1 / 55

CS 39: Symmetry and Topology

Learn about regular polytopes, Platonic solids, visualizing 4D shapes, and higher-dimensional polytopes through symmetry and topology. Discover different types of projections and build an understanding of structures in multiple dimensions.

ethanv
Download Presentation

CS 39: Symmetry and Topology

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related