1. CHS UCB BRIDGES, July 2002 3D Visualization Models of the Regular Polytopes in Four and Higher Dimensions. Carlo H. Séquin University of California, Berkeley

2. CHS UCB Goals of This Talk u Expand your thinking. u Teach you “hyper-seeing,” seeing things that one cannot ordinarily see, in particular: Four- and higher-dimensional objects.

3. CHS UCB What is the 4th Dimension ? Some people think: “it does not really exist,” “it’s just a philosophical notion,” “it is ‘TIME’,”... But, it is useful and quite real!

4. CHS UCB Higher-dimensional Spaces Mathematicians Have No Problem: u A point P(x, y, z) in this room is determined by: x = 2m, y = 5m, z = 1.5m; has 3 dimensions. u Positions in other data sets P = P(d1, d2, d3, d4,... dn). u Example #1: Telephone Numbers represent a 7- or 10-dimensional space. u Example #2: State Space: x, y, z, v x, v y, v z...

5. CHS UCB Seeing Mathematical Objects u Very big point u Large point u Small point u Tiny point u Mathematical point

6. CHS UCB Geometrical View of Dimensions u Read my hands … (inspired by Scott Kim, ca 1977).

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8. What Is a Regular Polytope u “Polytope” is the generalization of the terms “Polygon” (2D), “Polyhedron” (3D), … to arbitrary dimensions. u “Regular” means: All the vertices, edges, faces… are indistinguishable form each another. u Examples in 2D: Regular n-gons:

9. CHS UCB Regular Polytopes in 3D u The Platonic Solids: There are only 5. Why ? …

10. CHS UCB Why Only 5 Platonic Solids ? Lets try to build all possible ones: u from triangles: 3, 4, or 5 around a corner; u from squares: only 3 around a corner; u from pentagons: only 3 around a corner; u from hexagons:  floor tiling, does not close. u higher N-gons:  do not fit around vertex without undulations (forming saddles)  now the edges are no longer all alike!

11. CHS UCB Do All 5 Conceivable Objects Exist? I.e., do they all close around the back ? u Tetra  base of pyramid = equilateral triangle. u Octa  two 4-sided pyramids. u Cube  we all know it closes. u Icosahedron  antiprism + 2 pyramids (are vertices at the sides the same as on top ?) Another way: make it from a cube with six lines on the faces  split vertices symmetrically until all are separated evenly. u Dodecahedron  is the dual of the Icosahedron.

12. CHS UCB Constructing a (d+1)-D Polytope Angle-deficit = 90° creates a 3D cornercreates a 4D corner ? 2D 3D4D 3D Forcing closure:

13. CHS UCB “Seeing a Polytope” u I showed you the 3D Platonic Solids … But which ones have you actually seen ? u For some of them you have only seen projections. Did that bother you ?? u Good projections are almost as good as the real thing. Our visual input after all is only 2D. -- 3D viewing is a mental reconstruction in your brain, -- that is where the real "seeing" is going on ! u So you were able to see things that "didn't really exist" in physical 3-space, because you saw good enough “projections” into 2-space, yet you could still form a mental image ==> “Hyper-seeing.” u We will use this to see the 4D Polytopes.

14. CHS UCB Projections How do we make “projections” ? u Simplest approach: set the coordinate values of all unwanted dimensions to zero, e.g., drop z, retain x,y, and you get a parallel projection along the z-axis. i.e., a 2D shadow. u Alternatively, use a perspective projection: back features are smaller  depth queue. Can add other depth queues: width of beams, color, fuzziness, contrast (fog)...

15. CHS UCB Wire Frame Projections u Shadow of a solid object is mostly a blob. u Better to use wire frame, so we can also see what is going on on the back side.

16. CHS UCB Oblique Projections u Cavalier Projection 3D Cube  2D4D Cube  3D (  2D )

17. CHS UCB Projections : VERTEX / EDGE / FACE / CELL - First. u 3D Cube: Paralell proj. Persp. proj. u 4D Cube: Parallel proj. Persp. proj.

18. CHS UCB 3D Models Need Physical Edges Options: u Round dowels (balls and stick) u Profiled edges – edge flanges convey a sense of the attached face u Actual composition from flat tiles – with holes to make structure see-through.

19. CHS UCB Edge Treatments Leonardo DaVinci – George Hart

20. CHS UCB How Do We Find All 4D Polytopes? u Reasoning by analogy helps a lot: -- How did we find all the Platonic solids? u Use the Platonic solids as “tiles” and ask: l What can we build from tetrahedra? l From cubes? l From the other 3 Platonic solids?  Need to look at dihedral angles! Tetrahedron: 70.5°, Octahedron: 109.5°, Cube: 90°, Dodecahedron: 116.5°, Icosahedron: 138.2°.

