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- Rachel Fletcher is the author of Infinite Measure: Learning to Design in Geometric Harmony with Art, Architecture, an... moreRachel Fletcher is the author of Infinite Measure: Learning to Design in Geometric Harmony with Art, Architecture, and Nature. She taught theater design at Tufts University in the late 1970s and early 1980s, has been a faculty member of the New York School of Interior Design since 1996, and for ten years was a contributing editor to the Nexus Network Journal. Her professional work designing theatrical spaces led to an interest in the principles of geometric proportion and harmony as a design system, including time as a geometer and teacher of geometry and proportion at dozens of universities, museums, and institutions in the United States and Europe, as well as for school-age children and adult professionals. In this capacity, she received an International Center for Jefferson Studies Fellowship Award from the Thomas Jefferson Foundation to study geometric proportions in Jefferson’s architectural works.
Fletcher was the creator/curator of the museum exhibits “Infinite Measure,” “Design by Nature,” and “Harmony by Design: The Golden Mean” and the author of the latter’s exhibit catalog.
Her website is www.infinitemeasure.comedit
A creative workbook and an authoritative reference guide for teachers, students, and practitioners of design, including architecture, interior design, landscape architecture, painting, sculpture, the graphic arts, theater and stage... more
A creative workbook and an authoritative reference guide for teachers, students, and practitioners of design, including architecture, interior design, landscape architecture, painting, sculpture, the graphic arts, theater and stage design, and even musical instruments and crafts. Taking pages from books of nature, art, and architecture, Fletcher provides visual designers of all art forms and disciplines with geometric methods for composing harmonious spaces and places.
In Infinite Measure, Fletcher shares her professional knowledge and experience of geometry and proportion by offering practical techniques for design applications, including step-by-step elementary and advanced drawings for producing proportional schemes with a compass and rule; commentaries on geometric symbols and useful theorems; definitions; and etymologies of essential mathematical terms. A highlight of the book are Fletcher’s original studies that analyze harmonious proportions in world-famous art, architecture, landscape design, and other compositions. These include the South Rose Window at Cathédrale Notre Dame de Paris, Andrea Palladio’s Villa Emo and Teatro Olimpico, a Stradivari violin, Thomas Jefferson’s Poplar Forest, Beatrix Farrand’s garden courtyard for the Oriental Institute at the University of Chicago, the illuminated Lindisfarne Gospels, a Louis Sullivan stencil for the Chicago Stock Exchange, and Eero Saarinen’s North Christian Church.
In Infinite Measure, Fletcher shares her professional knowledge and experience of geometry and proportion by offering practical techniques for design applications, including step-by-step elementary and advanced drawings for producing proportional schemes with a compass and rule; commentaries on geometric symbols and useful theorems; definitions; and etymologies of essential mathematical terms. A highlight of the book are Fletcher’s original studies that analyze harmonious proportions in world-famous art, architecture, landscape design, and other compositions. These include the South Rose Window at Cathédrale Notre Dame de Paris, Andrea Palladio’s Villa Emo and Teatro Olimpico, a Stradivari violin, Thomas Jefferson’s Poplar Forest, Beatrix Farrand’s garden courtyard for the Oriental Institute at the University of Chicago, the illuminated Lindisfarne Gospels, a Louis Sullivan stencil for the Chicago Stock Exchange, and Eero Saarinen’s North Christian Church.
Research Interests:
An indispensable teaching tool for all design disciplines: industrial design; graphic design; interior design; architecture; landscape architecture; and textile design. The information includes five methods of drawing the golden mean with... more
An indispensable teaching tool for all design disciplines: industrial design; graphic design; interior design; architecture; landscape architecture; and textile design.
The information includes five methods of drawing the golden mean with a compass and rule.
Each method is illustrated with representational drawings showing the implementation of the golden section in architecture from 2800 B.C. with Stonehenge to the present day.
The information includes five methods of drawing the golden mean with a compass and rule.
Each method is illustrated with representational drawings showing the implementation of the golden section in architecture from 2800 B.C. with Stonehenge to the present day.
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Palladio created two distinct versions of Villa Emo at Fanzolo, the plan published in I quattro libri and the constructed villa that survives to this day. In both, harmony appears mathematically in the dimensions. The published plan is... more
Palladio created two distinct versions of Villa Emo at Fanzolo, the plan published in I quattro libri and the constructed villa that survives to this day. In both, harmony appears mathematically in the dimensions. The published plan is measured in Vicentine feet and is comprised of whole number measures that have been identified as among the terms of musical harmonies. Meanwhile, a geometric analysis based on a 1972 survey of the constructed Villa Emo reveals a plan of the central block that is not perfectly square, but proportioned to a circle that inscribes two smaller squares. Subsequent room dimensions and passageways follow in golden mean progression. The placement of doors and fireplaces derives from a regular pentagon whose base is drawn on the front edge of the portico. While proportions derived from other geometric shapes are also present, the golden mean also appears to dominate the front elevation.
