ^*.
= Π / 3, ψ ^* = π / 6, γ = π / 6, an explanatory view of a pseudo-conical wall having a development view (2Θ = π / 18),
FIG. 40A is a developed view, and FIG. 40B is a perspective view showing a state in which the conical tubular folding structure having the developed view of FIG. 40A is half-folded.

41 is a conical tubular fold structure having fold lines shown in FIG. 37, where N = 6, γ + ψ ^* =
FIG. 9 is a development view (2Θ = π / 6) when π / 3, ψ ^* = π / 4, γ = π / 12.

FIG. 42 shows a reduced number of stages in the development view of FIG. 40A.
Without each step ψ ^*In the expanded view when the value of is increased
is there.

43 is a development view of a conical wall having the same shape as FIG. 42. FIG.

44 is an explanatory view of the pseudo-conical structure having the fold line shown in FIG. 38, FIG. 44A is a development view, and FIG.
FIG. 44B is a perspective view showing a state in which the pseudo-conical structure having the development view of FIG. 44A is half-folded.

FIG. 45 shows that 2Θ = π / 6, ψ ^* = π / 6, γ
It is a development view (N = 6) of the repeating spiral type conical structure with a folding line obtained as = π / 6.

FIG. 46 is a development view of a folding conical wall with a folding line in which the spiral in the circumferential direction of FIG. 40A is moved up one step at the right end.

FIG. 47 is a diagram for explaining a folding method of a foldable / unfoldable plane paper, FIG. 47A is an explanatory view of a main part of a conventionally known foldable plane paper development view, and FIG. 47B is a cylinder after folding. It is a principal part explanatory view of the development view of the flat paper which becomes a shape.

48 is a development view of a foldable / unfoldable tubular structure, and FIG. 48A is a development view of a tubular foldable structure composed of the same triangular elements (parts), FIG.
B is a development view of a tubular folding structure composed of the same isosceles trapezoidal elements (parts).

FIG. 49 is an exploded view of a foldable / unfoldable tubular structure, and FIG. 49A shows different trapezoidal elements (parts).
FIG. 49B is a development view of the tubular folding structure configured by the above, and FIG. 49B is a development view of the tubular folding structure configured by different triangular elements (parts).

50 is an explanatory view of a tubular folding structure having a cross-shaped cross section, FIG. 50A is a developed view, and FIG. 50B is a structure formed by joining both ends of the developed view of FIG. 50A. FIG.

51 is an explanatory view of a tubular folding structure having a rhombic cross section, FIG. 51A is a development view, and FIG. 51B is a drawing.
It is a top view when the structure formed by joining both ends of the development view of A is folded.

52 is an explanatory view of a tubular folding structure having a pseudo-elliptical cross section, FIG. 52A shows a development view, and FIG. 52B shows a structure formed by joining both ends of the development view of FIG. 52A. It is a top view at the time of folding.

53 is an explanatory view of a tubular folding structure having a rectangular cross section, FIG. 53A is a development view, and FIG. 53B is FIG.
It is a top view when folding the structure formed by joining both ends of the development view of 3A.

54 is a development view of a cylindrical tubular folding structure having a main fold line along a spiral, and FIG. 54A is the same as FIG.
54B is a view corresponding to the development view of FIG. A, and FIG. 54B is a view corresponding to the development view of FIG. 51A.

55 is a development view of a cylindrical tubular folding structure having a main fold line along a spiral, and FIG. 55A is the same as FIG.
FIG. 55B is a view corresponding to the development view of A, and FIG. 55B is a view corresponding to the development view of FIG. 53A.

56 is a development view of a cylindrical tubular folding structure having a main fold line along a spiral and is a view corresponding to FIG. 49A. FIG.

[FIG. 57] FIG. 57 is a diagram illustrating an angular relationship of a fold line group configured by a fold line formed in zigzag in the horizontal direction and a fold line formed in zigzag in the vertical direction.

