From Shape Rules to Rule Schemata
and Back
Athanassios Economou and Sotirios Kotsopoulos
Abstract Shape rules and rule schemata are compared in terms of their expressive
and productive features in design inquiry. Two kinds of formal processes are
discussed to facilitate the comparison. The first proceeds from shape rule instances
and infers rule schemata that the shape rules can be defined in. The second proceeds
from rule schemata and postulates shape rule instances that can be defined within
the schemata. These two parallel processes mirror our intuition in design: the
conceptual need to frame explicit actions within general frameworks of principles,
and the productive need to supply general principles with an explicit system of
actions.
Introduction
Shape rules and rule schemata have always been at the center of shape computation
discourse [1, 2]. The algebraic foundations, mechanisms and conventions underlying both constructs have been carefully crafted over time and have provided a
formidable framework for the study of visual calculation with shapes and design at
large. Recently shape rules, parametric shape rules and the rule schemata within
which these are defined have all been generously recast to provide a more comprehensive approach to design formalism [3, 4]. Perhaps the most significant new idea
is the reformulation of the schema as an abstract symbolic expression that takes on
shapes in its variables. In prior discourse the schema was shape-specific (parametric
shape); all the shapes that were defined in this schema were determined by an
assignment of real values to the variables of the schema. The new extended
formulation of a shape schema allows for the representation of any parametric
shape as an assignment to the variables of the schema. Clearly the power of
A. Economou (*)
Georgia Institute of Technology, Atlanta, GA, USA
e-mail: economou@coa.gatech.edu
S. Kotsopoulos
Massachusetts Institute of Technology, Cambridge, MA, USA
© Springer International Publishing Switzerland 2015
J.S. Gero, S. Hanna (eds.), Design Computing and Cognition '14,
DOI 10.1007/978-3-319-14956-1_22
383
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A. Economou and S. Kotsopoulos
Table 1 Shape rules and rule schemata
Shape rules
Rule schemata
x
x
x
x
p(x)
x
b(x)
x
d(x)
x
x+t(x)
x
x
x
x
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
x
x
t(x)
t*(x)
p(x)
x
b(x)
x
d(x)
x
x+t(x)
x
d(t(x))
x+t*(x)
b(b(x))
X
xþ
tð x Þ
b(b(x))
x þ tR ð x Þ
!
!
x
x þ tðxÞ þ tR ðx þ tðxÞÞ
x
symbolic expressions in either side of a rule to function as variables that can
instantiate shapes suggests an entirely new way at looking at shape rules that nicely
complements the existing visual approach in shape grammars. A list of shape rules
and a list of rule schemata are juxtaposed in Table 1. Note that the list of shape rules
and the list of rule schemata are independent.
The juxtaposition of these two representations of rules – and the possibilities
they suggest when they are set one against the other is quite telling. The key
characteristics they foreground – one emphasizing shapes, geometry and visual
representation, the other emphasizing abstraction, and symbolic/discursive perspective, – together indeed suggest a rich structure to be explored and contrasted.
Intuitively the contrast between pictorial rules and symbolic rules given in shape
rules and rule schemata suggests the useful dichotomy between visual and discursive symbols [5]. The analogy is clear. Shape rules are given in terms of visual
means including specific shapes and other visual tokens as needed. Rule schemata
are given in terms of symbolic means including symbols, operations, and other
indexical tokens as needed. Shape rules come as visual devices devoid of any
From Shape Rules to Rule Schemata and Back
385
structure; shapes fuse and split in any way desired. Rule schemata appear as
conceptual devices devoid of any shape; schemata combine by sums and products
and are visualized by predicates and assignments (more on this later in the paper).
Furthermore, the symbolic forms that the recursive definitions of schemata assume
appear all as atomic units that can combine in specific and well-constrained ways –
a seemingly very different world from the world of shapes and the constant fusion
that shapes invite.
