CS 39: Symmetry and Topology

1. CS 39: Symmetry and Topology 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. 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 bend! • higher n-gons:  do not fit around a vertex without undulations (forming saddles);  Now the edges are no longer all alike!

6. Forming a 4D Polytope Corner Angle-deficit = 90° 2D 3D Forcing closure: ? 3D 4D creates a 3D corner creates a 4D corner

7. How Do We Find All 4D Polytopes? • Reasoning by analogy helps a lot:-- How did we find all the Platonic solids? • Now: Use the Platonic solids as “tiles” and ask: • What can we build from tetrahedra? • or from cubes? • or 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° > 120 °.

8. Possible 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) Cross-Polytope 5 around an edge (352.5°)  (600 cells) 600-Cell Using Cubes (90°): 3 around an edge (270.0°)  (8 cells) Hypercube Using Octahedra (109.5°): 3 around an edge (328.5°)  (24 cells) 24-Cell Using Dodecahedra (116.5°): 3 around an edge (349.5°)  (120 cells) 120-Cell Using Icosahedra (138.2°): None! : dihedral angle is too large ( 414.6°).

9. Visualizing 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 types of projections

10. Wire-Frame Projections • Project a 4D polytope from 4D space to 3D space: • Shadow of a solid object is mostly a “blob”. • Better to use wire frame, so we can also see what is going on at the back side.

11. Different Possible Projections 3D Cube 2D 4D Cube  3D ( 2D ) Oblique or Perspective Projections We may use color to give “depth” information:(front) (back) (transition)

12. Projections: VERTEX / EDGE / FACE / CELL - First. • 3D Cube: Paralell proj. Persp. proj. • 4D Cube: Parallel proj. Persp. proj.

13. 5-Cell or Simplex in 4D • 5 cells, 10 faces, 10 edges, 5 vertices. • (self-dual).

14. 16-Cell or “Cross Polytope” in 4D • 16 cells, 32 faces, 24 edges, 8 vertices.

15. Hypercube or Tessaract in 4D • 8 cells, 24 faces, 32 edges, 16 vertices. • (Dual of 16-Cell).

16. Hypercube, Perspective Projections

17. 4D Hypercube Vertex-first Parallel Projection

18. Corpus Hypercubus Salvador Dali “Unfolded”Hypercube

19. 24-Cell • 24 cells, 96 faces, 96 edges, 24 vertices. • (self-dual).

20. 120-Cell (1982) Thin face frames, Perspective projection.

21. 120-Cell • Cell-first,extremeperspectiveprojection • Z-Corp. model • (Things get really crunched together in the center! )

22. 120-Cell • 120 cells, 720 faces, 1200 edges, 600 vertices.Cell-first parallel projection,(shows less than half of the edges.)

23. 120-Cell Soap Bubble John Sullivan

24. 600-Cell • Dual of 120 cell. • Cell-first,extremeperspectiveprojection • Z-Corp. model • (Things get really crunched together in the center! )

25. 600-Cell • Cell-first,less extremeperspectiveprojection • Z-Corp. model • (Things still get crunched together in the center! )

26. 600-Cell • Dual of 120 cell. • 600 cells, 1200 faces, 720 edges, 120 vertices. • Cell-first parallel projection,shows less than half of the edges.

27. 600-Cell • David Richter

28. How About the Higher Dimensions? • For a 5D regular polytope, 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 … Corners form from 3 or 4 5-Cells, or from 3 Tessaracts. • There are always three methods by which we can generate regular polytopes for 5D and higher…

29. Hypercube Series • “Measure Polytope” Series • Consecutive perpendicular sweeps:(introductory pantomime) 1D 2D 3D 4D This series extents to arbitrary dimensions!

30. Simplex Series • Connect all the dots among n+1 equally spaced vertices:(Find new vertex above COG). 1D 2D 3D This series also goes on indefinitely!The issue is how to make “nice” projections.

31. 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

32. 5D and Beyond Always 3 polytopes that result from the: • Simplex series, • Cross polytope series, • Measure polytope series, This 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!

33. “Dihedral Angles in Higher Dim.” • Consider the angle through which one cell has to be rotated to be brought on top of an adjoining neighbor cell.

34. High-D Regular Polytopes 1. HYPERCUBES

35. Preferred Hypercube Projections • Use Cavalier Projections to maintain sense of parallel sweeps:

36. 6D Hypercube • Oblique Projection

37. 6D Zonohedron • Sweep symmetrically in 6 directions (in 3D)

38. Parade of Projections (cont.) 2. SIMPLICES

39. 3D Simplex Projections • Look for symmetrical projectionsfrom 3D to 2D, or … • How to put 4 vertices symmetrically in 2Dand so that edges do not intersect. Similarly for 4D and higher…

40. 4D Simplex Projection: 5 Vertices • “Edge-first” parallel projection:V5 in center of tetrahedron V5

41. Another 4D Simplex Model • 3-sided Bi-Pyramid

42. 2013: 125th Anniversary of AMS • 125 Tetrahedra in 25 Projected 5-Cells

43. Models of High-D Regular Polytopes • What is a “good” model ? • Maintain as much of the symmetry as possible; • Get a good feel for the structure of the polytope. • Avoid spurious edge intersections. • Simple projections will not do this! • Better: just place the appropriate number of vertices in a symmetrical manner, and connect them with the required edges. • (Maintain topology of edge graph)

44. 5D Simplex: 6 Vertices Based on Octahedron • Two methods: Avoid central intersection: Offset edges from middle. Based on Tetrahedron(plus 2 vertices inside).

45. 6D Simplex: 7 Vertices (Method A) Start from 5D arrangement that avoids central edge intersection (skewed octahedron). Then add point in center:

46. 6D Simplex (Method A) = skewed octahedron with center vertex

47. 6D Simplex: 7 Vertices (Method B) • Skinny Tetrahedron plusthree vertices around girth,(all vertices on same sphere):

48. 7D and 8D Simplices Use a warped cube to avoid intersecting diagonals

49. Parade of Projections (cont.) 3. CROSS POLYTOPES

50. 4D Cross Polytope Profiled edges, indicating attached faces.