FIG. 58 is an explanatory view of a specific example of the tubular folding structure in which the fold line connecting in the horizontal direction has a straight fold line, and FIG. 58A is a tubular shape in which the vertical fold line has a zigzag shape. FIG. 58B is a development view of the structure, and FIG. 58B is a development view of the cylindrical structure in which the vertical folding line is curved.

FIG. 59 is an explanatory view of a specific example of a tubular folding structure in which the fold line connecting in the horizontal direction has a straight fold line. In the development view of FIG. 58A, the development is made by reversing each stage. It is a figure.

[Fig. 60] Fig. 60 is a development view of a tubular folding structure configured by repetitive folding lines, and is a development view of a structure having a rectangular cross section after being folded.

FIG. 61 is a diagram illustrating an angular relationship of a tubular folding structure having a fold line group in which fold lines connected in the vertical direction form a straight line.

FIG. 62 is a diagram of a specific example of a tubular folding structure having a fold line group in which fold lines connected in the vertical direction form a straight line, and FIG. 62A shows a / b = 2.5, N 62B is a development view when = 4, and FIG. 62B is a development view when a / b = 2.5 and N = 6.

63 is a diagram of a modified model of an equiangular spiral, FIG. 63A is an explanatory diagram of spiral fold lines, and FIG. 63B is a diagram showing a specific example when N = 6 in FIG. 63A. .

64 is a diagram of a modified model of an iterative conformal helix, and FIG. 64A is an explanatory diagram of a spiral fold line, and FIG. 64B.
FIG. 64B is a diagram showing a specific example when N = 6 in FIG. 64A.

FIG. 65 is an explanatory diagram of an atypical pyramidal structure, FIG. 65A is a diagram illustrating an angular relationship, and FIG. 65B is a specific example of an atypical pyramidal structure.

66 is a diagram for explaining a development view of a foldable cylindrical tubular structure, FIG. 66A is a development view of a buckling pattern of a cylinder and corresponds to FIG. 9, and FIG. 66B is each element. 66C is an explanatory view of the angle of FIG. 66C, and FIG. 66C is a view inclined so that AB in FIG. 66A is horizontal.

67 is an explanatory view of a foldable cylindrical tubular structure, FIG. 67A is a drawing redrawing FIG. 66B and corresponds to FIG. 22A, and FIG. 66B is a diagram corresponding to FIG. 66B and is a diagram corresponding to FIG. 66B.

68 is an explanatory view of a foldable cylindrical tubular structure, FIG. 68A is a view in which GF in FIG. 67B is inclined so as to be horizontal, and FIG. 68B is a view in which FIG. 68A is redrawn. is there.

69 is a development view of a foldable conical cylindrical folding structure, FIG. 69A is a development view of a conical structure corresponding to the cylindrical structure of FIG. 66A, and FIG. 69B is a development view of FIG. 67A. It is a development view of a conical structure corresponding to a cylindrical structure.

70 is an exploded view of a foldable conical tubular folding structure and is an exploded view of a conical structure corresponding to the cylindrical structure of FIG. 68B.

71 is a development view of a foldable conical tubular fold structure, and FIG. 71A is a development view of a conical structure corresponding to the two-tiered cylindrical structure in FIG. 67B. 71B is a redrawing of FIG. 71A.

72 is an explanatory diagram of a drawing and an angle regarding a fold line group of a backward spiral, and is a view corresponding to FIG. 33.

FIG. 73 is a development view of a specific example of a conical tubular folding structure having a reverse spiral fold line group.

74 is a drawing and an explanatory view of angles regarding a folding line group of the same direction spiral, FIG. 74A is an overall view, and FIG.
B is an enlarged view of a main part of FIG. 74A.

[FIG. 75] FIG. 75 is a development view of a specific example of a conical tubular folding structure having a fold line group of spirals in the same direction.

76 is an explanatory view of a conical structure composed of trapezoidal elements, FIG. 76A is an explanatory view of a drawing method and an angular relationship, and FIG. 76B is a structure of a specific example formed by the drawing method of FIG. 76A. FIG.