But there are more ways that we can look at these sets of rules. Perhaps we could
look at their usage and fit in creative design settings and in particular at their
adaptation in studio, see for example [6–10]. Clearly some designers do things (that
is, draw or make three-dimensional models) without being able to exactly describe
what they are doing, i.e., whether a specific action they do is a particular transformation or operation. Still other designers opt for a more systematic approach (that
is, they outline general principles of action) without knowing in advance how
exactly they will use them to resolve the problem at hand. In that sense
the usefulness of shape rules and rule schemata to capture specific actions that
designers do (formal composition as visual process) or general principles that
designers discuss of (formal composition as conceptual process) may be quite
rewarding to pursue. And similarly the ability of shape rules and rule schemata to
model formal strategies in design including bottom up processes determined within
explicit, narrow contexts, or top down processes framed by open-ended principles
applicable to wide variety of contexts, could also be rewarding to pursue. Intuitively, both types of formal systems are deployed in design to solve particular kinds
of problems in spatial composition: shape rules are mostly deployed because of
their visual specificity; rule schemata because of their conceptual generality. And
still both shape rules and rule schemata are just alternative ways to describe the very
same thing from a different vantage point.
The work here provides a tentative comparison between shape rules and rule
schemata and attempts to show how these two modes of visual computation inform
one another. A brief account of both forms of rules is given and a series of pictorial
examples illustrate their similarities and differences. Both types of rules are seen
within a general theory of computational design structured around the notion of a
design algorithm <a, k> where the input a is information for construction and the
output k is the design description [11]. The account is necessarily fleeting and
impressionistic and it is used primarily as a scaffold for a brief examination of both
forms and the possibilities that each provides in design inquiry. The account is
structured around two vectors pointing from shapes to schemata and from schemata
to shapes. These two vectors are used here to structure the discourse and to suggest
possibilities for merging seeing and reflecting in visual computation. The work
concludes with a brief discussion of a computational framework that facilitates both
views of design inquiry – the pictorial redescription of existing shape rules as
explicit assignments in rule schemata and the modeling of rule schemata from
scratch.
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Two Directions
The underlying algebraic framework within which the shape rules and the rule
schemata are defined, including the algebras of shapes Uij, the algebras of labels Vij,
the algebras of weights Wij, their combinations and the ways all facilitate computations of all sorts, has been given in various sources (see for example [1, 3, 12,
13]. The following discussion provides a brief overview of shape representation in
shape grammars using shapes in a Euclidean space, but it equally well applies to
labeled shapes and weighted shapes as well as parametric labeled shapes and
parametric weighted shapes. An extended discussion of the formalism and especially the recent work on rule schemata is given in [3].
Shape Rules
A shape rule consists of a pair of shapes. Shapes consist of four basic elements:
points, lines, planes and solids and their combinations. The basic elements and the
shapes they define are readily described using linear equations and higher degree
equations and the resources and conventions of analytic geometry. These descriptive devices are enough to capture all shapes, from the rudimentary polygons and
polyhedra described by linear equations, to higher degree equations representing
curves, b-splines, NURBS and so forth. The basic elements are related one to
another through boundary conditions. A three-dimensional solid is bounded by
two-dimensional planes. A two-dimensional plane is bounded by one-dimensional
lines. An one-dimensional line is bounded by zero-dimensional points. And the
zero-dimensional points have no boundaries (and no parts). Moreover, shapes are
always defined in a Euclidean space that has a dimension equal or bigger than the
dimension of the basic elements that make the shapes – the shapes are always parts
of the Euclidean space they are defined in. The structure is quite elegant: A solid
can be a defined only in a three-dimensional Euclidean space. A plane can be
defined in a two- and/or three-dimensional Euclidean space. A line can be defined in
a one-, two- and/or three-dimensional Euclidean space. And a point can be defined
in any dimensional Euclidean space up to three dimensions. These elements
combine to produce a generous structure of ten spatial systems Uij, for i ¼ basic
elements and j ¼ dimension of space, with specific algebraic attributes [2]. The ten
algebras of shape are presented in Table 2.
A shape consisting of lines and defined in the two-dimensional Euclidean space
in the algebra U12 is given in Fig. 1.
Any pair of specific shape instances A and B determines a shape rule. The
notation of a shape rule follows the convention of an arrow (!) separating the
two shapes, on the left and right hand side of the rule, with the additional convention of registration marks (+) to fix the spatial relation between them. An example
of a shape rule is shown in Fig. 2.
