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The Other Polyhedra

The Other Polyhedra. Steven Janke Colorado College. Five Regular Polyhedra. Dodecahedron. Icosahedron. Tetrahedron Octahedron Cube. Prehistoric Scotland. Carved stones from about 2000 B.C.E. Roman Dice. ivory. stone. Roman Polyhedra. Bronze, unknown function.

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The Other Polyhedra

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  1. The Other Polyhedra Steven Janke Colorado College

  2. Five Regular Polyhedra Dodecahedron Icosahedron Tetrahedron Octahedron Cube

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

  4. Roman Dice ivory stone

  5. RomanPolyhedra 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: • The vertices of P all lie on a sphere. • All the dihedral angles of P are equal. • All the vertex figures are regular polygons. • All the solid angles are congruent. • 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 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 φ2 - 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 (1471-1528) Melancholia I, 1514

  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 BrücknerVielecke und Vielflache, 1900

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

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