ISAMA 2007, Texas A&M Hyper-Seeing the Regular Hendeca-choron. (= 11-Cell) Carlo H. Séquin & Jaron Lanier CS Division & CET; College of Engineering University of California, Berkeley

Jaron Lanier Visitor to the College of Engineering, U.C. Berkeley and the Center for Entrepreneurship & Technology

“Do you know about the 4-dimensional 11-Cell ? -- a regular polytope in 4-D space; can you help me visualize that thing ?” Ref. to some difficult group-theoretic math paper Phone call from Jaron Lanier, Dec. 15, 2006

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 form each another. u Examples in 2D: Regular n-gons:

Regular Polyhedra in 3D The Platonic Solids: There are only 5. Why ? …

Why Only 5 Platonic Solids ? Lets try to build all possible ones: 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)  now the edges are no longer all alike!

Let’s Build Some 4-D Polychora... By analogy with 3-D polyhedra: u each will be bounded by 3-D cells in the shape of some Platonic solid; u at every vertex (edge) the same number of Platonic cells will join together; u that number has to be small enough, so that some wedge of free space is left, u which then gets forcibly closed and thereby produces some bending into 4-D.

All Regular Polychora 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) 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) Using Icosahedra (138.2°):  NONE: angle too large (414.6°).

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.

Oblique Projections u Cavalier Projection 3-D Cube  2-D4-D Cube  3-D (  2-D )

Projections : VERTEX / EDGE / FACE / CELL - First. u 3-D Cube: Paralell proj. Persp. proj. u 4-D Cube: Parallel proj. Persp. proj.

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

The 6 Regular Polytopes 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

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!

Kepler-Poinsot Solids u Mutually intersecting faces (all) u Faces in the form of pentagrams (3,4) Gr. Dodeca, Gr. Icosa, Gr. Stell. Dodeca, Sm. Stell. Dodeca But we can do even worse things...

Hemicube (single-sided, not a solid any more!) u If we are only concerned with topological connectivity, we can do weird things ! 3 faces onlyvertex graph K 4 3 saddle faces Q

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 OUR BUILDING BLOCK ! connect opposite perimeter points connectivity: graph K 6 5-D Simplex; warped octahedron

Cross-cap Model of the Projective Plane u All these Hemi-polyhedra have the topology of the Projective Plane...

Cross-cap Model of the Projective Plane u Has one self-intersection crease, a so called Whitney Umbrella

Another Model of the Projective Plane: Steiner’s Roman Surface u Has 6 Whitney umbrellas; tetrahedral symmetry. u Polyhedral model: An octahedron with 4 tetrahedral faces removed, and 3 equatorial squares added.

Building Block: Hemi-icosahedron u The Projective Plane can also be modeled with Steiner’s Roman Surface. u This leads to a different set of triangles used (exhibiting more symmetry).

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

Adding More Cells... 2 Cells+ Yellow Cell = 3 Cells + Cyan, Magenta = 5 Cells u Must never add more than 3 faces around an edge!

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

How Much Further to Go ?? u So far we have assembled: 5 of 11 cells; but engaged all vertices and all edges, and 40 out of all 55 triangular faces! u It is going to look busy (messy)! u This object can only be “assembled” in your head ! You will not be able to “see” it ! (like learning a city by walking around in it).

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

What is the Grand Plan ? u We know from: H.S.M. Coxeter: A Symmetrical Arrangement of Eleven Hemi-Icosahedra. Annals of Discrete Mathematics 20 (1984), pp u The regular 4-D 11-Cell has 11 vertices, 55 edges, 55 faces, 11 cells. u 3 cells join around every single edge. u Every pair of cells shares exactly one face.

The Basic Framework: 10-D Simplex u 10-D Simplex also has 11 vertices, 55 edges. u In 10-D space they can all have equal length. u 11-Cell uses only 55 of 165 triangular faces. u Make a suitable projection from 10-D to 3-D; (maintain as much symmetry as possible). u Select 11 different colors for the 11 cells; (Color faces with the 2 colors of the 2 cells).

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

Symmetrical Arrangements of 11 Points 3-sided prism 4-sided prism 5-sided prism u Now just add all 55 edges and a suitable set of 55 faces.

Point Placement Based on Plato Shells u Try for even more symmetry ! verticesall 55 edges shown 10 vertices on a sphere Same scheme as derived from the Steiner cell !

The Full 11-Cell

Conclusions u The way to learn to “see” the hendecachoron is to try to understand its assembly process. u The way to do that is by pursuing several different approaches: l Bottom-up: understand the building-block cell, the hemi-icosahedron, and how a few of those fit together. l Top-down: understand the overall symmetry (K 11 ), and the global connectivity of the cells. u An excellent application of hyper-seeing !

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 am still trying to get my mind around it...

u Artistic coloring by Jaron Lanier Questions ?

Building Block: Hemi-icosahedron u Uses all the edges of the 5D simplex but only half of the available faces. u Has the topology of the Projective Plane (like the Cross-Cap ).