1. CS 39: Symmetry and Topology
Regular Polytopes in Four and Higher Dimensions
Carlo H. Séquin 1

2. 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:

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

4. 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!

5. 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 bend!
higher n-gons:  do not fit around a vertex without undulations (forming saddles);  Now the edges are no longer all alike!
Now we can see why there are only 5 Platonic solids: When we use triangles for the surface facets, we have a choice, we can use 3 or 4 or 5 around a point to form a regular corner. This leads to the Tetrahedron, the Octahedron, and the Icosahedron which uses 20 triangles. …
But if we use squares there is only one option: 3 squares around a corner – forming a cube.
And we have already seen that with 3 pentagons we can make a dodecahedron.
All larger n-gons are to “round” and are not able to make true 3D corners.

6. Forming a 4D Polytope Corner
Angle-deficit = 90° 2D 3D
Forcing closure: ? 3D 4D
creates a 3D corner
creates a 4D corner

7. How Do We Find All 4D Polytopes?
Reasoning by analogy helps a lot: -- How did we find all the Platonic solids?
Now: Use the Platonic solids as “tiles” and ask:
What can we build from tetrahedra?
or from cubes?
or 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° > 120 °.
Now we have to count in how many different ways we can do this. The crucial value now is the dihedral angle, i.e., the angle between two adjacent faces of a Platonic cell; it tells us how many of this particular solid can be fit around a shared edge, -- and how much space is left over! For the tetrahedron this critical angle is about 70 degrees, so we can fit 3, 4, or 5 tetrahedra around a common edge, and still have a little room to do some bending.

8. Possible 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) 600-Cell
Using Cubes (90°):
3 around an edge (270.0°)  (8 cells) Hypercube
Using Octahedra (109.5°):
3 around an edge (328.5°)  (24 cells) 24-Cell
Using Dodecahedra (116.5°):
3 around an edge (349.5°)  (120 cells) 120-Cell
Using Icosahedra (138.2°):
 None! : dihedral angle is too large ( 414.6°).
Here is the complete analysis: We can fit 3, 4, or 5 tetrahedra around and edge, and all these options yield valid regular 4D polytopes.
With cubes there is only one valid option: 3 cubes around an edge yield a hypercube corner.
We can also use 3 octahedra to make a valid corner, or 3 dotecahedra.
But the icosahedron is useless: its interior angle at each edge is more than 120 degrees, 3 of them don't fit around a common edge.
OK – these are just abstract numbers … Can we get an idea what these things might look like? Let’s use some “mathematical seeing.”

9. Visualizing 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 ?
Review of various types of projections 

10. Wire-Frame Projections
Project a 4D polytope from 4D space to 3D space:
Shadow of a solid object is mostly a “blob”.
Better to use wire frame, so we can also see what is going on at the back side.
We cannot directly look into 4-D space. But we can make a projection of these 4D objects down to 3D. How do we make a “good” projection?
Shadows = “blobs” -> uninformative! -- Better to just use wireframes of these objects, so we can see through them, and also see their backsides.
The figure on the right is only a 2D image, but your brain allows you to see it as a cube – a 3D object.
With a little practice you can also look at 3D projections and gain an understanding of the 4D object from which it was derived.

11. Different Possible Projections
Oblique or Perspective Projections
We have some options of how we want to do this projection – just as in engineering where we can use either an oblique or a perspective projection to show an ordinary 3D cube.
In addition, we can also use color to give depth information: Here, red is the front object, blue is the extruded back object, and green are the transition edges going from front to back.
3D Cube  2D D Cube  3D ( 2D )
We may use color to give “depth” information: (front) (back) (transition)

12. Projections: VERTEX / EDGE / FACE / CELL - First.
3D Cube:
Paralell proj.
Persp. proj.
4D Cube:
Parallel proj.
The direction from which we look at our polytope also matters!

13. 5-Cell or Simplex in 4D 5 cells, 10 faces, 10 edges, 5 vertices.
(self-dual).

