1. The Other Polyhedra Steven Janke Colorado College

2. Five Regular Polyhedra DodecahedronIcosahedron Tetrahedron Octahedron Cube

3. Prehistoric Scotland Carved stones from about 2000 B.C.E.

4. Roman Dice ivory stone

5. Roman Polyhedra Bronze, unknown function

6. Radiolaria drawn by Ernst Haeckel (1904)

7. Theorem: Let P be a convex polyhedron whose faces are congruent regular polygons. Then the following are equivalent: 1.The vertices of P all lie on a sphere. 2.All the dihedral angles of P are equal. 3.All the vertex figures are regular polygons. 4.All the solid angles are congruent. 5.All the vertices are surrounded by the same number of faces.

8. Plato’s Symbolism (Kepler’s sketches) Octahedron = Air Tetrahedron = Fire Cube = Earth Icosahedron = Water Dodecahedron = Universe

9. Theorem: There are only five convex regular polyhedra. (Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron) Proof: In a regular polygon of p sides, the angles are (1-2/p)π. With q faces at each vertex, the total of these angles must Be less than 2π: q(1-2/p)π < 2π 1/p + 1/q > 1/2 Only solutions are: (3,3) (3,4) (4,3) (3,5) (5,3)

10. Johannes Kepler (1571-1630) (detail of inner planets)

11. Golden Ratio in a Pentagon

12. Three golden rectangles inscribed in an icosahedron

13. Euler’s Formula: V + F = E + 2 Tetrahedron 4 4 6 Cube 8 6 12 Octahedron 6 8 12 Dodecahedron 20 12 30 Icosahedron 12 20 30 Vertices Faces Edges Duality: Vertices Faces

14. Regular Polyhedra Coordinates: Cube: (±1, ±1, ±1) Tetrahedron: (1, 1, 1) (1, -1, -1) (-1, 1, -1) (-1, -1, 1) Octahedron: (±1, 0, 0) (0, ±1, 0) (0, 0, ±1) Iscosahedron: (0, ±φ, ±1) (±1, 0, ±φ) (±φ, ±1, 0) Dodecahedron: (0, ±φ -1, ±φ) (±φ, 0, ±φ -1 ) (±φ -1, ±φ, 0) (±1, ±1, ±1) Where φ 2 - φ - 1 = 0 giving φ = 1.618 … (Golden Ratio)

15. Portrait of Luca Pacioli (1445-1514) (by Jacopo de Barbari (?) 1495)

16. Basilica of San Marco (Venice) (Floor Pattern in Marble) Possibly designed by Paolo Uccello in 1430

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

18. Church of Santa Maria in Organo, Verona (Fra Giovanni da Verona 1520’s)

19. Leonardo da Vinci (1452-1519) Illustrations for Luca Pacioli's 1509 book The Divine Proportion

20. Leonardo da Vinci “Elevated” Forms

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

22. Wentzel Jamnitzer (1508-1585) Perspectiva Corporum Regularium, 1568

23. Wentzel Jamnitzer

24. Theorem: The only finite rotation groups are: Cyclic Dihedral Tetrahedral (alternating group of degree 4) Octahedral Icosahedral (alternating group of degree 5)

25. Lorenz Stoer Geometria et Perspectiva, 1567

26. Lorenz Stoer Geometria et Perspectiva, 1567

27. Jean Cousin Livre de Perspective, 1560

28. Jean-Francois Niceron Thaumaturgus Opticus, 1638

29. Tomb of Sir Thomas Gorges Salisbury Cathedral, 1635

30. M.C. Escher (1898-1972) Stars, 1948

31. M.C. Escher Waterfall, 1961

32. M.C. Escher Reptiles, 1943

33. Order and Chaos M.C. Escher

34. Regular Polygon with 5 sides

35. Johannes Kepler Harmonice Mundi, 1619

36. Theorem: There are only four regular star polyhedra. Small Stellated Dodecahedron (5/2, 5) Great Dodecahedron (5, 5/2) Great Stellated Dodecahedron (5/2, 3) Great Icosahedron (3, 5/2)

37. Kepler: Archimedean Solids Faces regular, vertices identical, but faces need not be identical

38. Lemma: Only three different kinds of faces can occur at each vertex of a convex polyhedra with regular faces. Theorem: The set of convex polyhedra with regular faces and congruent vertices contains only the 13 Archimedean polyhedra plus two infinite families: the prisms and antiprisms.

39. Max Brückner Vielecke und Vielflache, 1900

40. Historical Milestones 1.Theatetus (415 – 369 B.C.): Octahedron and Icosahedron. 2.Plato (427 – 347 B.C.): Timaeus dialog (five regular polyhedra). 3.Euclid (323-285 B.C.): Constructs five regular polyhedra in Book XIII. 4.Archimedes (287-212 B.C.): Lost treatise on 13 semi-regular solids. 5.Kepler (1571- 1630): Proves only 13 Archimedean solids. 6.Euler (1707-1783): V+F=E+2 7.Poinsot (1777-1859): Four regular star polyhedra. Cauchy proved. 8.Coxeter (1907 – 2003): Regular Polytopes. 9.Johnson, Grunbaum, Zalgaller (1969): Prove 92 polyhedra with regular faces. 10.Skilling (1975): Proves there are 75 uniform polyhedra.

41. Retrosnub Ditrigonal Icosidodecahedron (a.k.a. Yog Sothoth) (Vertices: 60; Edges:180; Faces: 100 triangles + 12 pentagrams

42. References: Coxeter, H.S.M. – Regular Polytopes 1963 Cromwell, Peter – Polyhedra 1997 Senechal, Marjorie, et. al. – Shaping Space 1988 Wenninger, Magnus – Polyhedron Models 1971 Cundy,H. and Rollett, A. – Mathematical Models 1961

43. Polyhedra inscribed in other Polyhedra