1. Regular Polytopes in Four and Higher Dimensions
Draft for BRIDGES 2002
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. Constructing an (n+1)D Polytope
Angle-deficit = 90° 2D 3D
Forcing closure: ? 3D 4D
creates a 3D corner
creates a 4D corner

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

7. 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 ? Review of various projections 

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

9. Oblique Projections
Cavalier Projection
3D Cube  2D
4D Cube  3D  2D

10. 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°.

11. 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)
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°).

12. 5-Cell or Simplex in 4D 5 cells, 10 faces, 10 edges, 5 vertices.
(self-dual).

13. 16-Cell or “Cross Polytope” in 4D
16 cells, 32 faces, 24 edges, 8 vertices.

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

15. 24-Cell in 4D
24 cells, 96 faces, 96 edges, 24 vertices.
(self-dual).

16. 120-Cell in 4D
120 cells, faces, edges, vertices. Cell-first parallel projection, (shows less than half of the edges.)

17. 120-Cell
(1982)
Thin face frames, Perspective projection.

18. 120-Cell
Cell-first, extreme perspective projection
Z-Corp. model

19. 600-Cell in 4D
Dual of 120 cell.
600 cells, faces, edges, vertices.
Cell-first parallel projection, shows less than half of the edges.

20. 600-Cell
Cell-first, parallel projection,
Z-Corp. model

21. How About the 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 always three methods by which we can generate regular polytopes for 5D and higher…

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

23. 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.

24. 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

25. 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!