From Shape Rules to Rule Schemata and Back
Table 2 The ten algebras of
shape
387
U00
U01
U11
U02
U12
U22
U03
U13
U23
U33
Fig. 1 A two-dimensional
shape containing lines
Fig. 2 A shape rule
Symbolically, for the two shapes A, B the shape rule is expressed as:
A!B
Shape rules apply in a design process when there is a match of the rule to a part
of the design at hand. The left hand side of the rule shows the shape that is matched
in the design. The right hand side shows the shape that substitutes the shape that has
been matched by the left hand side of the rule. If there is no match, then the rule
cannot be applied in the particular design context. More technically, for shapes A,
B, C, the shape rule A ! B can apply to a shape C whenever there is a transformation t that makes the shape t(A) part of the shape C. If the shape t(A) is part of the
shape C the rule subtracts the shape t(A) away from the shape C and replaces it by
the shape t(B). The resulting shape C0 and the corresponding computation are given
then as:
C0 ¼ ½C
tðAÞ þ tðBÞ
The application of the rule is distinguished from the expression of the rule itself
by the convention of a double arrow (¼>) showing to the left the initial shape C and
to the right the derived shape C0 , after the application of the rule. The sequence of
shapes (designs) generated by the rule A ! B in the manner shown above is
symbolically expressed as C ¼> C0 ¼> C00 ¼> . . . ¼> C0 ... 0 . The same sequence
of shapes may be taken as a finite set of shapes that all are productions of the rule
A ! B. In this sense, the finite sequence (derivation) of the shapes C, C 0 , C 00 , . . .,
C 0 . . . 0 has as members shapes that all share as their common property that they are
all productions of the same shape rule. All sequences of shape rule applications
bring to the foreground the compositional machinery (rules) used to produce the
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Fig. 3 A visual computation with the shape rule of Fig. 2
design. For example, a sequence of applications of the shape rule in Fig. 2 can
generate a sequence of designs shown in Fig. 3.
Rule Schemata
A rule schema consists of a pair of schemata. Schemata are comprised by variables,
and/or combinations of sets of variables. Specific parametric shape instances can be
determined when values are assigned to these variables by an assignment
g restricted in some way by a predicate. The values assigned to the variables are
shapes consisting of points, lines, planes and solids and any combination of them
(as well as labeled and/or weighted shapes). Different assignments define different
shapes and additional conditions can be added at will to define families of shapes
with specific attributes. On the other extreme, constant values may be assigned to
variables to define a single representation. For example, the shape in Fig. 1 can be
defined in a schema that is restricted by the predicate g(S1): S1 is a triangle
the set of variables
S1 (L1, L2, L3)
L1 (V1, V2)
L2 (V2, V3)
L3 (V3, V1)
and the list of values of the assignment g
V1 (0, 0)
V2 (3.46, 2)
V3 (0, 8)
Clearly different predicates can change the attributes and the number of assignments on the schema, thus specifying different shapes. And furthermore, different
assignments of numeric values to the variables can specify different shape
instances. Three shape instances for different assignments are given in Fig. 4.
The assignments are: g1: V1 (0, 0); V2 (3.46, 2); V3 (0, 8); g2: V1 (0, 0); V2 (3, 6);
V3 (0, 8); and g3: V1 (0, 0); V2 (0, 9.24); V3 (4.62, 8) respectively. The initial one is
the one illustrated in Fig. 1.
From Shape Rules to Rule Schemata and Back
389
Fig. 4 Three different
instances of the schema
Any pair of schemata can determine a rule schema. A symbolic expression of a
parametric rule schema is given as:
x!y
Rule schemata are formal generalizations of rules that specify a particular
treatment for an entire family of shapes instead of specific shape instances. The
variables in the pair of parametric shapes x and y are assigned values by an
assignment g, the properties of which are determined by a predicate, specifying
a certain class of shapes. When specific shapes are defined by g the schemata x and
y become the shapes g(x) and shape g( y) respectively, and the rule schema is
recast as a shape rule. Different constraints expressed in the predicate may lead to
the formation of more or less constraint parametric shape rules. The constraints
determining the instantiation of schemata to shapes do not affect the shape
instances themselves, and do not determine how these shapes partake in spatial
composition. Hence, visual ambiguity is preserved, shapes remain structure-less,
rules unconstrained, and their productions open to interpretation. This makes
descriptive (symbolic) precision and spatial (visual) ambiguity simultaneously
possible in the same process of calculating. A symbolic expression of the resulting
rule is given as:
gð x Þ ! gð y Þ
For example, the shape rule in Fig. 2 can be defined in a rule schema
x ! x þ tðxÞ, whereas x is a parametric triangle and t(x) an isometric copy of the
parametric triangle in a specific spatial relation to the initial parametric triangle.
