Draft for BRIDGES 2002 Regular Polytopes in Four and Higher Dimensions Carlo H. Séquin.
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Transcript of Draft for BRIDGES 2002 Regular Polytopes in Four and Higher Dimensions Carlo H. Séquin.
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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 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 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 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 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