1. No category

CSE325
Computer Science
and Sculpture
Prof. George Hart
2. Polyhedra in Art + Sculpture
A Historical View
Polyhedra
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From Greek: poly=many + hedra=seats
Singular: Polyhedron
Def: 3D object bounded by flat surfaces
Many types:
– Platonic solids
– Archimedean solids
– Convex / concave
• Long history of use in 3D design
In Two Dimensions: Polygons
• Greek gon = “knee”
• Regular polygon:
equal lengths
equal angles
• Allow “stars”
• Terminology:
– corner = vertex
– plural: vertices
• Number prefixes:
3) tri4) tetra5) penta6) hexa7) hepta8) octa9) ennea10) deca…
Examples: Regular Polygons
Five “Regular” Polyhedra
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Every face identical
Every face regular
Every vertex identical
Only 5 are possible
– Euclid gives proof
• “Platonic Solids”
• Plato described them
– (known earlier)
Dodecahedron=12 sides
Icosahedron=20
tetrahedron octahedron cube
Some Dodecahedra
12 isosceles triangles
12 rhombi
Regular: 12 pentagons
“rhombic dodechedron”
12 isosceles triangles
12 kites
12 irregular pentagons
Some Non-convex Dodecahedra
concave dodecahedron
“small stellated
dodecahedron”
(12 pentagrams)
A torus is not convex
Historical Examples
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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?
Prehistoric Scotland
Carved stone from circa 2000 B.C.E.
Hundreds known.
Most are cube-based.
I don’t know of any icosahedron-based examples.
Roman Dice
ivory
stone
Roman Dodecahedra
Bronze, unknown function
Roman Icosahedron
Paolo Uccello (1397-1475)
Small stellated dodecahedron mosaic
mazzocchio (donut hat)
Piero della Francesca (1410? - 1492)
Truncated tetrahedron
Icosahedron in cube
Leonardo da Vinci (1452-1519)
Illustrations for Luca Pacioli's 1509 book The Divine Proportion
Leonardo da Vinci
Illustrations for Luca Pacioli's 1509 book The Divine Proportion
Compare “Solid Edges” to Lines
Leonardo da Vinci
“Elevated” Forms
Leonardo Doodles
Leonardo Doodles
Leonardo
Cube structure
Leonardo’s Ludo Geometrico
ludo geometrico = “geometry game”
= “make systematic modifications”
Leonardo
Torus variations
Luca Pacioli (1445-1514)
Portrait of Pacioli, by Jacopo de Barbari, 1495
Luca Pacioli
The Divine Proportion
“Golden ratio”
Luca Pacioli
Pacioli + Leonardo
Printed as woodcuts in 1509
Fra Giovanni da Verona, 1520’s
Intarsia by Giovanni da Verona
Albrecht Durer (1471-1528)
Melancholia I, 1514
Albrecht Durer
Painter’s Manual, 1525
Net of snub cube
Albrecht Durer
Find the error!
Painter’s Manual, 1525
Daniele Barbaro (1513-1570)
La Practica della
Perspectiva, 1568
Wentzel Jamnitzer (1508-1585)
Perspectiva Corporum Regularium, 1568
Wentzel Jamnitzer
Wentzel Jamnitzer
Wentzel Jamnitzer
(oldest chiral icosahedral image)
Johannes Kepler (1571-1630)
(detail of inner planets)
Johannes Kepler
Harmonice
Mundi, 1619
Kepler: Archimedean Solids
Faces regular, vertices identical, but faces need not be identical
Johannes Kepler
Regular Dodecahedron
Rhombic Dodecahedron
Johannes Kepler
Symbolism from Plato:
Octahedron = air
Tetrahedron = fire
Cube = earth
Icosahedron = water
Dodecahedron =
the universe
Augustin Hirschvogel (1503-1553)
Lorenz Stoer
Geometria et Perspectiva, 1567
Lorenz Stoer
Geometria et Perspectiva, 1567
Jean Cousin
Livre de Perspective, 1560
Nicolas Neufchatel
Portrait of Johann Neudorfer and his Son, 1561
Hans Lencker
Perspectiva, 1571
Hans Lencker
Perspectiva, 1571
Lorenzo Sirigatti
La pratica di prospettiva, 1596
Paul Pfinzing
Optica, 1616
Jean-Francois Niceron
Thaumaturgus Opticus, 1638
Jean Dubreuil
La Perspective Pratiq, 1642
Jean Dubreuil
La Perspective Pratiq, 1642
Tomb of Sir Thomas Gorges
Salisbury
Cathedral,
1635
Lorenz Zick
(1594-1666)
Turned Ivory Spheres
Modern Asian example
Jacques Ozanam
Geometrie pratique,
1684
Alain Manesson Mallet
La Geometrie
Pratique, 1702
Abraham Sharp (1651-1742)
Geometry Improv'd, 1718
Brook Taylor
New Principles of Linear Perspective, 1719
Brook Taylor
New Principles of Linear Perspective, 1719
Paul Heinecken
Lucidum Prospectivae
Speculum
1727
Thomas Malton
Compleat Treatise
on Perspective,
1779
Christoph Nilson
Anleitung zur
Linearperspective,
c. 1800
Max Brückner
Vielecke und Vielflache, 1900
M.C. Escher (1898-1972)
Stars, 1948
M.C. Escher
Double Planetoid, 1949
M.C. Escher
Waterfall, 1961
M.C. Escher
Reptiles, 1943
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