ISAMA 2007, Texas A&MISAMA 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 LanierJaron Lanier

Visitor to the College of Engineering, U.C. Berkeleyand the Center for Entrepreneurship & Technology

““Do you know about the Do you know about the 4-dimensional 11-Cell ? 4-dimensional 11-Cell ?

-- a regular polytope in 4-D space;-- a regular polytope in 4-D space;

can you help me visualize that thing ?”can you help me visualize that thing ?”

Ref. to some difficult group-theoretic math paperRef. to some difficult group-theoretic math paper

Phone call from Jaron Lanier, Dec. 15, 2006

What Is a Regular Polytope ?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 form each another.

Examples in 2D: Regular n-gons:

Regular Polyhedra in 3DRegular Polyhedra in 3D

The Platonic Solids:

There are only 5. Why ? …

Why Only 5 Platonic Solids ?Why Only 5 Platonic Solids ?

Lets try to build all possible ones: 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) now the edges are no longer all alike!

Let’s Build Some 4-D Polychora ...Let’s Build Some 4-D Polychora ...

By analogy with 3-D polyhedra:

each will be bounded by 3-D cellsin the shape of some Platonic solid;

at every vertex (edge) the same numberof Platonic cells will join together;

that number has to be small enough,so that some wedge of free space is left,

which then gets forcibly closedand thereby produces some bending into 4-D.

AllAll Regular Polychora in 4D 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 ?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.

Oblique ProjectionsOblique Projections

Cavalier Projection

3-D Cube 2-D 4-D Cube 3-D ( 2-D )

ProjectionsProjections: : VERTEXVERTEX / / EDGEEDGE / / FACEFACE // CELLCELL - First.- First.

3-D Cube:

Paralell proj.

Persp. proj.

4-D Cube:

Parallel proj.

Persp. proj.

Projections of a Hypercube to 3-DProjections 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-DThe 6 Regular Polytopes in 4-D

120-Cell 120-Cell ( 600V, 1200E, 720F )( 600V, 1200E, 720F )

Cell-first,extremeperspectiveprojection

Z-Corp. model

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

David Richter

An 11-Cell ???An 11-Cell ???

Another Regular 4-D Polychoron ?Another Regular 4-D Polychoron ?

I have just shown that there are only 6.

“11” feels like a weird number;typical numbers are: 8, 16, 24, 120, 600.

The notion of a 4-D 11-Cell seems bizarre!

Kepler-Poinsot SolidsKepler-Poinsot Solids

Mutually intersecting faces (all)

Faces in the form of pentagrams (3,4)

Gr. Dodeca, Gr. Icosa, Gr. Stell. Dodeca, Sm. Stell. Dodeca

1 2 3 4

But we can do even worse things ...

Hemicube Hemicube ((single-sidedsingle-sided, not a solid any more!), not a solid any more!)

If we are only concerned with topological connectivity, we can do weird things !

3 faces only vertex graph K4 3 saddle faces

Q

Hemi-dodecahedronHemi-dodecahedron

A self-intersecting, single-sided 3D cell Is only geometrically regular in 9D space

connect oppositeperimeter points

connectivity: Petersen graph

six warped pentagons

Hemi-icosahedronHemi-icosahedron

A self-intersecting, single-sided 3D cell Is only geometrically regular in 5D

THIS IS OUR BUILDING BLOCK !

connect oppositeperimeter points

connectivity: graph K6

5-D Simplex;warped octahedron

Cross-cap Model of the Projective PlaneCross-cap Model of the Projective Plane

All these Hemi-polyhedra have the topology of the Projective Plane ...

Cross-cap Model of the Projective PlaneCross-cap Model of the Projective Plane

Has one self-intersection crease,a so called Whitney Umbrella

Another Model of the Projective Plane:Another Model of the Projective Plane:Steiner’s Steiner’s Roman SurfaceRoman Surface

Has 6 Whitney umbrellas;tetrahedral symmetry.

Polyhedral model: An octahedronwith 4 tetrahedral faces removed, and 3 equatorial squares added.

Building Block: Hemi-icosahedronBuilding Block: Hemi-icosahedron

The Projective Plane can also be modeled with Steiner’s Roman Surface.

This leads to a different set of triangles used(exhibiting more symmetry).

Gluing Two Steiner-Cells TogetherGluing Two Steiner-Cells Together

Two cells share one triangle face

Together they use 9 vertices

Hemi-icosahedron

Adding More Cells . . .Adding More Cells . . .

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

Adding Cells SequentiallyAdding Cells Sequentially

1 cell 2 cells inner faces 3rd cell 4th cell 5th cell

How Much Further to Go ??How Much Further to Go ??

So far we have assembled: 5 of 11 cells;but engaged all vertices and all edges,and 40 out of all 55 triangular faces!

It is going to look busy (messy)!

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 ConstructionA More Symmetrical Construction Exploit the symmetry of the Steiner cell !

One Steiner cell 2nd cell added on “inside”Two cells with cut-out faces

4th white vertex used by next 3 cells

(central) 11th vertex used by last 6 cells

What is the Grand Plan ?What is the Grand 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.

3 cells join around every single edge.

Every pair of cells shares exactly one face.

The Basic Framework: 10-D SimplexThe Basic Framework: 10-D Simplex

10-D Simplex also has 11 vertices, 55 edges.

In 10-D space they can all have equal length.

11-Cell uses only 55 of 165 triangular faces.

Make a suitable projection from 10-D to 3-D;(maintain as much symmetry as possible).

Select 11 different colors for the 11 cells;(Color faces with the 2 colors of the 2 cells).

The Complete Connectivity DiagramThe Complete Connectivity Diagram

From: Coxeter [2], colored by Tom Ruen

Symmetrical Arrangements of 11 PointsSymmetrical Arrangements of 11 Points

3-sided prism 4-sided prism 5-sided prism

Now just add all 55 edges and a suitable set of 55 faces.

Point Placement Based on Plato ShellsPoint Placement Based on Plato Shells Try for even more symmetry !

1 + 4 + 6 vertices all 55 edges shown10 vertices on a sphere

Same scheme as derived from the Steiner cell !

The Full 11-CellThe Full 11-Cell

ConclusionsConclusions The way to learn to “see” the hendecachoron

is to try to understand its assembly process.

The way to do that is by pursuing several different approaches: Bottom-up: understand the building-block cell,

the hemi-icosahedron, and how a few of those fit together.

Top-down: understand the overall symmetry (K11),and the global connectivity of the cells.

An excellent application of hyper-seeing !

What Is the 11-Cell Good For ?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)

Are there More Polychora Like This ?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 am still trying to get my mind around it . . .

Artistic coloring by Jaron Lanier

Questions ?Questions ?

Building Block: Hemi-icosahedronBuilding Block: Hemi-icosahedron

Uses all the edges of the 5D simplexbut only half of the available faces.

Has the topology of the Projective Plane(like the Cross-Cap ).