Different rule schemata can be formed after spatial relations between any type of
triangles, or parallelograms, trapezoids, quadrilaterals, pentagons, hexagons and
any other shape desired.
Rule schemata apply in a design process when there is a match of the assignment
of the parametric shape at the left hand side of the rule to a part of the design at
hand. If there is a match then the assignment of the parametric shape is substituted
with the corresponding transformation of the assignment of this shape in the right
hand side of the rule. More technically, for parametric shapes x, y, and a shape C,
the rule schema x ! y applies to the shape C whenever there is a transformation
t that makes the shape g(x) – for some assignment g that assigns values to the
variables of x – part of the shape C. If the t(g(x)) is in fact part of shape C, then the
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rule subtracts the shape t(g(x)) from C and replaces it by the shape t(g( y)). The
corresponding computation is given as:
C0 ¼ ½C
tðgðxÞÞ þ tðgð yÞÞ
As before, the application of the rule schema is distinguished from the expression of the rule itself by the convention of a double arrow (¼>) showing to the left
the shape C and to the right the derived shape C0 , after the application of the rule.
The design process and the productions of the rule schemata share the same
conventions that apply in shape rule computation.
Back and Forth
There is a strong affinity between the two formal devices – and a tension too. The
formal structure of both types of rules is identical. They are both determined by a
pair of things in a relation: a pair of shape instances in the case of shape rules and a
pair of schemata whose constraints and variables instantiate shapes. Intuitively this
extra layer of abstraction, involving predicates and variables, suggests and invites a
closer look.
The pivotal role and significance of predicates and variables become evident
when a rule is given in a recursive form. In this form the variables of x and y may be
recursively related to produce an indefinite number of symbolic expressions that
associate x and y in desired ways. For example, if y and its variables in the right
hand of the rule is a transformation t of x and its variables in the left hand side of the
rule, then the rule x ! y can be rewritten as x ! t(x). Alternatively, if y and its
variables in the right hand of the rule is related through some operation, say a
division d, with x and its variables in the left hand side of the rule, then the rule can
be rewritten as x ! d(x). In general, if the variables of x and y can be associated
through some design operation f, then y becomes a function f(x) and the rule schema
can be rewritten in the form:
x ! f ðxÞ
The question then is what are the possible operators f that can relate the two
variables of x and y in meaningful and constructive ways. Clear candidates are:
(a) the transformation operation t; (b) the boundary operation b; (c) the part operator
p; and the division operation d. More could be envisioned, but more productively,
more could be constructed from those through compositions and additions. A nice
set of rule schemata to start the discussion is found in [4]. The rule schemata
presented there illustrate a set of discrete design processes that can be taken
individually, reversed when possible, and combined under addition and composition. A list of basic rule schemata and their inverses is shown in Table 3.
From Shape Rules to Rule Schemata and Back
391
Table 3 A list of basic rule schemata
Schema
Inverse
x!
!x
x!x
x ! t(x)
t(x) ! x
x ! b(x)
b(x) ! x
x ! p(x)
p(x) ! x
x ! d(x)
d(x) ! x
Fig. 5 A shape rule
generating spiral patterns
The possibilities are bewildering. New combinations and products can be produced to structure shape rules that can be defined within them and to suggest new
trajectories in design. For example, a rule schema like
x þ tR ðxÞ ! x þ tðxÞ þ tR ðx þ tðxÞÞ
could be very useful to account for the generative specification of a bilateral or
rotational growth of modular patterns. And any other combination or product of
variables might provide a useful structure to model shape rules. Rule schemata
appear indeed to have a generative power because of their ability to form compositions and combinations in sequences that are potentially novel and meaningful in
terms of the shape rules that these might be defined in. Still shape rules appear to
resist the design interpretation that the rule schemata endow them. A constructive
comparison between these two formal devices is briefly discussed below, in two
sets of exercises. The first looks at an existing shape rule and infers rule schemata
that this rule could be defined in. The second looks at an existing rule schema and
postulates shape rules that can be defined within the schema.
