1. University of California, Berkeley
Florida 1999
BRIDGES, July 2002
3D Visualization Models of the Regular Polytopes in Four and Higher Dimensions .
Carlo H. Séquin
University of California, Berkeley

2. Goals of This Talk Expand your thinking.
Teach you “hyper-seeing,” seeing things that one cannot ordinarily see, in particular: Four- and higher-dimensional objects.
NOT an original math research paper ! (facts have been known for >100 years) NOT a review paper on literature … (browse with “regular polyhedra” “120-Cell”)
Also: Use of Rapid Prototyping in math.

3. A Few Key References …
Ludwig Schläfli: “Theorie der vielfachen Kontinuität,” Schweizer Naturforschende Gesellschaft, 1901.
H. S. M. Coxeter: “Regular Polytopes,” Methuen, London, 1948.
John Sullivan: “Generating and rendering four-dimensional polytopes,” The Mathematica Journal, 1(3): pp76-85, 1991.
Thanks to George Hart for data on 120-Cell, 600-Cell, inspiration.

4. 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!

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

6. Seeing Mathematical Objects
Very big point
Large point
Small point
Tiny point
Mathematical point

7. Geometrical View of Dimensions
Read my hands … (inspired by Scott Kim, ca 1977).

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

10. Regular Polytopes in 3D
The Platonic Solids:
There are only 5. Why ? …

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

12. Do All 5 Conceivable Objects Exist?
I.e., do they all close around the back ?
Tetra  base of pyramid = equilateral triangle.
Octa  two 4-sided pyramids.
Cube  we all know it closes.
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.
Dodecahedron  is the dual of the Icosahedron.

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

14. “Seeing a Polytope”
I showed you the 3D Platonic Solids … But which ones have you actually seen ?
For some of them you have only seen projections. Did that bother you ??
Good projections are almost as good as the real thing. Our visual input after all is only 2D D viewing is a mental reconstruction in your brain, -- that is where the real "seeing" is going on !
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.”
We will use this to see the 4D Polytopes.

15. Projections How do we make “projections” ?
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.
Alternatively, use a perspective projection: back features are smaller  depth queue. Can add other depth queues: width of beams, color, fuzziness, contrast (fog) ...

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

17. Oblique Projections Cavalier Projection 3D Cube  2D
4D Cube  3D ( 2D )

18. Projections: VERTEX / EDGE / FACE / CELL - First.
3D Cube:
Paralell proj.
Persp. proj.
4D Cube:
Parallel proj.

19. 3D Models Need Physical Edges
Options:
Round dowels (balls and stick)
Profiled edges – edge flanges convey a sense of the attached face
Actual composition from flat tiles – with holes to make structure see-through.

20. Edge Treatments
Leonardo DaVinci – George Hart

21. How Do We Find All 4D Polytopes?
Reasoning by analogy helps a lot: -- How did we find all the Platonic solids?
Use the Platonic solids as “tiles” and ask:
What can we build from tetrahedra?
From cubes?
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°.

22. 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°).

23. 5-Cell or Simplex in 4D 5 cells, 10 faces, 10 edges, 5 vertices.
(self-dual).

24. 4D Simplex Additional tiles made on our FDM machine.
Using Polymorf TM Tiles

25. 16-Cell or “Cross Polytope” in 4D
16 cells, 32 faces, 24 edges, 8 vertices.

26. 4D Cross Polytope
Highlighting the eight tetrahedra from which it is composed.

27. 4D Cross Polytope

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

29. 4D Hypercube
Using PolymorfTM Tiles made by Kiha Lee on FDM.

30. Corpus Hypercubus
“Unfolded” Hypercube
Salvador Dali

31. 24-Cell in 4D
24 cells, 96 faces, 96 edges, 24 vertices.
(self-dual).

32. 24-Cell, showing 3-fold symmetry

33. 24-Cell “Fold-out” in 3D
Andrew Weimholt

34. 120-Cell in 4D
120 cells, faces, edges, vertices. Cell-first parallel projection, (shows less than half of the edges.)

35. 120 Cell
Hands-on workshop with George Hart

36. 120-Cell
Séquin (1982)
Thin face frames, Perspective projection.

37. 120-Cell
Cell-first, extreme perspective projection
Z-Corp. model

38. (smallest ?) 120-Cell
Wax model, made on Sanders machine

39. Radial Projections of the 120-Cell
Onto a sphere, and onto a dodecahedron:

40. 120-Cell, “exploded”
Russell Towle

41. 120-Cell Soap Bubble
John Sullivan

42. 600-Cell, A Classical Rendering
Total: tetra-cells, faces, edges, vertices.
At each Vertex: tetra-cells, faces, edges.
Oss, 1901
Frontispiece of Coxeter’s 1948 book “Regular Polytopes,”
and John Sullivan’s Paper “The Story of the 120-Cell.”

43. 600-Cell
Cross-eye Stereo Picture by Tony Smith

44. 600-Cell in 4D
Dual of 120 cell.
600 cells, faces, edges, vertices.
Cell-first parallel projection, shows less than half of the edges.

