1. 4D Polytopes and 3D Models of Them
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4D Polytopes and 3D Models of Them
George W. Hart
Stony Brook University

2. Goals of This Talk Expand your thinking.
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Goals of This Talk
Expand your thinking.
Visualization of four- and higher-dimensional objects.
Show Rapid Prototyping of complex structures.
Note: Some Material and images adapted from Carlo Sequin

3. What is the 4th Dimension ?
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What is the 4th Dimension ?
Some people think: “it does not really exist” “it’s just a philosophical notion” “it is ‘TIME’ ”
But, a geometric fourth dimension is as useful and as real as 2D or 3D.

4. Higher-dimensional Spaces
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Higher-dimensional Spaces
Coordinate Approach:
A point (x, y, z) has 3 dimensions.
n-dimensional point: (d1, d2, d3, d4, ..., dn).
Axiomatic Approach:
Definition, theorem, proof...
Descriptive Geometry Approach:
Compass, straightedge, two sheets of paper.

5. What Is a Regular Polytope?
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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 equivalent.
Assume convexity for now.
Examples in 2D: Regular n-gons:

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

7. Why Only 5 Platonic Solids ?
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Why Only 5 Platonic Solids ?
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 (not convex)

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

9. Florida 1999
“Seeing a Polytope”
Real “planes”, “lines”, “points”, “spheres”, …, do not exist physically.
We understand their properties and relationships as ideal mental models.
Good projections are very useful. Our visual input is only 2D, but we understand as 3D via mental construction in the brain.
You are able to “see” things that don't really exist in physical 3-space, because you “understand” projections into 2-space, and you form a mental model.
We will use this to visualize 4D Polytopes.

10. Florida 1999
Projections
Set the coordinate values of all unwanted dimensions to zero, e.g., drop z, retain x,y, and you get a orthogonal projection along the z-axis. i.e., a 2D shadow.
Linear algebra allows arbitrary direction.
Alternatively, use a perspective projection: rays of light form cone to eye.
Can add other depth queues: width of lines, color, fuzziness, contrast (fog) ...

11. Wire Frame Projections
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Wire Frame Projections
Shadow of a solid object is mostly a blob.
Better to use wire frame, so we can see components.

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

13. Projections: VERTEX / EDGE / FACE / CELL – centered
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Projections: VERTEX / EDGE / FACE / CELL – centered
3D Cube:
Paralell proj.
Persp. proj.
4D Cube:
Parallel proj.

14. 3D Objects Need Physical Edges
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3D Objects Need Physical Edges
Options:
Round dowels (balls and stick)
Profiled edges – edge flanges convey a sense of the attached face
Flat tiles for faces – with holes to make structure see-through.

15. Florida 1999
Edge Treatments
(Leonardo Da Vinci)

16. How Do We Find All 4D Polytopes?
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How Do We Find All 4D Polytopes?
Sum of dihedral angles around each edge must be less than 360 degrees.
Use the Platonic solids as “cells”
Tetrahedron: 70.5°
Octahedron: 109.5°
Cube: 90°
Dodecahedron: 116.5°
Icosahedron: 138.2°.

17. All Regular Convex 4D Polytopes
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All Regular Convex 4D Polytopes
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)
Using Dodecahedra (116.5°):
3 around an edge (349.5°)  (120 cells)
Using Icosahedra (138.2°):
 none: angle too large.

18. 5-Cell or 4D Simplex 5 cells, 10 faces, 10 edges, 5 vertices.
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5-Cell or 4D Simplex
5 cells, 10 faces, 10 edges, 5 vertices.
Carlo Sequin
Can make with Zometool also

19. 16-Cell or “4D Cross Polytope”
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16-Cell or “4D Cross Polytope”
16 cells, 32 faces, 24 edges, 8 vertices.

20. 4D Hypercube or “Tessaract”
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4D Hypercube or “Tessaract”
8 cells, 24 faces, 32 edges, 16 vertices.

