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Draft for BRIDGES 2002

This text provides an introduction to regular polytopes in various dimensions, focusing on their construction and properties. It discusses the concept of regularity, explores the five Platonic solids in 3D, and then delves into the construction of regular polytopes in 4D. The text also touches on projections and various methods of generating regular polytopes in higher dimensions. It concludes by highlighting the three types of regular polytopes that exist in 5D and beyond.

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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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