45. 600-Cell
David Richter

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

47. 600-Cell
Cell-first, parallel projection,
Z-Corp. model

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

49. Fused Deposition Modeling

50. Zooming into the FDM Machine

51. SFF: 3D Printing -- Principle
Selectively deposit binder droplets onto a bed of powder to form locally solid parts.
Head
Powder Spreading
Printing
Powder
Feeder
Build

52. 3D Printing: Z Corporation

53. 3D Printing: Z Corporation
Cleaning up in the de-powdering station

54. Designing 3D Edge Models
Is not totally trivial … because of shortcomings of CAD tools:
Limited Rotations – weird angles
Poor Booleans – need water tight shells

55. How We Did It … SLIDE (Jordan Smith, U.C.Berkeley) Some “cheating” …
Exploiting the strength and weaknesses of the specific programs that drive the various rapid prototyping machines.

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

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

58. Hypercube Series
“Measure Polytope” Series (introduced in the pantomime)
Consecutive perpendicular sweeps:
1D D D D
This series extents to arbitrary dimensions!

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

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

61. 5D and Beyond The three polytopes that result from the
Simplex series,
Cross polytope series,
Measure polytope series,
. . . is all there is in 5D and beyond!
2D 3D 4D 5D 6D 7D 8D 9D … 
Luckily, we live in one of the interesting dimensions!
Duals !
Dim. #

62. “Dihedral Angles in Higher Dim.”
Consider the angle through which one cell has to be rotated to be brought on top of an adjoining neighbor cell.
Space 2D 3D 4D 5D 6D 
Simplex Series
60°
70.5°
75.5°
78.5°
80.4°
90°
Cross Polytopes
109.5°
120°
126.9°
131.8°
180°
Measure Polytopes

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

64. Parade of Projections …
1. HYPERCUBES

65. Hypercube, Perspective Projections

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

67. 4D Hypercube
Vertex-first Projection

68. Preferred Hypercube Projections
Use Cavalier Projections to maintain sense of parallel sweeps:

69. 6D Hypercube
Oblique Projection

70. 6D Zonohedron
Sweep symmetrically in 6 directions (in 3D)

71. Modular Zonohedron Construction
Injection Molded Tiles:
Kiha Lee, CS 285, Spring 2002

72. 4D Hypercube – “squished”…
… to serve as basis for the 6D Hypercube

73. Composed of 3D Zonohedra Cells
The “flat” and the “pointy” cell:

74. 5D Zonohedron
Extrude by an extra story …
Extrusion

75. 5D Zonohedron  6D Zonohedron
Another extrusion
Triacontrahedral Shell

76. Parade of Projections (cont.)
2. SIMPLICES

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

78. 4D Simplex Projection: 5 Vertices
“Edge-first” parallel projection: V5 in center of tetrahedron V5

79. 5D Simplex: 6 Vertices Two methods: Based on Octahedron
Avoid central intersection:
Offset edges from middle.
Based on Tetrahedron (plus 2 vertices inside).

80. 5D Simplex with 3 Internal Tetras
With 3 internal tetrahedra; the 12 outer ones assumed to be transparent.

81. 6D Simplex: 7 Vertices (Method A)
Start from 5D arrangement that
avoids central edge intersection,
Then add point in center:

82. 6D Simplex (Method A)
= skewed octahedron with center vertex

83. 6D Simplex: 7 Vertices (Method B)
Skinny Tetrahedron plus three vertices around girth, (all vertices on same sphere):

84. 7D and 8D Simplices
Use a warped cube to avoid intersecting diagonals

85. Parade of Projections (cont.)
3. CROSS POLYTOPES

86. 4D Cross Polytope
Profiled edges, indicating attached faces.

87. 5D Cross Polytope
FDM --- SLIDE

88. 5D Cross Polytope with Symmetry
Octahedron + Tetrahedron (10 vertices)

89. 6D Cross Polytope
12 vertices  icosahedral symmetry

90. 7D Cross Polytope
14 vertices  cube + octahedron

91. New Work – in progress
other ways to color these edges …

92. Coloring with Hamiltonian Paths
Graph Colorings:
Euler Path: visiting all edges
Hamiltonian Paths: visiting all vertices
Hamiltonian Cycles: closed paths
Can we visit all edges with multiple Hamiltonian paths ?
Exploit symmetry of the edge graphs of the regular polytopes!

93. 4D Simplex: 2 Hamiltonian Paths
Two identical paths, complementing each other C2

94. 4D Cross Polytopes: 3 Paths
All vertices have valence 6 !

95. Hypercube: 2 Hamiltonian Paths
C4 (C2)
4-fold (2-fold) rotational symmetry around z-axis.

96. 24-Cell: 4 Hamiltonian Paths
Aligned:
 4-fold symmetry

97. The Big Ones … ?
. . . to be done !

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

101. What is a Regular Polytope?
How do we know that we have a completely regular polytope ? I show you a vertex ( or edge or face) and then spin the object -- can you still identify which one it was ? -- demo with irregular object -- demo with symmetrical object. Notion of a symmetry group -- all the transformations rotations (mirroring) that bring object back into cover with itself.