77: FIG. 77: FIG. 7 constituted by trapezoidal elements
77A is an explanatory view of a conical structure different from FIG. 6, FIG. 77A is an explanatory view of a drawing method and an angular relationship, and FIG. 77B is a developed view of a structure of a specific example formed by the drawing method of FIG. 77A.

FIG. 78 is an explanatory diagram of a drawing method and an angular relationship of a conical tubular folding structure configured with rectangular elements different from those in FIG. 74.

79 is an illustration of a specific example of a foldable conical tubular fold structure having fold lines formed based on the drawing method of FIG. 78, and FIG. 79A is a development view of one step up. , FIG. 79B is a development view of two steps up.

FIG. 80 is a development view of a foldable / unfoldable conical food container as a cylindrical folding structure according to the first embodiment of the present invention.

81 is an explanatory view of the unfolded (unfolded) state of the food container of FIG. 80, FIG. 81A is a plan view of the unfolded food container as seen from above, and FIG. 81B is the unfolded food container. FIG.

82 is an explanatory view of the food container of FIG. 80 in a half-folded state (during folding), FIG. 82A is a plan view of the half-folded food container seen from above, and FIG. 82B is a half-folded state. It is a side view of the food container of FIG.

83 is an explanatory view of the food container of FIG. 80 in a completely folded state, FIG. 83A is a plan view of the food container in a completely folded state seen from above, and FIG. 83B is in a completely folded state. It is a side view of the food container in the open state.

FIG. 84 is a cross-sectional view of the apparatus for manufacturing a foldable / unfoldable food container according to the first embodiment of the present invention, which is a diagram showing a pre-expansion step of the vacuum forming method.

FIG. 85 is a cross-sectional view of the apparatus for manufacturing a foldable / unfoldable food container according to the first embodiment of the present invention, showing a press-fitting step of the vacuum forming method.

FIG. 86 is a cross-sectional view of the apparatus for manufacturing a foldable / unfoldable food container according to the first embodiment of the present invention, which is a diagram showing a forming step of a vacuum forming method.

FIG. 87 is a development view of a foldable / unfoldable conical container according to a second embodiment of the present invention, which is a tubular folding structure.

88 is an explanatory view of the unfolded (unfolded) state of the container of FIG. 87, FIG. 87A is a plan view of the unfolded container seen from above, and FIG. 87B is a side view of the unfolded container. Is.

89 is an explanatory view of the container of FIG. 87 in a half-folded state (in the middle of folding), FIG. 89A is a plan view of the half-folded container seen from above, and FIG. 89B is a half-folded container. FIG.

90 is an explanatory view of the container of FIG. 87 in a completely folded state, FIG. 90A is a plan view of the container in a completely folded state seen from above, and FIG. 90B is in a completely folded state. It is a side view of the container of FIG.

FIG. 91 is a development view of a foldable / unfoldable conical container according to a third embodiment of the present invention, which is a cylindrical folding structure.

92 is an explanatory view of the unfolded (before folding) state of the container of FIG. 91, FIG. 92A is a plan view of the unfolded container seen from above, and FIG. 92B is a side view of the unfolded container. Is.

93 is an explanatory view of the container of FIG. 91 in a half-folded state (during folding), FIG. 93A is a plan view of the half-folded container seen from above, and FIG. 93B is a half-folded container. FIG.

94 is an explanatory view of the container of FIG. 91 in a completely folded state, FIG. 94A is a plan view of the container in a completely folded state seen from above, and FIG. 94B is in a completely folded state. It is a side view of the container of FIG.

FIG. 95 is a development view of a foldable / unfoldable cylindrical container as a tubular folding structure according to a fourth embodiment of the present invention, and FIG. 95A shows a central portion that can be folded in two stages as in the second embodiment. FIG. 95B is a development view of the cylindrical container, and FIG. 95B is FIG. 95A.
FIG. 4 is a development view in which both end faces of are formed to be linear.

