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CSE325 Computer Science and Sculpture

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

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CSE325 Computer Science and Sculpture

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  1. CSE325 Computer Science and Sculpture Prof. George Hart

  2. 2. Polyhedra in Art + Sculpture A Historical View

  3. 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

  4. 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

  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) Dodecahedron=12 sides Icosahedron=20 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 “small stellated dodecahedron” (12 pentagrams) A torus is not convex 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 don’t 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. Leonardo’s 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, 1520’s

  31. Intarsia by Giovanni da Verona

  32. Albrecht Durer (1471-1528) Melancholia I, 1514

  33. Albrecht Durer Painter’s Manual, 1525 Net of snub cube

  34. Albrecht Durer Find the error! Painter’s 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

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