From Shapes Rules to Rule Schemata
The trajectory from shape rules to rule schemata is straightforward. In this inquiry
shape rules are considered as instances of particular assignments in rule schemata,
and in extension as pictorial instances of particular rule schemata. Any shape rule
from existing shape grammars and any shape rule constructed from scratch could do
to illustrate this inference. A nice set of shape rules to start the discussion is found in
[14]. The shape rules presented in this work are divided in two sets. The first intents
to produce existing designs (plus some additional designs that potentially belong in
the same set). The second intents to produce novel things without paying attention
to existing designs. The former types of rules are illustrated using squares and the
latter triangles. A shape rule selected from this work is given in Fig. 5. All labels
associated with the original shape rules are omitted here or rather are substituted by
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A. Economou and S. Kotsopoulos
Fig. 6 A description of the
shape rule of Fig. 4 in terms
of the schema x ! x þ tðxÞ
Fig. 7 A description of the
shape rule of Fig. 4 in terms
of the schema x ! x + y
a singular cross to denote the fixing of the application of the rule in the Cartesian
plane.
Clearly this shape rule can be described in a variety of ways. An intuitive reading
could result in a description given by the rule schema x ! x þ tðxÞ. Here x is a
right-angle triangle, t a similarity transformation including scales, reflections and
rotations about edges and/or vertices of the triangle and t(x), a similar copy of the
initial triangle. The context of the original paper clearly suggests that the shape rule
above (and the rest of the shape rules in the paper) all illustrate aspects of an
additive process in design. For every shape in the left hand side of the rule, an
isometric and/or scaled copy of the shape is added in the right hand side in a specific
spatial relation to the former one. The shape rule of the example shows a possible
way that a triangle x can be combined with a similar copy of itself t(x) so that the
short side of the large copy matches the long (hypotenuse) of the original triangle.
The description of this shape rule in terms of the schema x ! x þ tðxÞ is given
diagrammatically in Fig. 6.
The description of the shape rule in terms of a rule schema can easily be cast in
alternative ways. For example, the rule can be recast as x ! x þ y, whereas y some
other shape arbitrarily related to the initial shape x. A possible interpretation of the
added shape y could be a concave quadrilateral carefully chosen to match two of its
edges to the small and medium sides of the initial triangle x. The description of the
shape rule of Fig. 4 in terms of the schema x ! x þ y is given diagrammatically in
Fig. 7.
It is interesting to note that the added shape y need not be a gestalt shape, say, the
concave quadrilateral above. The two longer lines of this quadrilateral could do it
too. In this sense the emphasis seems to shift from the addition of a closed
polygonal shape to the addition of an open polygonal shape y that does not share
any edges or part of edges with the initial triangle in the left hand side of the rule. A
different way of casting this rule could start from the selection of a point outside the
initial triangle and its joint with two lines y and t( y) with two of the vertices of the
triangle in the left hand side of the rule. The description of the shape rule of Fig. 4 in
terms of the schema x ! x þ y þ tð yÞ is given diagrammatically in Fig. 8.
From Shape Rules to Rule Schemata and Back
393
Fig. 8 A description of the
shape rule of Fig. 4 in terms
of the schema
x ! x þ y þ tð y Þ
Fig. 9 A description of the
set of shape rule of Fig. 4 in
terms of the schema pðxÞ !
x or alternatively
x ! p 1 ð xÞ
And this is not all. The rule can also be recast as p(x) ! x, for p(x) a part of a
shape x, and x a shape. In this case, the shape p(x) in the left hand side of the rule is
the right-angle triangle, and the shape x in the right-hand of the rule is a shape that
has the shape illustrated in the left hand side of the rule as it s proper part. This rule
schema can be alternatively cast as x ! p 1(x), for p 1(x) the inverse of p(x),
meaning that a shape x goes to a shape with x as a part [3]. The algebraic notation of
p 1(x) might alienate the visual thinkers but it provides a uniform treatment in the
classification of the basic schemata and consistency in notation too. In this case, the
shape x is the right-angle triangle and the p 1(x) is the shape that has the right-angle
triangle as its proper part. The description of the shape rule of Fig. 5 in terms of the
schema x ! p 1(x) is given diagrammatically in Fig. 9.