96 is a development view of the conical container of Example 5 of the present invention, and is a development view of the conical container in which the main fold line is formed along the spiral. FIG.

[Explanation of symbols]

1 ... Tube wall, 2 ... Bottom wall, M ... Mountain fold line, M, V ... Fold line,
P ... parts, V ... valley fold line.

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Claims (6)

[Claims] 1. A tubular folding structure having the following constituent requirements (A01) to (A07), (A01) a plurality of polygonal parts, and linear parts for connecting the outer sides of the respective parts to each other. A fold line having a connecting portion and a linear fold line that can be folded along the linear part connecting portion is formed, and the fold line is a mountain on the one surface side when viewed from one surface side of the cylinder wall. The cylinder wall having a plurality of mountain fold lines that form a fold and one or more valley fold lines that form a valley fold, (A02) a bottom wall that closes one axial end of the cylinder wall, and (A03) a side wall. A plurality of fold lines having fold lines that satisfy a continuation condition that is a condition for the fold lines to be continuous at both ends of the development view;
A plurality of nodes, which are the intersections of the mountain fold lines and the valley fold lines, are arranged at a predetermined interval, and are formed so that the difference between the number of the mountain fold lines and the number of the valley fold lines that intersect at one node is 2. A plurality of fold lines having fold lines satisfying a folding condition that is a condition that each part including a plurality of fold lines intersecting at the one node can be folded in close contact with each other, (A05) folded along the fold lines Sometimes, the plurality of fold lines having a fold line that satisfies a closing condition, which is a condition for the parts to come into close contact with each other and the cylindrical wall to be folded in the axial direction, (A
06) The cylindrical wall having a cross section perpendicular to the central axis formed to have a constant diameter along the central axis direction, and (A07) the cylindrical wall having a polygonal or triangular shape with a pentagon or more, or a circular cross section. 2. The following constituent requirements (A01) to (A05), (A
(A01) a plurality of polygonal parts, and a linear part connecting part that connects the outer sides of the parts to each other, the linear part connecting part A fold line that is formed along a straight line that is foldable along the fold line. The cylinder wall having one or more valley fold lines, (A02) the bottom wall closing one end of the cylinder wall in the axial direction, and (A03) because the fold lines are continuous at both ends of the side wall development view. (A04) A plurality of nodes, which are intersections of the mountain fold line and the valley fold line, are arranged at a predetermined interval and intersect at one node. The difference between the number of mountain fold lines and the number of valley fold lines is 2, and the one node A plurality of fold lines having fold lines that satisfy the folding condition, which is a condition that each part including a plurality of fold lines intersecting with each other can be folded in a close contact state,
(A05) The plurality of fold lines having fold lines that satisfy a closing condition, which is a condition for the parts to come into close contact with each other and to fold the tubular wall in the axial direction when folded along the fold lines. ′) Cylindrical wall whose cross-sectional diameter decreases toward one end in the central axis direction 3. The tubular folding structure according to claim 1 or 2, characterized by comprising the following constituent features (A08): (A08) formed on only a part of the tubular wall in the central axis direction. The plurality of fold lines. 4. The tubular folding structure according to claim 1 or 2, characterized by comprising the following constituent features (A09): (A09) along a spiral about the central axis of the tubular wall The plurality of fold lines having formed fold lines. 5. The cylindrical fold line structure according to claim 1, further comprising the following constituent features (A010) and (A011), and (A010) seen from the outside of the cylindrical wall. (A011) Of the two mountain fold lines, the plurality of polygonal parts having only the triangular polygonal parts formed by two mountain fold lines and one valley fold line. The two mountain fold lines in which one mountain fold line is arranged along a horizontal plane of the cylinder wall, and the other mountain fold line is arranged at a certain angle with respect to a generatrix of the cylinder wall. 6. The structure with a cylindrical fold line according to claim 1, wherein the structure (A012) has the following constituent features (A012): The plurality of polygonal parts having only parts.