21. Hypercube, Perspective Projections
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22. Nets: 11 Unfoldings of Cube
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Nets: 11 Unfoldings of Cube

23. Hypercube Unfolded -- “Net”
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Hypercube Unfolded -- “Net”
One of the 261 different unfoldings

24. Florida 1999
Corpus Hypercubus
“Unfolded” Hypercube
Salvador Dali

25. 24-Cell 24 cells, 96 faces, 96 edges, 24 vertices. (self-dual).
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24-Cell
24 cells, 96 faces, 96 edges, 24 vertices.
(self-dual).

26. Florida 1999
24-Cell “Net” in 3D
Andrew Weimholt

27. 120-Cell 120 cells, 720 faces, 1200 edges, 600 vertices.
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120-Cell
120 cells, faces, edges, vertices.
Cell-first parallel projection, (shows less than half of the edges.)

28. Florida 1999
120-cell Model
Marc Pelletier

29. 120-Cell Thin face frames, Perspective projection. Carlo Séquin
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120-Cell
Carlo Séquin
Thin face frames, Perspective projection.

30. 120-Cell – perspective projection
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120-Cell – perspective projection

31. Florida 1999
(smallest ?) 120-Cell
Wax model, made on Sanders machine

32. 120-Cell – perspective projection
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120-Cell – perspective projection
Selective laser sintering

33. 3D Printing — Zcorp
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34. Florida 1999
120-Cell, “exploded”
Russell Towle

35. 120-Cell Soap Bubble John Sullivan
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120-Cell Soap Bubble
John Sullivan
Stereographic projection preserves 120 degree angles

36. 120-Cell “Net” with stack of 10 dodecahedra George Olshevski
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120-Cell “Net”
with stack of 10 dodecahedra
George Olshevski

37. 600-Cell -- 2D projection Oss, 1901
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600-Cell D projection
Total: tetra-cells, faces, edges, vertices.
At each Vertex: tetra-cells, faces, edges.
Oss, 1901
Frontispiece of Coxeter’s book “Regular Polytopes,”

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

39. Florida 1999
600-Cell
Dual of 120 cell.
600 cells, faces, edges, vertices.
Cell-first parallel projection, shows less than half of the edges.
Can make with Zometool

40. Florida 1999
600-Cell
Straw model by David Richter

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

42. History 3D Models of 4D Polytopes
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History 3D Models of 4D Polytopes
Ludwig Schlafli discovered them in Worked algebraically, no pictures in his paper. Partly published in 1858 and 1862 (translation by Cayley) but not appreciated.
Many independent rediscoveries and models.

43. Stringham (1880) First to rediscover all six
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Stringham (1880)
First to rediscover all six
His paper shows cardboard models of layers
3 layers of 120-cell
(45 dodecahedra)

44. Victor Schlegel (1880’s) Five regular polytopes
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Victor Schlegel (1880’s)
Invented “Schlegel Diagram”
3D  2D perspective transf.
Used analogous 4D  3D
projection in educational
models.
Built wire and thread models.
Advertised and sold models via commercial catalogs: Dyck (1892) and Schilling (1911).
Some stored at Smithsonian.
Five regular polytopes

45. Sommerville’s Description of Models
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Sommerville’s Description of Models
“In each case, the external boundary of the projection represents one of the solid boundaries of the figure. Thus the 600-cell, which is the figure bounded by 600 congruent regular tetrahedra, is represented by a tetrahedron divided into 599 other tetrahedra … At the center of the model there is a tetrahedron, and surrounding this are successive zones of tetrahedra. The boundaries of these zones are more or less complicated polyhedral forms, cardboard models of which, constructed after Schlegel's drawings, are also to be obtained by the same firm.”

46. Cardboard Models of 120-Cell
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Cardboard Models of 120-Cell
From Walther Dyck’s 1892 Math and Physics Catalog

47. Paul S. Donchian’s Wire Models
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Paul S. Donchian’s Wire Models
1930’s
Rug Salesman with
“visions”
Wires doubled to show how front overlays back
Widely displayed
Currently on view at the Franklin Institute

48. Florida 1999
Zometool
1970 Steve Baer designed and produced "Zometool" for architectural modeling
Marc Pelletier discovered (when 17 years old) that the lengths and directions allowed by this kit permit the construction of accurate models of the 120-cell and related polytopes.
The kit went out of production however, until redesigned in plastic in 1992.