14. 16-Cell or “Cross Polytope” in 4D
16 cells, 32 faces, 24 edges, 8 vertices.

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

16. Hypercube, Perspective Projections
Here 3D projections work just fine.

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

18. 4D Hypercube
Vertex-first Parallel Projection

19. Corpus Hypercubus
“Unfolded” Hypercube
Salvador Dali

20. 24-Cell
24 cells, 96 faces, 96 edges, 24 vertices.
(self-dual).

21. 120-Cell
(1982)
Thin face frames, Perspective projection.

22. 120-Cell Cell-first, extreme perspective projection Z-Corp. model
(Things get really crunched together in the center! )

23. 120-Cell
120 cells, faces, edges, vertices. Cell-first parallel projection, (shows less than half of the edges.)
No crunching in the center!

24. 120-Cell Soap Bubble
John Sullivan

25. 600-Cell Dual of 120 cell. Cell-first, extreme perspective projection
Z-Corp. model
(Things get really crunched together in the center! )

26. 600-Cell Cell-first, less extreme perspective projection Z-Corp. model
(Things still get crunched together in the center! )

27. 600-Cell
Dual of 120 cell.
600 cells, faces, edges, vertices.
Cell-first parallel projection, shows less than half of the edges.

28. 600-Cell
David Richter

29. How About the Higher Dimensions?
For a 5D regular polytope, 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 …
Corners form from 3 or 4 5-Cells, or from 3 Tessaracts.
There are always three methods by which we can generate regular polytopes for 5D and higher…

30. Hypercube Series “Measure Polytope” Series
Consecutive perpendicular sweeps: (introductory pantomime)
1D D D D
This series extents to arbitrary dimensions!

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

32. 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

33. 5D and Beyond Always 3 polytopes that result from the:
Simplex series,
Cross polytope series,
Measure polytope series,
This 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!

34. “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

35. High-D Regular Polytopes
1. HYPERCUBES

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

37. 6D Hypercube
Oblique Projection

38. 6D Zonohedron
Sweep symmetrically in 6 directions (in 3D)

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

40. 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…

41. 4D Simplex Projection: 5 Vertices
“Edge-first” parallel projection: V5 in center of tetrahedron V5

42. Another 4D Simplex Model
3-sided Bi-Pyramid
I used this to make this sculpture …

43. 2013: 125th Anniversary of AMS 125 Tetrahedra in 25 Projected 5-Cells
To celebrate the 125th Anniversary of the American Mathematics Sosicety.
125 Tetrahedra in 25 Projected 5-Cells

44. Models of High-D Regular Polytopes
What is a “good” model ?
Maintain as much of the symmetry as possible;
Get a good feel for the structure of the polytope.
Avoid spurious edge intersections.
Simple projections will not do this!
Better: just place the appropriate number of vertices in a symmetrical manner, and connect them with the required edges.
(Maintain topology of edge graph)

45. 5D Simplex: 6 Vertices Two methods: Based on Octahedron
Avoid central intersection:
Offset edges from middle.
Based on Tetrahedron (plus 2 vertices inside).

46. 6D Simplex: 7 Vertices (Method A)
Start from 5D arrangement that
avoids central edge intersection
(skewed octahedron).
Then add point in center:

47. 6D Simplex (Method A)
= skewed octahedron with center vertex

48. 6D Simplex: 7 Vertices (Method B)
Skinny Tetrahedron plus three vertices around girth, (all vertices on same sphere):

49. 7D and 8D Simplices
Use a warped cube to avoid intersecting diagonals

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

51. 4D Cross Polytope
Profiled edges, indicating attached faces.

52. 5D Cross Polytope
3D Print (FDM)
CAD Model

53. 5D Cross Polytope with Symmetry
Octahedron + Tetrahedron (10 vertices)

54. 12 vertices  icosahedral symmetry
6D Cross Polytope
12 vertices  icosahedral symmetry

55. 7D Cross Polytope
14 vertices  cube + octahedron

56. 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.”