21. CHS UCB All Regular Polytopes in 4D Using Tetrahedra (70.5°): 3 around an edge (211.5°)  (5 cells) Simplex 4 around an edge (282.0°)  (16 cells) Cross polytope 5 around an edge (352.5°)  (600 cells) Using Cubes (90°): 3 around an edge (270.0°)  (8 cells) Hypercube Using Octahedra (109.5°): 3 around an edge (328.5°)  (24 cells) Hyper-octahedron Using Dodecahedra (116.5°): 3 around an edge (349.5°)  (120 cells) Using Icosahedra (138.2°):  none: angle too large (414.6°).

22. CHS UCB 5-Cell or Simplex in 4D u 5 cells, 10 faces, 10 edges, 5 vertices. u (self-dual).

23. CHS UCB 4D Simplex u Using Polymorf TM Tiles Additional tiles made on our FDM machine.

24. CHS UCB 16-Cell or “Cross Polytope” in 4D u 16 cells, 32 faces, 24 edges, 8 vertices.

25. CHS UCB 4D Cross Polytope u Highlighting the eight tetrahedra from which it is composed.

26. CHS UCB 4D Cross Polytope

27. CHS UCB Hypercube or Tessaract in 4D u 8 cells, 24 faces, 32 edges, 16 vertices. u (Dual of 16-Cell).

28. CHS UCB 4D Hypercube u Using Polymorf TM Tiles made by Kiha Lee on FDM.

29. CHS UCB Corpus Hypercubus Salvador Dali “Unfolded” Hypercube

30. CHS UCB 24-Cell in 4D u 24 cells, 96 faces, 96 edges, 24 vertices. u (self-dual).

31. CHS UCB 24-Cell, showing 3-fold symmetry

32. CHS UCB 24-Cell “Fold-out” in 3D Andrew Weimholt

33. CHS UCB 120-Cell in 4D u 120 cells, 720 faces, 1200 edges, 600 vertices. Cell-first parallel projection, (shows less than half of the edges.)

34. CHS UCB 120 Cell u Hands-on workshop with George Hart

35. CHS UCB 120-Cell Thin face frames, Perspective projection. Séquin (1982)

36. CHS UCB 120-Cell u Cell-first, extreme perspective projection u Z-Corp. model

37. CHS UCB (smallest ?) 120-Cell Wax model, made on Sanders machine

38. CHS UCB Radial Projections of the 120-Cell u Onto a sphere, and onto a dodecahedron:

39. CHS UCB 120-Cell, “exploded” Russell Towle

40. CHS UCB 120-Cell Soap Bubble John Sullivan

41. CHS UCB 600-Cell, A Classical Rendering u Oss, 1901 Frontispiece of Coxeter’s 1948 book “Regular Polytopes,” and John Sullivan’s Paper “The Story of the 120-Cell.” u Total: 600 tetra-cells, 1200 faces, 720 edges, 120 vertices. u At each Vertex: 20 tetra-cells, 30 faces, 12 edges.

42. CHS UCB 600-Cell Cross-eye Stereo Picture by Tony Smith

43. CHS UCB 600-Cell in 4D u Dual of 120 cell. u 600 cells, 1200 faces, 720 edges, 120 vertices. u Cell-first parallel projection, shows less than half of the edges.

44. CHS UCB 600-Cell u David Richter

45. CHS UCB Slices through the 600-Cell At each Vertex: 20 tetra-cells, 30 faces, 12 edges. Gordon Kindlmann

46. CHS UCB 600-Cell u Cell-first, parallel projection, u Z-Corp. model

47. CHS UCB Model Fabrication Commercial Rapid Prototyping Machines: u Fused Deposition Modeling (Stratasys) u 3D-Color Printing (Z-corporation)

48. CHS UCB Fused Deposition Modeling

49. CHS UCB Zooming into the FDM Machine

50. CHS UCB SFF: 3D Printing -- Principle u Selectively deposit binder droplets onto a bed of powder to form locally solid parts. Powder SpreadingPrinting Build Feeder Powder Head

51. CHS UCB 3D Printing: Z Corporation

52. CHS UCB 3D Printing: Z Corporation Cleaning up in the de-powdering station

53. CHS UCB Beyond 4 Dimensions … u What happens in higher dimensions ? u How many regular polytopes are there in 5, 6, 7, … dimensions ?

