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

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

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

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

The 6 Regular Polychora in 4-D...

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

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

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

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

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

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

Hemi-Cube  Single-sided; not a solid any more! u Has the connectivity of the projective plane! 3 faces onlyvertex graph K 4 3 saddle faces

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

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

Hemi-Icosahedron u A self-intersecting, single-sided 3D cell u Is only geometrically regular in  5D  This is the BUILDING BLOCK for the 10D JEWEL ! connect opposite perimeter points connectivity: graph K 6 5-D simplex; warped octahedron

The Complete Connectivity Diagram u From: Coxeter [2], colored by Tom Ruen

Combining Two Cells  A highly confusing, intersecting mess! Add new cells on the inside ! All the edges of the first 5 cells. Starter cell with 4 tetra faces

Six More Cells !

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

The Full 11-Cell

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

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

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

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