Download Siggraph 2007 - People @ EECS at UC Berkeley

SIGGRAPH 2007, San Diego
The Regular 4-Dimensional
11-Cell & 57-Cell
Carlo H. Séquin & James F. Hamlin
University of California, Berkeley
4 Dimensions ??
4th dimension exists !
and it is NOT “time” !
 The

The 57-Cell is a complex, self-intersecting
4-dimensional geometrical object.

It cannot be explained
with a single image / model.
San Francisco

Cannot be understood from one single shot !
To Get to Know San Francisco

need a rich assembly of impressions,

then form an “image” in your mind...
Regular Polygons in 2 Dimensions

“Regular”
means: All the vertices and edges
are indistinguishable from each another.

There are infinitely many regular n-gons !
...

Use them to build regular 3D objects 
Regular Polyhedra in 3-D
(made from regular 2-D n-gons)
The Platonic Solids:
There are only 5. Why ? …
Why Only 5 Platonic Solids ?
Ways to build a regular convex corner:



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).
Let’s Build Some 4-D Polychora
“multi-cell”
By analogy with 3-D polyhedra:

Each will be bounded by 3-D cells
in the shape of some Platonic solid.

Around every edge the same small number
of Platonic cells will join together.
(That number has to be small enough,
so that some wedge of free space is left.)

This gap then gets forcibly closed,
thereby producing bending into 4-D.
All Regular “Platonic” Polychora in 4-D
Using Tetrahedra (Dihedral angle = 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) “600-Cell”
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) “120-Cell”
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:

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 Projections
 Cavalier
Projection
3-D Cube  2-D
4-D Cube  3-D ( 2-D )
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 Polychora in 4-D
120-Cell
( 600V, 1200E, 720F )
 Cell-first,
extreme
perspective
projection
 Z-Corp.
model
600-Cell
( 120V, 720E, 1200F ) (parallel proj.)
 David
Richter
Kepler-Poinsot “Solids” in 3-D
1
Gr. Dodeca,
2
3
4
Gr. Icosa,
Gr. Stell. Dodeca,
Sm. Stell. Dodeca

Mutually intersecting faces (all above)

Faces in the form of pentagrams (#3,4)
But in 4-D we can do even “crazier” things ...
Even “Weirder” Building Blocks:
Non-orientable, self-intersecting 2D manifolds
Cross-cap
Steiner’s Roman Surface
Models of the 2D Projective Plane
Klein bottle
 Construct 2 regular 4D objects:
the 11-Cell & the 57-Cell
Hemi-icosahedron
connect opposite
perimeter points
connectivity:
graph K6
5-D Simplex;
warped octahedron

A self-intersecting, single-sided 3D cell

Is only geometrically regular in 5D
 BUILDING BLOCK FOR THE 11-CELL
The Hemi-icosahedral Building Block
10 triangles – 15 edges – 6 vertices
Steiner’s
Roman Surface
Polyhedral
model with
10 triangles
with cut-out
face centers
Gluing Two Steiner-Cells Together
Hemi-icosahedron
 Two
cells share one triangle face
 Together
they use 9 vertices
Adding Cells Sequentially
1 cell 2 cells
inner faces 3rd cell 4th cell 5th cell
How Much Further to Go ?

We have assembled only 5 of 11 cells
and it is already looking busy (messy)!

This object cannot be “seen” in one model.
It must be “assembled” in your head.

Use different ways to understand it:
 Now try a “top-down” approach.
Start With the Overall 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-Cell
has 11 vertices, 55 edges, 55 faces, 11 cells.

Its edges form the complete graph K11 .
Start: Highly Symmetrical Vertex-Set
Center Vertex + Tetrahedron + Octahedron
1 + 4 + 6 vertices
all 55 edges shown
The Complete Connectivity Diagram
762

Based on [ Coxeter 1984, Ann. Disc. Math 20 ]
Views of the 11-Cell
Solid faces
Transparency
The Full 11-Cell
660 automorphisms
– a building block of our universe ?
On to the 57-Cell . . .
 It
has a much more complex connectivity!
 It
is also self-dual: 57 V, 171 E, 171 F, 57 C.
 Built
5
from 57 Hemi-dodecahedra
such single-sided cells join around edges
Hemi-dodecahedron
connect opposite
perimeter points

connectivity:
Petersen graph
six warped
pentagons
A self-intersecting, single-sided 3D cell
 BUILDING BLOCK FOR THE 57-CELL
Bottom-up Assembly of the 57-Cell (1)
5 cells around a common edge (black)
Bottom-up Assembly of the 57-Cell (2)
10 cells around a common (central) vertex
Vertex
Cluster
(v0)
 10
cells
with one
corner
at v0
Edge Cluster
around v1-v0
+ vertex clusters
at both ends.
Connectivity Graph of the 57-Cell

57-Cell is self-dual. Thus the graph of all its edges
also represents the adjacency diagram of its cells.
Six edges join
at each vertex
Each cell has
six neighbors
Connectivity Graph of the 57-Cell (2)
 Thirty
 No
2nd-nearest neighbors
loops yet (graph girth is 5)
Connectivity Graph of the 57-Cell (3)
Graph
projected
into plane

Every possible combination of 2 primary
edges is used in a pentagonal face
Connectivity Graph of the 57-Cell (4)
Connectivity in shell 2 :  truncated hemi-icosahedron
Connectivity Graph of the 57-Cell (5)
20 vertices
30 vertices
6 vertices
1 vertex
57 vertices
total
 The
3 “shells” around a vertex
 Diameter
of graph is 3
Connectivity Graph of the 57-Cell (6)

The 20 vertices in the outermost shell
are connected as in a dodecahedron.
An “Aerial Shot” of the 57-Cell
A “Deconstruction” of the 57-Cell
EXTRA
Hemi-cube
3 faces only
(single-sided, not a solid any more!)
vertex graph K4
3 saddle faces
Simplest object with the connectivity of the projective plane,
(But too simple to form 4-D polychora)
Physical Model of a Hemi-cube
Made on a Fused-Deposition Modeling Machine