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The zodiac is widely known as a band of twelve celestial constellations. It also contains a mathematical model for cosmographic depiction, based on observations of the sun, moon, and visible planets as they traverse the celestial... more
The zodiac is widely known as a band of twelve celestial constellations. It also contains a mathematical model for cosmographic depiction, based on observations of the sun, moon, and visible planets as they traverse the celestial ecliptic. Here wee consider the zodiac as a timepiece or calendar; how this system of planets and constellations emerges from elementary geometric patterns; and how these patterns inform the symbols of the zodiac and frame our world view.
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Eero Saarinen’s North Christian Church, an important contribution to post-war liturgical church architecture, serves a community of Disciples of Christ in Columbus, Indiana. Early design sketches illustrate elementary geometric shapes and... more
Eero Saarinen’s North Christian Church, an important contribution to post-war liturgical church architecture, serves a community of Disciples of Christ in Columbus, Indiana. Early design sketches illustrate elementary geometric shapes and symbols – triangle, square, cross, hexagon, and octagon – whose proportions appear in the plan and section of the completed structure.
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“Dynamic symmetry” is the name given by Jay Hambidge for the proportioning principle that appears in “root rectangles” where a single incommensurable ratio persists through endless spatial divisions. In Part One of a continuing series... more
“Dynamic symmetry” is the name given by Jay Hambidge for the proportioning principle that appears in “root rectangles” where a single incommensurable ratio persists through endless spatial divisions. In Part One of a continuing series [Fletcher 2007], we explored the relative characteristics of root-two, -three, -four, and - five systems of proportion and became familiar with diagonals, reciprocals, complementary areas,
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It is impossible to construct circles and squares of equal areas or perimeters precisely, for circles are measured by the incommensurable value pi (S) and squares by rational whole numbers. But from early times, geometers have attempted... more
It is impossible to construct circles and squares of equal areas or perimeters precisely, for circles are measured by the incommensurable value pi (S) and squares by rational whole numbers. But from early times, geometers have attempted to reconcile these two orders of geometry. "Squaring the circle" can represent the union of opposing eternal and finite qualities, symbolizing the fusion of matter and spirit and the marriage of heaven and earth. In this column, we consider various methods for squaring the circle and related geometric constructions. I Introduction From the domed Pantheon of ancient Rome, if not before, architects have fashioned sacred dwellings after conceptions of the universe, utilizing circle and square geometries to depict spirit and matter united. Circular domes evoke the spherical cosmos and the descent of heavenly spirit to the material plane. Squares and cubes delineate the spatial directions of our physical world and portray the lifting up of mater...
Research Interests: Mathematics and Heaven
The zodiac is widely known as a band of twelve celestial constellations. It also contains a mathematical model for cosmographic depiction, based on observations of the sun, moon, and visible planets as they traverse the celestial... more
The zodiac is widely known as a band of twelve celestial constellations. It also contains a mathematical model for cosmographic depiction, based on observations of the sun, moon, and visible planets as they traverse the celestial ecliptic. Here wee consider the zodiac as a timepiece or calendar; how this system of planets and constellations emerges from elementary geometric patterns; and how these patterns inform the symbols of the zodiac and frame our world view.
Research Interests:
Thomas Jefferson dedicated his later years to establishing the University of Virginia, believing that the availability of a public liberal education was essential to national prosperity and individual happiness. His design for the... more
Thomas Jefferson dedicated his later years to establishing the
University of Virginia, believing that the availability of a public
liberal education was essential to national prosperity and individual
happiness. His design for the University stands as one of his greatest
accomplishments and has been called “the proudest achievement of
American architecture.” Taking Jefferson’s design drawings as a basis
for study, this paper explores the possibility that he incorporated
incommensurable geometric proportions in his designs for the
Rotunda. Without actual drawings to illustrate specific geometric
constructions, it cannot be said definitively that Jefferson utilized
such proportions. But a comparative analysis between Jefferson’s
plans and Palladio’s renderings of the Pantheon (Jefferson’s primary
design source) suggests that both designs developed from similar
geometric techniques.
