720 likes | 936 Views
CSE325 Computer Science and Sculpture. Prof. George Hart. 2. Polyhedra in Art + Sculpture. A Historical View. Polyhedra. From Greek: poly =many + hedra =seats Singular: Polyhedron Def: 3D object bounded by flat surfaces Many types: Platonic solids Archimedean solids
E N D
CSE325 Computer Science and Sculpture Prof. George Hart
2. Polyhedra in Art + Sculpture A Historical View
Polyhedra • 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
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- … In Two Dimensions: Polygons
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) 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 “small stellated dodecahedron” (12 pentagrams) A torus is not convex concave dodecahedron
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?
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
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
Leonardo da Vinci “Elevated” Forms
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
The Divine Proportion “Golden ratio”
Pacioli + Leonardo Printed as woodcuts in 1509
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 (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
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