From Rule Schemata to Shape Rules
The trajectory from rule schemata to shape rules is straightforward too. In this
inquiry rule schemata are considered individually or in various combinations and/or
compositions and shape rules are introduced that are restricted in some way by a
predicate and a set of assignments to pictorially instantiate the rule schemata. Any
rule schema from existing rule schemata classifications could do to illustrate this
inference. A nice set of rule schemata to start the discussion is found in [4]. The rule
schemata presented there encapsulate a set of discrete processes in design – that can
be taken individually, reversed when possible, and combined under addition and
composition. In the following example, a schema already encountered in the
previous section is selected to help us draw comparisons between the exercise in
the previous section and the exercise in this section.
x ! x þ tðxÞ
This schema is perhaps the most frequent deployed in shape grammar discourse:
it specifies to add a transformed copy t(x) of a shape x in a specific way constrained
by some predicate contained in the schema. The schema could be intuitively cast as:
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A. Economou and S. Kotsopoulos
“Look at a design, find a part x that is of interest to you, and repeat it in some way”.
More formally, the rule schema could be recast as: “look at a design A and if there is
a transformation T such that the shape x is part of A, then replace the occurrence of
shape T(x) in A with the shape T ðxÞ þ T ðtðxÞÞ or better, with the shape T ðx þ tðxÞÞ.
An indefinite number of shape rules can be defined based on this rule schema.
One way to look at the possible classification of shape rules fixed within this rule
schema is to look at the spatial relation between the initial shape x and the
transformed copy of the shape t(x), and the transformation t under which the copy
t(x) was constructed. The possible spatial relations between the shape x and its copy
t(x) can be classified in families of spatial relations with respect to the dimensionality of the basic elements that comprise the spatial relation between the two shapes.
For example, for a shape x a right triangle and an isometric copy of itself t(x), there
are four sub-families of spatial relations that can be defined between the right
triangle x and its copy t(x): the two triangles x and t(x) may share the empty
shape, a point, a line or a plane. Clearly for each of these conditions there are
many more sub-conditions to discern with respect to the spatial transformations that
specify the spatial relation; i.e., whether the transformation is, say, a translation, a
rotation, a reflection, a glide reflection and so forth [15, 16]. In all cases, these
spatial relations provide the blueprints for the specifications of the shape rule in the
rule schema. One instance of a shape rule for each of these four families of spatial
relations is shown in Fig. 10.
Fig. 10 Four shape rules
defined with the rule
schema x ! x þ tððxÞ that
satisfy the predicate x is a
3-gon and t an isometric
transformation
From Shape Rules to Rule Schemata and Back
395
Fig. 11 Four shape rules
defined with the rule
schema x ! x þ tððxÞ that
satisfy the predicate x is a
3-gon and t a similarity
transformation
Note that these four families of spatial relations can be nicely expressed as
intersections (.) of the basic spatial elements of the shapes, their boundaries and
their boundary inverses too. More specifically, in the first spatial relation, the
intersection x.t(x) of the shapes x and t(x) is the empty shape. In the second spatial
relation the intersection b(x). b(t(x)) of the boundaries of the shapes x and t(x) is a
single point (basic element in the algebra U02). In the third spatial relation the
intersection x.t(x) of the shapes x and t(x) is a single line (basic element in the
algebra U12). And in the fourth spatial relation, the intersection b 1 ðxÞ:b 1 ðtðxÞÞ of
the inverses of the boundaries of the shapes x and t(x), is a single plane (basic
element in the algebra U22). It should be noted that in the last case the shape b 1(x)
is part of b 1(t(x)), a condition that implies the definition of the part relation .
The families of shape rules that can be defined within the rule schema x ! x + t
(x) can be significantly extended with respect to the transformation t that specified
the geometry of the t(x). Figure 11 shows samples of the four families of shape rules
defined in this schema for t a similarity transformation, that is, a scale transformation combined with any isometric transformation including any combinations of
translations, rotations, and reflections.
The next families of transformations that may be added on the similarity transformations of the Euclidean space are the affine, linear, and topological transformations of the so-called non-Euclidean space [17]. Figure 12 shows samples
of four families of shape rules defined in this schema for t an affine transformation,
that is, a stretch/compress transformation combined with any similarity transformation, and t(x) an affine copy of the shape x.
