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UCBUCB Draft for BRIDGES 2002Draft for BRIDGES 2002

Regular Polytopes in Four and Higher Dimensions

Carlo H. Séquin

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UCBUCB What Is a Regular PolytopeWhat 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:

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UCBUCB Regular Polytopes in 3DRegular Polytopes in 3D

The Platonic Solids:

There are only 5. Why ? …

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UCBUCB Why Only 5 Platonic Solids ?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!

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UCBUCB Constructing an (n+1)D PolytopeConstructing an (n+1)D Polytope

Angle-deficit = 90°

creates a 3D corner creates a 4D corner

?

2D

3D 4D

3D

Forcing closure:

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UCBUCB Wire Frame ProjectionsWire 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.

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UCBUCB Constructing 4D Regular PolytopesConstructing 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

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UCBUCB ProjectionsProjections: : VERTEXVERTEX / / EDGE EDGE / / FACEFACE / / CELL CELL - First.- First.

3D Cube:

Paralell proj.

Persp. proj.

4D Cube:

Parallel proj.

Persp. proj.

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UCBUCB Oblique ProjectionsOblique Projections

Cavalier Projection

3D Cube 2D 4D Cube 3D 2D

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UCBUCB How Do We Find All 4D Polytopes?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°.

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UCBUCB All Regular Polytopes in 4DAll 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°).

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UCBUCB 5-Cell or Simplex in 4D5-Cell or Simplex in 4D

5 cells, 10 faces, 10 edges, 5 vertices. (self-dual).

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UCBUCB 16-Cell or “Cross Polytope” in 4D16-Cell or “Cross Polytope” in 4D

16 cells, 32 faces, 24 edges, 8 vertices.

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UCBUCB Hypercube or Tessaract in 4DHypercube or Tessaract in 4D

8 cells, 24 faces, 32 edges, 16 vertices.

(Dual of 16-Cell).

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UCBUCB 24-Cell in 4D24-Cell in 4D

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

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UCBUCB 120-Cell in 4D120-Cell in 4D

120 cells, 720 faces, 1200 edges, 600 vertices.

Cell-first parallel projection,(shows less than half of the edges.)

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UCBUCB 120-Cell120-Cell

Thin face frames, Perspective projection.

(1982)

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UCBUCB 120-Cell120-Cell

Cell-first,extremeperspectiveprojection

Z-Corp. model

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UCBUCB 600-Cell in 4D600-Cell in 4D

Dual of 120 cell.

600 cells, 1200 faces, 720 edges, 120 vertices.

Cell-first parallel projection,shows less than half of the edges.

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UCBUCB 600-Cell600-Cell

Cell-first, parallel projection,

Z-Corp. model

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UCBUCB How About the Higher Dimensions?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°, 600-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…

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UCBUCB Hypercube SeriesHypercube Series

“Measure Polytope” Series(introduced in the pantomime)

Consecutive perpendicular sweeps:

1D 2D 3D 4D

This series extents to arbitrary dimensions!

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UCBUCBSimplex SeriesSimplex Series

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.

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UCBUCB Cross Polytope SeriesCross 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 2D 3D 4D

A square frame for every pair of axes

6 square frames= 24 edges

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UCBUCB 5D and Beyond5D 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 … 5 6 3 3 3 3 3 3

Luckily, we live in one of the interesting dimensions!