University of Virginia, believing that the availability of a public
liberal education was essential to national prosperity and individual
happiness. His design for the University stands as one of his greatest
accomplishments and has been called “the proudest achievement of
American architecture.” Taking Jefferson’s design drawings as a basis
for study, this paper explores the possibility that he incorporated
incommensurable geometric proportions in his designs for the
Rotunda. Without actual drawings to illustrate specific geometric
constructions, it cannot be said definitively that Jefferson utilized
such proportions. But a comparative analysis between Jefferson’s
plans and Palladio’s renderings of the Pantheon (Jefferson’s primary
design source) suggests that both designs developed from similar
geometric techniques.
Research Interests:
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The Pantheon in Rome has been depicted in countless paintings and measured drawings. This paper considers how the building and its subsequent representations express meaning through elementary geometric symbols, patterns, and proportions.... more
The Pantheon in Rome has been depicted in countless paintings and measured drawings. This paper considers how the building and its subsequent representations express meaning through elementary geometric symbols, patterns, and proportions. The author observes notable discrepancies between a sampling of measured plans from Sebastiano Serlio’s woodcut engravings in the Renaissance to current laser campaigns. She analyzes the different drawings for underlying geometric patterns. A pattern of rotated squares, in root-two proportion, appears consistently in the horizontal plan of each measured set and complements Mark Wilson Jones’s proposed scheme of a conjoined sphere and cube. This comparative method of analysis offers students and scholars of descriptive geometry a useful tool for interpretation.
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Thomas Jefferson dedicated his later years to establishing the University of Virginia, believing that the availability of a public liberal education was essential to national prosperity and individual happiness. His design for the... more
Thomas Jefferson dedicated his later years to establishing the University of Virginia, believing that the availability of a public liberal education was essential to national prosperity and individual happiness. His design for the University stands as one of his greatest accomplishments and has been called “the proudest achievement of American architecture.” Taking Jefferson’s design drawings as a basis for study, this paper explores the possibility that he incorporated incommensurable geometric proportions in his designs for the Rotunda. Without actual drawings to illustrate specific geometric constructions, it cannot be said definitively that Jefferson utilized such proportions. But a comparative analysis between Jefferson’s plans and Palladio’s renderings of the Pantheon (Jefferson’s primary design source) suggests that both designs developed from similar geometric techniques.
Research Interests:
A unique geometric construction known to Thomas Jefferson reveals a rich interplay of root-two geometric elements when applied to Jefferson’s octagonal plan of Poplar Forest, his eighteenth-century villa retreat.
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Palladio created two distinct versions of Villa Emo at Fanzolo, the plan published in I quattro libri and the constructed villa that survives to this day. In both, harmony appears mathematically in the dimensions. The published plan is... more
Palladio created two distinct versions of Villa Emo at Fanzolo, the plan published in I quattro libri and the constructed villa that survives to this day. In both, harmony appears mathematically in the dimensions. The published plan is measured in Vicentine feet and is comprised of whole number measures that have been identified as among the terms of musical harmonies. Meanwhile, a geometric analysis based on a 1972 survey of the constructed Villa Emo reveals a plan of the central block that is not perfectly square, but proportioned to a circle that inscribes two smaller squares. Subsequent room dimensions and passageways follow in golden mean progression. The placement of doors and fireplaces derives from a regular pentagon whose base is drawn on the front edge of the portico. While proportions derived from other geometric shapes are also present, the golden mean also appears to dominate the front elevation.
Research Interests:
At Nexus 2000, Rachel Fletcher argued that Palladio may well have made use of the ‘golden section’, or extreme and mean ratio, in the design of the Villa Emo at Fanzolo. In this issue of Nexus Network Journal, Lionel March argued that the... more
At Nexus 2000, Rachel Fletcher argued that Palladio may well have made use of the ‘golden section’, or extreme and mean ratio, in the design of the Villa Emo at Fanzolo. In this issue of Nexus Network Journal, Lionel March argued that the Golden Section is nowhere to be found in the Villa Emo as described in I quattro libri dell’archittetura. In the present paper, Rachel Fletcher defends her original thesis, comparing the Villa Emo as actually built to the project for it that Palladio published in his book.
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Eero Saarinen’s North Christian Church, an important contribution to post-war liturgical church architecture, serves a community of Disciples of Christ in Columbus, Indiana. Early design sketches illustrate elementary geometric shapes and... more
Eero Saarinen’s North Christian Church, an important contribution to post-war liturgical church architecture, serves a community of Disciples of Christ in Columbus, Indiana. Early design sketches illustrate elementary geometric shapes and symbols – triangle, square, cross, hexagon, and octagon – whose proportions appear in the plan and section of the completed structure.
