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Draft for BRIDGES 2002

Regular Polytopes

in Four and Higher Dimensions

Carlo H. Séquin

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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 indistinguishable form each another.



Examples in 2D: Regular n-gons:

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Regular Polytopes in 3D



The Platonic Solids: There are only 5. Why ? …

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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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Constructing an (n+1)D Polytope

Angle-deficit = 90 ° 2D 3D 3D creates a 3D corner Forcing closure: 4D ?

creates a 4D corner

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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.

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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



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Projections : VERTEX / EDGE / FACE / CELL - First.



3D Cube: Paralell proj.

Persp. proj.



4D Cube: Parallel proj.

Persp. proj.

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Oblique Projections



Cavalier Projection

3D Cube  2D 4D Cube  3D  2D

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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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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

°).

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5-Cell or Simplex in 4D



5 cells, 10 faces, 10 edges, 5 vertices.



(self-dual).

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16 Cell or “Cross Polytope” in 4D



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

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Hypercube or Tessaract in 4D



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



(Dual of 16-Cell).

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24-Cell in 4D



24 cells, 96 faces, 96 edges, 24 vertices.



(self-dual).

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

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120-Cell



Cell-first, extreme perspective projection



Z-Corp. model

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600-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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600-Cell



Cell-first, parallel projection,



Z-Corp. model

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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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Hypercube 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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Simplex 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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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 2D 3D 4D

A square frame for every pair of axes 6 square frames = 24 edges

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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 …



5 6 3 3 3 3 3 3 Luckily, we live in one of the interesting dimensions!