1. Graphics Lunch, Nov. 15, 2007 The Regular 4-Dimensional 11-Cell & 57-Cell Carlo H. Séquin University of California, Berkeley

2. 4 Dimensions ?? u The 4 th dimension exists ! and it is NOT “time” ! u The 11-Cell and 57-Cell are complex, self-intersecting, 4-dimensional geometrical objects. u They cannot be visualized / explained with a single image / model.

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

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

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

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: u from triangles: 3, 4, or 5 around a corner;  3 u from squares: only 3 around a corner;  1... u from pentagons: only 3 around a corner;  1 u from hexagons:  planar tiling, does not close.  0 u 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: u Each will be bounded by 3-D cells in the shape of some Platonic solid. u Around every edge the same small number of Platonic cells will join together. (That number has to be small enough, so that some wedge of free 3D space is left.) u 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° > 120° ):  NONE: angle too large (414.6°).

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

11. Oblique Projections u Cavalier Projection 3-D Cube  2-D4-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 ) u Cell-first, extreme perspective projection u Z-Corp. model

15. 600-Cell ( 120V, 720E, 1200F ) u Cell-first, extreme perspective projection u Z-Corp. model

16. 600-Cell (Parallel Projection) u David Richter

17. An 11-Cell ??? Another Regular 4-D Polychoron ? u I have just shown that there are only 6. u “11” feels like a weird number; typical numbers are: 8, 16, 24, 120, 600. u The notion of a 4-D 11-Cell seems bizarre!

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

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

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

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

22. Hemi-icosahedron u A self-intersecting, single-sided 3D cell u Is only geometrically regular in  5D  BUILDING BLOCK FOR THE 11-CELL connect opposite perimeter points connectivity: graph K 6 5-D Simplex; warped octahedron

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

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

25. Adding Cells Sequentially 1 cell2 cellsinner faces3 rd cell 4 th cell 5 th cell

26. A More Symmetrical Construction u Exploit the symmetry of the Steiner cell ! One Steiner cell2 nd cell added on “inside” Two cells with cut-out faces 4 th white vertex used by next 3 cells (central) 11 th vertex used by last 6 cells

27. How Much Further to Go ? u We have seen at most 5 of 11 cells and it already looked busy (messy)! u This object cannot be “seen” in one model. It must be “assembled” in your head. u Use different ways to understand it:  Now try a “top-down” approach.

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

29. Start: Highly Symmetrical Vertex-Set Center Vertex + Tetrahedron + Octahedron 1 + 4 + 6 verticesall 55 edges shown

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

31. Views of the 11-Cell Solid faces Transparency

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

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

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

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

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

37. Vertex Cluster (v0) u 10 cells with one corner at v0

38. Edge Cluster around v1-v0 + vertex clusters at both ends.

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

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

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

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

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

44. Connectivity Graph of the 57-Cell (6) u The 20 vertices in the outermost shell are connected as in a dodecahedron.

45. An “Aerial Shot” of the 57-Cell

46. A “Deconstruction” of the 57-Cell

47. Questions ?