Research Interests:
Geometer Rachel Fletcher explains the geometry, symbolism, and applications of the vesica piscis
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It is impossible to construct circles and squares of equal areas or perimeters precisely, for circles are measured by the incommensurable value pi (π) and squares by rational whole numbers. But from early times, geometers have attempted... more
It is impossible to construct circles and squares of equal areas or perimeters precisely, for circles are measured by the incommensurable value pi (π) and squares by rational whole numbers. But from early times, geometers have attempted to reconcile these two orders of geometry. “Squaring the circle” can represent the union of opposing eternal and finite qualities, symbolizing the fusion of matter and spirit and the marriage of heaven and earth. In this column, we consider various methods for squaring the circle and related geometric constructions.
Research Interests:
The zodiac is widely known as a band of twelve celestial constellations. It also contains a mathematical model for cosmographic depiction, based on observations of the sun, moon, and visible planets as they traverse the celestial... more
The zodiac is widely known as a band of twelve celestial constellations. It also contains a mathematical model for cosmographic depiction, based on observations of the sun, moon, and visible planets as they traverse the celestial ecliptic. Here wee consider the zodiac as a timepiece or calendar; how this system of planets and constellations emerges from elementary geometric patterns; and how these patterns inform the symbols of the zodiac and frame our world view.
Research Interests:
Incommensurable ratios cannot be stated in finite, whole number fractions. But such ratios can organize spatial compositions so that the same ratio persists through endless divisions. We explore this proportioning principle, which Jay... more
Incommensurable ratios cannot be stated in finite, whole number fractions. But such ratios can organize spatial compositions so that the same ratio persists through endless divisions. We explore this proportioning principle, which Jay Hambidge calls “dynamic symmetry,” as it appears in “root rectangles” of incommensurable proportions.
Research Interests:
“Dynamic symmetry” is the name given by Jay Hambidge for the proportioning principle that appears in “root rectangles” where a single incommensurable ratio persists through endless spatial divisions. In Part One of a continuing series... more
“Dynamic symmetry” is the name given by Jay Hambidge for the proportioning principle that appears in “root rectangles” where a single incommensurable ratio persists through endless spatial divisions. In Part One of a continuing series [Fletcher 2007], we explored the relative characteristics of root-two, -three, -four, and -five systems of proportion and became familiar with diagonals, reciprocals, complementary areas, and
Research Interests:
. “Dynamic symmetry” is the name given by Jay Hambidge to describe a system of incommensurable ratios for proportioning areas within design compositions. In Parts One and Two of a continuing series, we surveyed the elements of root-two,... more
. “Dynamic symmetry” is the name given by Jay Hambidge to describe a system of incommensurable ratios for proportioning areas within design compositions. In Parts One and Two of a continuing series, we surveyed the elements of root-two, -three, -four, and -five rectangular systems and, using the root-two rectangle, explored diagonals, reciprocals, complementary areas, and other techniques for composing dynamic space
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. In Hebrew, Christian and Islamic revelation, the world is created in six days of physical activity, followed by a single day of stillness and rest. Geometry provides a fitting metaphor, for the radii of six equal circles mark out the... more
. In Hebrew, Christian and Islamic revelation, the world is created in six days of physical activity, followed by a single day of stillness and rest. Geometry provides a fitting metaphor, for the radii of six equal circles mark out the circumference of an identical circle placed in the center
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Geometer Rachel Fletcher explores the 1:2–√1:2 ratio associated with the regular quadrilateral figure known as the square, looking at the square’s inherent symbolism and the four-ness of the cross and the tetractys, as she constructs... more
Geometer Rachel Fletcher explores the 1:2–√1:2 ratio associated with the regular quadrilateral figure known as the square, looking at the square’s inherent symbolism and the four-ness of the cross and the tetractys, as she constructs ad quadratum and other geometric techniques.
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To Renaissance mathematician Luca Pacioli, it was the Divine Proportion. To German astronomer Johannes Kepler, it was a precious jewel. The only proportion to increase simultaneously by geometric progression and by simple addition, the... more
To Renaissance mathematician Luca Pacioli, it was the Divine Proportion. To German astronomer Johannes Kepler, it was a precious jewel. The only proportion to increase simultaneously by geometric progression and by simple addition, the Golden Section achieves unity among diverse elements in remarkably efficient ways. We explore the Golden Ratio 1: ϕ , also known as the Golden Mean, and its appearance in the regular pentagon and other geometric constructions.