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 ).
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