49. Florida 1999
120 Cell
Zome Model
Orthogonal projection

50. Uniform 4D Polytopes Analogous to the 13 Archimedean Solids
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Uniform 4D Polytopes
Analogous to the 13 Archimedean Solids
Allow more than one type of cell
All vertices equivalent
Alicia Boole Stott listed many in 1910
Now over 8000 known
Cataloged by George Olshevski and Jonathan Bowers

51. Truncated 120-Cell
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52. Truncated 120-Cell - Stereolithography
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53. Zometool Truncated 120-Cell
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Zometool Truncated 120-Cell
MathCamp 2000

54. Florida 1999
Ambo 600-Cell
Bridges Conference, 2001

55. Ambo 120-Cell Orthogonal projection Stereolithography Can do with Zome
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Ambo 120-Cell
Orthogonal projection
Stereolithography
Can do with Zome

56. Florida 1999
Expanded 120-Cell
Mira Bernstein,
Vin de Silva, et al.

57. Expanded Truncated 120-Cell
Florida 1999

58. Florida 1999
Big Polytope “Net”
George Olshevski

59. Big Polytope Zome Model
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Big Polytope Zome Model
Steve Rogers

60. Florida 1999
48 Truncated Cubes
Poorly designed FDM model

61. Prism on a Snub Cube – “Net”
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Prism on a Snub Cube – “Net”
George Olshevski

62. Duo-Prisms - “Nets” Andrew Weimholt George Olshevski Andrew Weimholt
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Duo-Prisms - “Nets”
Andrew Weimholt
George Olshevski
Andrew Weimholt
Robert Webb

63. Grand Antiprism “Net” with stack of 10 pentagonal antiprisms
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Grand Antiprism “Net”
with stack of 10 pentagonal antiprisms
George Olshevski

64. Non-Convex Polytopes Components may pass through each other
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Non-Convex Polytopes
Components may pass
through each other
Slices may be useful
for visualization
Slices may be
disconnected
Jonathan Bowers

65. Beyond 4 Dimensions … What happens in higher dimensions ?
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Beyond 4 Dimensions …
What happens in higher dimensions ?
How many regular polytopes are there in 5, 6, 7, … dimensions ?
Only three regular types:
Hypercubes — e.g., cube
Simplexes — e.g., tetrahedron
Cross polytope — e.g., octahedron

66. Hypercubes A.k.a. “Measure Polytope”
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Hypercubes
A.k.a. “Measure Polytope”
Perpendicular extrusion in nth direction:
1D D D D

67. Orthographic Projections
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Orthographic Projections
Parallel lines remain parallel

68. Florida 1999
Simplex Series
Connect all the dots among n+1 equally spaced vertices: (Put next one “above” center of gravity) D D D
This series also goes on indefinitely.

69. 7D Simplex A warped cube avoids intersecting diagonals.
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7D Simplex
A warped cube avoids intersecting diagonals.
Up to 6D can be constructed with Zometool.
Open problem: 7D constructible with Zometool?

70. A square frame for every pair of axes
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Cross Polytope Series
Place vertex in + and – direction on each axis, 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

71. 6D Cross Polytope 12 vertices suggests using icosahedron
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6D Cross Polytope
12 vertices suggests using icosahedron
Can do with Zometool.

72. Florida 1999
6D Cross Polytope
Chris Kling

73. Florida 1999
Some References
Ludwig Schläfli: “Theorie der vielfachen Kontinuität,” 1858, (published in 1901).
H. S. M. Coxeter: “Regular Polytopes,” 1963, (Dover reprint).
Tom Banchoff, Beyond the Third Dimension, 1990.
G.W. Hart, “4D Polytope Projection Models by 3D Printing” to appear in Hyperspace.
Carlo Sequin, “3D Visualization Models of the Regular Polytopes…”, Bridges 2002.

74. Florida 1999
Puzzle
Which of these shapes can / cannot be folded into a 4D hypercube?
Hint: Hold the red cube still and fold the others around it.
Scott Kim