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Draft for BRIDGES 2002

Draft for BRIDGES 2002. Regular Polytopes in Four and Higher Dimensions Carlo H. Séquin. What Is a Regular Polytope. “Polytope” is the generalization of the terms “Polygon” (2D), “Polyhedron” (3D), … to arbitrary dimensions.

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Draft for BRIDGES 2002

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  1. Draft for BRIDGES 2002 Regular Polytopesin Four and Higher Dimensions Carlo H. Séquin

  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. Persp. 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, 720 faces, 1200 edges, 600 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,extremeperspectiveprojection • Z-Corp. model

  19. 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.

  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°, 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…

  22. Hypercube Series • “Measure Polytope” Series(introduced in the pantomime) • Consecutive perpendicular sweeps: 1D 2D 3D 4D 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 2D 3D This series also goes on indefinitely!The issue is how to make “nice” projections.

  24. 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

  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 …5 63 3 3 3 3 3 Luckily, we live in one of the interesting dimensions!

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