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Exploring 4-D Space: The Regular 11-Cell and 57-Cell Polytopes

Dive into the fascinating world of 4-dimensional geometry as we unveil the mysteries of the Regular 11-Cell and 57-Cell polytopes, revealing their intricate structures and properties.

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Exploring 4-D Space: The Regular 11-Cell and 57-Cell Polytopes

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  1. SIGGRAPH 2007, San Diego Carlo H. Séquin & James F. Hamlin University of California, Berkeley The Regular 4-Dimensional 11-Cell & 57-Cell

  2. 4 Dimensions ?? • The 4th dimension exists !and it is NOT “time” ! • The 57-Cell is a complex, self-intersecting4-dimensional geometrical object. • It cannot be explained with a single image / model.

  3. San Francisco • Cannot be understood from one single shot !

  4. To Get to Know San Francisco • need a rich assembly of impressions, • then form an “image” in your mind...

  5. Regular Polygons in 2 Dimensions . . . • “Regular”means: All the vertices and edgesare indistinguishable from each another. • There are infinitely many regular n-gons ! • Use them to build regular 3D objects 

  6. Regular Polyhedra in 3-D(made from regular 2-D n-gons) The Platonic Solids: There are only 5. Why ? …

  7. Why Only 5 Platonic Solids ? Ways to build a regular convex corner: • from triangles: 3, 4, or 5 around a corner;  3 • from squares: only 3 around a corner;  1 . . . • from pentagons: only 3 around a corner;  1 • from hexagons:  planar tiling, does not close.  0 • higher N-gons:  do not fit around vertex without undulations (forming saddles).

  8. Let’s Build Some 4-D Polychora “multi-cell” By analogy with 3-D polyhedra: • Each will be bounded by 3-D cellsin the shape of some Platonic solid. • Around every edge the same small numberof Platonic cells will join together.(That number has to be small enough,so that some wedge of free space is left.) • This gap then gets forcibly closed,thereby producing bending into 4-D.

  9. All Regular “Platonic” Polychora in 4-D Using Tetrahedra (Dihedral angle = 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) Hyper-octahedron Using Dodecahedra (116.5°): 3 around an edge (349.5°)  (120 cells) “120-Cell” Using Icosahedra (138.2°):  NONE: angle too large (414.6°).

  10. How to View a Higher-D Polytope ? For a 3-D object on a 2-D screen: • Shadow of a solid object is mostly a blob. • Better to use wire frame, so we can also see what is going on on the back side.

  11. Oblique Projections • Cavalier Projection 3-D Cube  2-D 4-D Cube  3-D ( 2-D )

  12. Projections of a Hypercube to 3-D Cell-first Face-first Edge-first Vertex-first Use Cell-first: High symmetry; no coinciding vertices/edges

  13. The 6 Regular Polychora in 4-D

  14. 120-Cell ( 600V, 1200E, 720F ) • Cell-first,extremeperspectiveprojection • Z-Corp. model

  15. 600-Cell ( 120V, 720E, 1200F ) (parallel proj.) • David Richter

  16. Kepler-Poinsot “Solids” in 3-D 1 2 3 4 Gr. Dodeca, Gr. Icosa, Gr. Stell. Dodeca, Sm. Stell. Dodeca • Mutually intersecting faces (all above) • Faces in the form of pentagrams (#3,4) But in 4-D we can do even “crazier” things ...

  17. Even “Weirder” Building Blocks: Cross-cap Steiner’s Roman Surface Non-orientable, self-intersecting 2D manifolds Models of the 2D Projective Plane Construct 2 regular 4D objects:the 11-Cell & the 57-Cell Klein bottle

  18. Hemi-icosahedron connect oppositeperimeter points connectivity: graph K6 5-D Simplex;warped octahedron • A self-intersecting, single-sided 3D cell • Is only geometrically regular in 5D  BUILDING BLOCK FOR THE 11-CELL

  19. The Hemi-icosahedral Building Block 10 triangles – 15 edges – 6 vertices Steiner’sRoman Surface Polyhedral model with 10 triangles with cut-out face centers

  20. Gluing Two Steiner-Cells Together Hemi-icosahedron • Two cells share one triangle face • Together they use 9 vertices

  21. 2 cells inner faces 3rd cell 4th cell 1 cell 5th cell Adding Cells Sequentially

  22. How Much Further to Go ? • We have assembled only 5 of 11 cellsand it is already looking busy (messy)! • This object cannot be “seen” in one model.It must be “assembled” in your head. • Use different ways to understand it:  Now try a “top-down” approach.

  23. Start With the Overall Plan ... • We know from:H.S.M. Coxeter: A Symmetrical Arrangement of Eleven Hemi-Icosahedra.Annals of Discrete Mathematics 20 (1984), pp 103-114. • The regular 4-D 11-Cellhas 11 vertices, 55 edges, 55 faces, 11 cells. • Its edges form the complete graph K11 .

  24. Start: Highly Symmetrical Vertex-Set Center Vertex+Tetrahedron+Octahedron 1 + 4 + 6 vertices all 55 edges shown

  25. The Complete Connectivity Diagram 7 6 2 • Based on [ Coxeter 1984, Ann. Disc. Math 20 ]

  26. Views of the 11-Cell Solid faces Transparency

  27. The Full 11-Cell 660 automorphisms – a building block of our universe ?

  28. On to the 57-Cell . . . • It has a much more complex connectivity! • It is also self-dual: 57 V, 171 E, 171 F, 57 C. • Built from 57 Hemi-dodecahedra • 5 such single-sided cells join around edges

  29. Hemi-dodecahedron connect oppositeperimeter points connectivity: Petersen graph six warped pentagons • A self-intersecting, single-sided 3D cell  BUILDING BLOCK FOR THE 57-CELL

  30. Bottom-up Assembly of the 57-Cell (1) 5 cells around a common edge (black)

  31. Bottom-up Assembly of the 57-Cell (2) 10 cells around a common (central) vertex

  32. Vertex Cluster(v0) • 10 cells with one corner at v0

  33. Edge Clusteraround v1-v0 + vertex clusters at both ends.

  34. Connectivity Graph of the 57-Cell • 57-Cell is self-dual. Thus the graph of all its edges also represents the adjacency diagram of its cells. Six edges joinat each vertex Each cell has six neighbors

  35. Connectivity Graph of the 57-Cell (2) • Thirty 2nd-nearest neighbors • No loops yet (graph girth is 5)

  36. Connectivity Graph of the 57-Cell (3) Graph projected into plane • Every possible combination of 2 primary edges is used in a pentagonal face

  37. Connectivity Graph of the 57-Cell (4) Connectivity in shell 2 :  truncated hemi-icosahedron

  38. Connectivity Graph of the 57-Cell (5) 20 vertices 30 vertices 6 vertices 1 vertex 57 vertices total • The 3 “shells” around a vertex • Diameter of graph is 3

  39. Connectivity Graph of the 57-Cell (6) • The 20 vertices in the outermost shellare connected as in a dodecahedron.

  40. An “Aerial Shot” of the 57-Cell

  41. A “Deconstruction” of the 57-Cell

  42. E X T R A

  43. Hemi-cube (single-sided, not a solid any more!) 3 faces only vertex graph K4 3 saddle faces Simplest object with the connectivity of the projective plane, (But too simple to form 4-D polychora)

  44. Physical Model of a Hemi-cube Made on a Fused-Deposition Modeling Machine

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