54. CHS UCB Polytopes in Higher Dimensions u Use 4D tiles, look at “dihedral” angles between cells: 5-Cell: 75.5°, Tessaract: 90°, 16-Cell: 120°, 24-Cell: 120°, 120-Cell: 144°, 600-Cell: 164.5°. u Most 4D polytopes are too round … But we can use 3 or 4 5-Cells, and 3 Tessaracts. u There are three methods by which we can generate regular polytopes for 5D and all higher dimensions.

55. CHS UCB Hypercube Series u “Measure Polytope” Series (introduced in the pantomime) u Consecutive perpendicular sweeps: 1D 2D 3D 4D This series extents to arbitrary dimensions!

56. CHS UCB Simplex Series u Connect all the dots among n+1 equally spaced vertices: (Find next one above COG). 1D 2D 3D This series also goes on indefinitely! The issue is how to make “nice” projections.

57. CHS UCB Cross Polytope Series u Place vertices on all coordinate half-axes, a unit-distance away from origin. u Connect all vertex pairs that lie on different axes. 1D 2D 3D 4D A square frame for every pair of axes 6 square frames = 24 edges

58. CHS UCB 5D and Beyond The three polytopes that result from the l Simplex series, l Cross polytope series, l Measure polytope series,... is all there is in 5D and beyond! 2D 3D 4D 5D 6D 7D 8D 9D …  5 6 3 3 3 3 3 3 Luckily, we live in one of the interesting dimensions! Dim. # Duals !

59. CHS UCB “Dihedral Angles in Higher Dim.” u Consider the angle through which one cell has to be rotated to be brought on top of an adjoining neighbor cell. Space2D3D4D5D6D  Simplex Series 60°70.5°75.5°78.5°80.4°90° Cross Polytopes 90°109.5°120°126.9°131.8°180° Measure Polytopes 90°

60. CHS UCB Constructing 4D Regular Polytopes u Let's construct all 4D regular polytopes -- or rather, “good” projections of them. u What is a “good”projection ? l Maintain as much of the symmetry as possible; l Get a good feel for the structure of the polytope. u What are our options ? A parade of various projections 

61. CHS UCB Parade of Projections … 1. HYPERCUBES

62. CHS UCB Hypercube, Perspective Projections

63. CHS UCB Tiled Models of 4D Hypercube Cell-first - - - - - - - - - Vertex-first U.C. Berkeley, CS 285, Spring 2002,

64. CHS UCB 4D Hypercube Vertex-first Projection

65. CHS UCB Preferred Hypercube Projections u Use Cavalier Projections to maintain sense of parallel sweeps:

66. CHS UCB 6D Hypercube u Oblique Projection

67. CHS UCB 6D Zonohedron u Sweep symmetrically in 6 directions (in 3D)

68. CHS UCB Parade of Projections (cont.) 2. SIMPLICES

69. CHS UCB 3D Simplex Projections u Look for symmetrical projections from 3D to 2D, or … u How to put 4 vertices symmetrically in 2D and so that edges do not intersect. Similarly for 4D and higher…

70. CHS UCB 4D Simplex Projection: 5 Vertices u “Edge-first” parallel projection: V5 in center of tetrahedron V5

71. CHS UCB 5D Simplex: 6 Vertices u Two methods: Avoid central intersection: Offset edges from middle. Based on Tetrahedron (plus 2 vertices inside). Based on Octahedron

72. CHS UCB 5D Simplex with 3 Internal Tetras u With 3 internal tetrahedra; the 12 outer ones assumed to be transparent.

73. CHS UCB 6D Simplex: 7 Vertices (Method A) Start from 5D arrangement that avoids central edge intersection, Then add point in center:

74. CHS UCB 6D Simplex (Method A) = skewed octahedron with center vertex

75. CHS UCB 6D Simplex: 7 Vertices (Method B) u Skinny Tetrahedron plus three vertices around girth, (all vertices on same sphere):

76. CHS UCB 7D and 8D Simplices Use a warped cube to avoid intersecting diagonals

77. CHS UCB Parade of Projections (cont.) 3. CROSS POLYTOPES

78. CHS UCB 4D Cross Polytope Profiled edges, indicating attached faces.

79. CHS UCB 5D Cross Polytope u FDM --- SLIDE

80. CHS UCB 5D Cross Polytope with Symmetry Octahedron + Tetrahedron (10 vertices)

81. CHS UCB 6D Cross Polytope 12 vertices  icosahedral symmetry

82. CHS UCB 7D Cross Polytope 14 vertices  cube + octahedron

83. CHS UCB Conclusions -- Questions ? u Hopefully, I was able to make you see some of these fascinating objects in higher dimensions, and to make them appear somewhat less “alien.”

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