CSE325 Computer Science and Sculpture

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CSE325 Computer Science and Sculpture

• Prof. George Hart

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2. Polyhedra in Art Sculpture
A Historical View
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Polyhedra

• From Greek polymany hedraseats
• Singular Polyhedron
• Def 3D object bounded by flat surfaces
• Many types
• Platonic solids
• Archimedean solids
• Convex / concave
• Long history of use in 3D design

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In Two Dimensions Polygons

• Greek gon knee
• Regular polygon
• equal lengths
• equal angles
• Allow stars
• Terminology
• corner vertex
• plural vertices

• Number prefixes
• 3) tri-
• 4) tetra-
• 5) penta-
• 6) hexa-
• 7) hepta-
• 8) octa-
• 9) ennea-
• 10) deca-

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Examples Regular Polygons
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Five Regular Polyhedra

• Every face identical
• Every face regular
• Every vertex identical
• Only 5 are possible
• Euclid gives proof
• Platonic Solids
• Plato described them
• (known earlier)

Dodecahedron12 sides
Icosahedron20
tetrahedron octahedron cube
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Some Dodecahedra
12 isosceles triangles
12 rhombi
Regular 12 pentagons
rhombic dodechedron
12 isosceles triangles
12 kites
12 irregular pentagons
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Some Non-convex Dodecahedra
A torus is not convex
small stellated dodecahedron (12 pentagrams)
concave dodecahedron
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Historical Examples

• Stone, ivory, wood carving
• Bronze casting
• Drawing, woodcut, engraving, etc
• Painting
• Stone or wood tiling (mosaics intarsia)
• Wood, glass, or metal assembly

Guess How old is the oldest existing
dodecahedron?
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Prehistoric Scotland
Carved stone from circa 2000 B.C.E. Hundreds
known. Most are cube-based. I dont know of any
icosahedron-based examples.
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Roman Dice
ivory
stone
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Roman Dodecahedra
Bronze, unknown function
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Roman Icosahedron
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Paolo Uccello (1397-1475)
Small stellated dodecahedron mosaic
mazzocchio (donut hat)
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Piero della Francesca (1410? - 1492)
Truncated tetrahedron
Icosahedron in cube
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Leonardo da Vinci (1452-1519)
Illustrations for Luca Pacioli's 1509 book The
Divine Proportion
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Leonardo da Vinci
Illustrations for Luca Pacioli's 1509 book The
Divine Proportion
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Compare Solid Edges to Lines
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Leonardo da Vinci
Elevated Forms
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Leonardo Doodles
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Leonardo Doodles
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Leonardo
Cube structure
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Leonardos Ludo Geometrico
ludo geometrico geometry game
make systematic modifications
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Leonardo
Torus variations
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Luca Pacioli (1445-1514)
Portrait of Pacioli, by Jacopo de Barbari, 1495
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Luca Pacioli
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The Divine Proportion
Golden ratio
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Luca Pacioli
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Pacioli Leonardo
Printed as woodcuts in 1509
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Fra Giovanni da Verona, 1520s
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Intarsia by Giovanni da Verona
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Albrecht Durer (1471-1528)
Melancholia I, 1514
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Albrecht Durer
Painters Manual, 1525
Net of snub cube
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Albrecht Durer
Find the error!
Painters Manual, 1525
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Daniele Barbaro (1513-1570)
La Practica della Perspectiva, 1568
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Wentzel Jamnitzer (1508-1585)
Perspectiva Corporum Regularium, 1568
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Wentzel Jamnitzer
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Wentzel Jamnitzer
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Wentzel Jamnitzer
(oldest chiral icosahedral image)
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Johannes Kepler (1571-1630)
(detail of inner planets)
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Johannes Kepler
Harmonice Mundi, 1619
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Kepler Archimedean Solids
Faces regular, vertices identical, but faces need
not be identical
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Johannes Kepler
Regular Dodecahedron
Rhombic Dodecahedron
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Johannes Kepler

• Symbolism from Plato
• Octahedron air
• Tetrahedron fire
• Cube earth
• Icosahedron water
• Dodecahedron
• the universe

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Augustin Hirschvogel (1503-1553)
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Lorenz Stoer
Geometria et Perspectiva, 1567
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Lorenz Stoer
Geometria et Perspectiva, 1567
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Jean Cousin
Livre de Perspective, 1560
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Nicolas Neufchatel
Portrait of Johann Neudorfer and his Son, 1561
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Hans Lencker
Perspectiva, 1571
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Hans Lencker
Perspectiva, 1571
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Lorenzo Sirigatti
La pratica di prospettiva, 1596
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Paul Pfinzing
Optica, 1616
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Jean-Francois Niceron
Thaumaturgus Opticus, 1638
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Jean Dubreuil
La Perspective Pratiq, 1642
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Jean Dubreuil
La Perspective Pratiq, 1642
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Tomb of Sir Thomas Gorges
Salisbury Cathedral, 1635
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Lorenz Zick (1594-1666)Turned Ivory Spheres
Modern Asian example
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Jacques Ozanam
Geometrie pratique, 1684
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Alain Manesson Mallet
La Geometrie Pratique, 1702
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Abraham Sharp (1651-1742)
Geometry Improv'd, 1718
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Brook Taylor
New Principles of Linear Perspective, 1719
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Brook Taylor
New Principles of Linear Perspective, 1719
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Paul Heinecken
Lucidum Prospectivae Speculum 1727
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Thomas Malton
Compleat Treatise on Perspective, 1779
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Christoph Nilson
Anleitung zur Linearperspective, c. 1800
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Max BrücknerVielecke und Vielflache, 1900
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M.C. Escher (1898-1972)
Stars, 1948
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M.C. Escher
Double Planetoid, 1949
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M.C. Escher
Waterfall, 1961
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M.C. Escher
Reptiles, 1943
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Conclusions

• Polyhedra, especially the five Platonic solids,
  have been an element of Western art for
  centuries.
• Beauty of symmetry
• Challenging models to show mastery of perspective
• Symbolic meaning assigned by Plato
• Mathematical foundation for artistry
• Good starting point for computer constructions