Exploring Regular Polytopes in Higher Dimensions: The 10-Dimensional Jewel

1. G4G9 A 10-dimensional Jewel Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

2. What Is a Regular Polytope ? • “Polytope”is the generalization of the terms “Polygon” (2D), “Polyhedron” (3D), “Polychoron” (4D), …to arbitrary dimensions. • “Regular”means: All the vertices, edges, faces, cells…are indistinguishable from each another. • Examples in 2D: Regular n-gons:There are infinitely many of them!

3. In 3 Dimensions . . . There are only 5 Platonic solids: They are composed from the regular 2D polygons. Only triangles, squares, and pentagons are useful: other n-gons are too ”round”;they cannot form nice 3D corners.

4. In 4D Space . . . The same constructive approach continues: We can use the Platonic solids as building blocks to form the “crust” of regular 4D polychora. Only 6 constructions are successful. Only 4 of the 5 Platonic solids can be used; the icosahedron is too round (dihedral angle > 120°). This is the result . . . 

5. The 6 Regular Polychorain 4-D . . .

6. 120-Cell ( 600V, 1200E, 720F ) • Cell-first,extremeperspectiveprojection • Z-Corp. model

7. 600-Cell ( 120V, 720E, 1200F ) (parallel proj.) • David Richter

8. In Higher-Dimensional Spaces . . . We can recursively construct new regular polytopes Using the ones from one dimension lower spceas the boundary (“surface”) element. But from dimension 5 onwards, there are just 3 each: N-Simplices (like tetrahedron) N-Cubes (hypercubes, measure-polytopes) N-Orthoplexes (cross-polytopes = duals of n-cube)

9. Thinking Outside the Box . . . Allow polyhedron faces to intersect . . . or even to be self-intersecting:

10. In 3D:  Kepler-PoinsotSolids • Mutually intersecting faces: (all) • Faces in the form of pentagrams: (3,4) • + 10 such objects in 4D space ! 1 2 3 4 Gr. Dodeca, Gr. Icosa, Gr. Stell. Dodeca, Sm. Stell. Dodeca

11. Single-Sided Polychora in 4D But we can do even wilder things ... Let’s allow single-sided polytope constructions like a Möbius band or a Klein bottle. In 4D we can make objects that close on themselves;they have the topology of the projective plane. The simplest one is the hemi-cube . . .

12. Hemi-Cube • Single-sided;not a solid any more! • Has the connectivity of the projective plane! 3 faces only vertex graph K4 3 saddle faces

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

14. Hemi-Dodecahedron • A self-intersecting, single-sided 3D cell • Is only geometrically regular in 9D space connect oppositeperimeter points connectivity: Petersen graph six warped pentagons

15. Hemi-Icosahedron • A self-intersecting, single-sided 3D cell • Is only geometrically regular in 5D  This is the BUILDING BLOCK for the 10D JEWEL ! connect oppositeperimeter points connectivity: graph K6 5-D simplex;warped octahedron

16. The Complete Connectivity Diagram • From: Coxeter [2], colored by Tom Ruen

17. Combining Two Cells Starter cell with4 tetra faces Add new cells on the inside ! All the edges of the first 5 cells.  A highly confusing, intersecting mess!

18. Six More Cells !

19. Regular Hendecachoron (11-Cell) Solid faces Transparency 11 vertices, 55 edges, 55 faces, 11 cells  self dual

20. The Full 11-Cell

21. The 10D Jewel 660 automorphisms – a building block of our universe ?

22. Hands-on Construction Project This afternoon we will build card-board models of the hemi-icosahedron. Thanks to Chris Palmer (now at U.C. Berkeley):for designing the parameterized template and for laser cutting the 30 colored parts.

23. What Is the 11-Cell Good For ? • A neat mathematical object ! • A piece of “absolute truth”:(Does not change with style, new experiments) • A 10-dimensional building block …(Physicists believe Universe may be 10-D)

24. Are there More Polychora Like This ? • Yes – one more: the 57-Cell • Built from 57 Hemi-dodecahedra • 5 such single-sided cells join around edges • It is also self-dual: 57 V, 171 E, 171 F, 57 C. • I may talk about it at G4G57 . . .