SIGGRAPH 2007, San Diego

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SIGGRAPH 2007, San Diego. Carlo H. Séquin &amp; James F. Hamlin University of California, Berkeley. The Regular 4-Dimensional 11-Cell &amp; 57-Cell. 4 Dimensions ??. The 4 th dimension exists ! and it is NOT “time” !

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SIGGRAPH 2007, San Diego

Carlo H. Séquin & James F. Hamlin

University of California, Berkeley

The Regular 4-Dimensional

11-Cell & 57-Cell

4 Dimensions ??
• The 4th dimension exists !and it is NOT “time” !
• The 57-Cell is a complex, self-intersecting4-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 edgesare 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 cellsin the shape of some Platonic solid.
• Around every edge the same small numberof 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

120-Cell ( 600V, 1200E, 720F )
• Cell-first,extremeperspectiveprojection
• Z-Corp. model
Kepler-Poinsot “Solids” in 3-D

1 2 3 4

Gr. Dodeca, 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:

Cross-cap Steiner’s Roman Surface

Non-orientable, self-intersecting 2D manifolds

Models of the 2D Projective Plane Construct 2 regular 4D objects:the 11-Cell & the 57-Cell

Klein bottle

Hemi-icosahedron

connect oppositeperimeter 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’sRoman 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

2 cells

inner faces

3rd cell

4th cell

1 cell

5th cell

How Much Further to Go ?
• We have assembled only 5 of 11 cellsand 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.

• 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.
• 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

7 6 2

• 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 from 57 Hemi-dodecahedra
• 5 such single-sided cells join around edges
Hemi-dodecahedron

connect oppositeperimeter 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 Clusteraround 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 joinat each vertex

Each cell has six neighbors

Connectivity Graph of the 57-Cell (2)
• Thirty 2nd-nearest neighbors
• No 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 shellare connected as in a dodecahedron.
Hemi-cube (single-sided, not a solid any more!)

3 faces only

vertex graph K4