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CSE325 Computer Science and Sculpture

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CSE325 Computer Science and Sculpture Prof. George Hart 2. Polyhedra in Art + Sculpture Polyhedra From Greek: poly=many + hedra=seats Singular: Polyhedron Def: 3D ... – PowerPoint PPT presentation

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Title: CSE325 Computer Science and Sculpture


1
CSE325 Computer Science and Sculpture
  • Prof. George Hart

2
2. Polyhedra in Art Sculpture
A Historical View
3
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

4
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-

5
Examples Regular Polygons
6
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
7
Some Dodecahedra
12 isosceles triangles
12 rhombi
Regular 12 pentagons
rhombic dodechedron
12 isosceles triangles
12 kites
12 irregular pentagons
8
Some Non-convex Dodecahedra
A torus is not convex
small stellated dodecahedron (12 pentagrams)
concave dodecahedron
9
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?
10
Prehistoric Scotland
Carved stone from circa 2000 B.C.E. Hundreds
known. Most are cube-based. I dont know of any
icosahedron-based examples.
11
Roman Dice
ivory
stone
12
Roman Dodecahedra
Bronze, unknown function
13
Roman Icosahedron
14
Paolo Uccello (1397-1475)
Small stellated dodecahedron mosaic
mazzocchio (donut hat)
15
Piero della Francesca (1410? - 1492)
Truncated tetrahedron
Icosahedron in cube
16
Leonardo da Vinci (1452-1519)
Illustrations for Luca Pacioli's 1509 book The
Divine Proportion
17
Leonardo da Vinci
Illustrations for Luca Pacioli's 1509 book The
Divine Proportion
18
Compare Solid Edges to Lines
19
Leonardo da Vinci
Elevated Forms
20
Leonardo Doodles
21
Leonardo Doodles
22
Leonardo
Cube structure
23
Leonardos Ludo Geometrico
ludo geometrico geometry game
make systematic modifications
24
Leonardo
Torus variations
25
Luca Pacioli (1445-1514)
Portrait of Pacioli, by Jacopo de Barbari, 1495
26
Luca Pacioli
27
The Divine Proportion
Golden ratio
28
Luca Pacioli
29
Pacioli Leonardo
Printed as woodcuts in 1509
30
Fra Giovanni da Verona, 1520s
31
Intarsia by Giovanni da Verona
32
Albrecht Durer (1471-1528)
Melancholia I, 1514
33
Albrecht Durer
Painters Manual, 1525
Net of snub cube
34
Albrecht Durer
Find the error!
Painters Manual, 1525
35
Daniele Barbaro (1513-1570)
La Practica della Perspectiva, 1568
36
Wentzel Jamnitzer (1508-1585)
Perspectiva Corporum Regularium, 1568
37
Wentzel Jamnitzer
38
Wentzel Jamnitzer
39
Wentzel Jamnitzer
(oldest chiral icosahedral image)
40
Johannes Kepler (1571-1630)
(detail of inner planets)
41
Johannes Kepler
Harmonice Mundi, 1619
42
Kepler Archimedean Solids
Faces regular, vertices identical, but faces need
not be identical
43
Johannes Kepler
Regular Dodecahedron
Rhombic Dodecahedron
44
Johannes Kepler
  • Symbolism from Plato
  • Octahedron air
  • Tetrahedron fire
  • Cube earth
  • Icosahedron water
  • Dodecahedron
  • the universe

45
Augustin Hirschvogel (1503-1553)
46
Lorenz Stoer
Geometria et Perspectiva, 1567
47
Lorenz Stoer
Geometria et Perspectiva, 1567
48
Jean Cousin
Livre de Perspective, 1560
49
Nicolas Neufchatel
Portrait of Johann Neudorfer and his Son, 1561
50
Hans Lencker
Perspectiva, 1571
51
Hans Lencker
Perspectiva, 1571
52
Lorenzo Sirigatti
La pratica di prospettiva, 1596
53
Paul Pfinzing
Optica, 1616
54
Jean-Francois Niceron
Thaumaturgus Opticus, 1638
55
Jean Dubreuil
La Perspective Pratiq, 1642
56
Jean Dubreuil
La Perspective Pratiq, 1642
57
Tomb of Sir Thomas Gorges
Salisbury Cathedral, 1635
58
Lorenz Zick (1594-1666) Turned Ivory Spheres
Modern Asian example
59
Jacques Ozanam
Geometrie pratique, 1684
60
Alain Manesson Mallet
La Geometrie Pratique, 1702
61
Abraham Sharp (1651-1742)
Geometry Improv'd, 1718
62
Brook Taylor
New Principles of Linear Perspective, 1719
63
Brook Taylor
New Principles of Linear Perspective, 1719
64
Paul Heinecken
Lucidum Prospectivae Speculum 1727
65
Thomas Malton
Compleat Treatise on Perspective, 1779
66
Christoph Nilson
Anleitung zur Linearperspective, c. 1800
67
Max Brückner Vielecke und Vielflache, 1900
68
M.C. Escher (1898-1972)
Stars, 1948
69
M.C. Escher
Double Planetoid, 1949
70
M.C. Escher
Waterfall, 1961
71
M.C. Escher
Reptiles, 1943
72
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
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