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Fig. 12 Four shape rules
defined with the rule
schema x ! x þ tððxÞ that
satisfy the predicate x is a
3-gon and t an affine
transformation
Still the expressive power of the rule schema can go beyond the transformational
property of the spatial relations between the shape x and its copy t(x). If for
example, the shape x is defined as any n-gon, then a long list of possible families
of shapes can be used to define shape rules within this rule schema, all very different
from the ones seen so far. Within this framework, spatial relations between say,
squares, rectangles, parallelograms, rhombi, kites, trapezoids, quadrilaterals of all
sorts, and so forth, are all spatial relations between a shape x and a shape t(x), for a
shape x and t(x) any of those and t a Euclidean or parametric transformation.
Pentagons and hexagons and heptagons and so forth, all wait to be tried for they
provide an inexhaustible really list of visual conditions to explore. Figure 13 shows
samples of the four families of shape rules defined in this schema for x a square, t a
scale transformation combined with any isometric transformations and t(x) a similar
copy of x.
Clearly, the predicate can be as elaborate as desired. The quest for provision of
tools to facilitate the construction and instantiation of such spatial relations is an
ongoing project in the design of software packages geared for visual composition. A
comprehensive treatment of such rules and the taxonomies they will produce as
pictorial illustrations of schemata and their products and sums is a welcome project
for design inquiry. The goal here was to suggest such an inquiry and illustrate some
initial first steps towards this direction.
From Shape Rules to Rule Schemata and Back
397
Fig. 13 Four shape rules
defined with the rule
schema x ! x þ tððxÞ that
satisfy the predicate x is a
4-gon and t a similarity
transformation
Discussion
Shape rules and rules schemata are useful to work with because of the compositional relations they foreground and the ways they facilitate distinct views of design
inquiry. Both formal devices provide a rich repertory of means to support expressive and productive calculation in design respectively. And both provide powerful
insight when they are contrasted one against the other and suggest new ways of
interpretation. Intuitively, both types of formal devices are deployed in design to
solve particular kinds of problems in spatial composition: shape rules are mostly
deployed because of their visual specificity; rule schemata because of their conceptual generality. And still both shape rules and rule schemata are just alternative
ways to describe the very same thing from a different vantage point.
An exciting aspect of the exercise of looking at existing shape rules and
attempting to infer rule schemata that these shape rules could be defined in, is
that such redescriptions of an existing rule, or set of rules, as predicates and
assignments in rule schemata, provide novel descriptions of the given corpus of
shape grammars. They also suggest interpretations of the existing sets of grammatical rules that may be potentially diverse and distinct from those envisioned from
the authors of the grammars. In this sense this act of redescription of the pictorial
rules of shape grammars as assignments in different schemata facilitates their novel
re-appropriation and re-usage in alternative contexts. A possible corollary of this
conclusion is that this shift in representation allows for the rules and the grammars
to emerge above specific domains such as residential architecture, public
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A. Economou and S. Kotsopoulos
architecture, ecclesiastical architecture, landscape architecture, ornamental design,
furniture design, product design, automobile design and any subcategories within
these fields. Instead this account focuses on the compositional schemata that can
discursively explain what the shape rules do, and the problems they address.
An exciting aspect of the exercise of looking at existing rule schemata and
attempting to define shape rules within them, is that such illustrations of the
schemata in terms of shape rule instances, provide concrete descriptions of the
given corpus of schemata. They may also suggest interpretations of these schemata
that are potentially diverse, and even non-intuitive, with respect to the schemata. In
this sense this act of redescription of the symbolic schemata as pictorial assignments facilitates their novel re-appropriation and re-usage in alternative contexts.
The major goal in this work has been to look at the pair of the shape rules and the
rules schemata from either side foregrounding each in the relation. This back-andforth between show and tell is what this is all about. In fact it is suggested here that
such pictorial redescriptions of shape rules as assignments in rule schemata, and
instantiations of schemata in visual symbols is the heart of design inquiry and that it
should underlie any computational framework for design.
A significant motivation underlying this work has been the systematic inquiry on
both aspects of rules so that they can both be implemented in shape rules and/or in
assignments in rule schemata and be freely instantiated, edited, used and tested in
an automated computer setting. The technical problems associated with the design
of a framework to implement shape recognition and shape rules are formidable. A
recent model based on an underlying graph theoretical representation of shape has
successfully managed to address a great deal of these problems [18, 19]. The next
step is the seamless implementation of shape rules in terms of rule schemata in an
interactive framework that allows for the free definition of any shape rule none so
ever and the immediate testing of its expressive power in the design at hand.
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