SIGGRAPH 2007, San DiegoSIGGRAPH 2007, San Diego

The Regular 4-Dimensional

11-Cell & 57-CellCarlo H. Séquin & James F. Hamlin

University of California, Berkeley

4 Dimensions ??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 FranciscoSan Francisco

Cannot be understood from one single shot !

To Get to Know San FranciscoTo Get to Know San Francisco

need a rich assembly of impressions, then form an “image” in your mind...

“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 Polygons in 2 DimensionsRegular Polygons in 2 Dimensions

. . .

Regular Polyhedra in 3-DRegular Polyhedra in 3-D(made from regular 2-D n-gons)(made from regular 2-D n-gons)

The Platonic Solids:

There are only 5. Why ? …

Why Only 5 Platonic Solids ?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 Let’s Build Some 4-D Polychora “multi-cell”“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.

AllAll Regular “Platonic” Polychora in 4-D Regular “Platonic” Polychora in 4-DUsing Tetrahedra (Dihedral angle = 70.5°):

3 around an edge (211.5°) (5 cells) Simplex 4 around an edge (282.0°) (16 cells) Cross polytope5 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 ?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 )

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 Polychora in 4-DThe 6 Regular Polychora 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

Kepler-Poinsot “Solids” in 3-DKepler-Poinsot “Solids” in 3-D

Mutually intersecting faces (all above) Faces in the form of pentagrams (#3,4)

Gr. Dodeca, Gr. Icosa, Gr. Stell. Dodeca, Sm. Stell. Dodeca1 2 3 4

But in 4-D we can do even “crazier” things ...

Even “Weirder” Building Blocks:Even “Weirder” Building Blocks:Non-orientable, self-intersecting 2D manifolds

Cross-cap Steiner’s Roman Surface

Klein bottle

Models of the 2D Projective Plane

Construct 2 regular 4D objects:the 11-Cell & the 57-Cell

Hemi-icosahedronHemi-icosahedron

A self-intersecting, single-sided 3D cell Is only geometrically regular in 5D BUILDING BLOCK FOR THE 11-CELL

connect oppositeperimeter points

connectivity: graph K6

5-D Simplex;warped octahedron

The Hemi-icosahedral Building BlockThe Hemi-icosahedral Building Block

Steiner’sRoman Surface

Polyhedral model with 10 triangles

with cut-out face centers

10 triangles – 15 edges – 6 vertices

Gluing Two Steiner-Cells TogetherGluing Two Steiner-Cells Together

Two cells share one triangle face Together they use 9 vertices

Hemi-icosahedron

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 ?

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.

Start With the Overall Plan ...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-Cellhas 11 vertices, 55 edges, 55 faces, 11 cells.

Its edges form the complete graph K11 .

Start: Highly Symmetrical Vertex-SetStart: Highly Symmetrical Vertex-SetCenter Vertex + Tetrahedron + Octahedron

1 + 4 + 6 vertices all 55 edges shown

The Complete Connectivity DiagramThe Complete Connectivity Diagram

Based on [ Coxeter 1984, Ann. Disc. Math 20 ]

7 6 2

Views of the 11-CellViews of the 11-Cell

Solid faces Transparency

The Full 11-Cell The Full 11-Cell

– a building block of our universe ?

660 automorphisms

On to the 57-Cell . . .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-dodecahedronHemi-dodecahedron

A self-intersecting, single-sided 3D cell

BUILDING BLOCK FOR THE 57-CELL

connect oppositeperimeter points

connectivity: Petersen graph

six warped pentagons

Bottom-up Assembly of the 57-Cell (1)Bottom-up Assembly of the 57-Cell (1)

5 cells around a common edge (black)

Bottom-up Assembly of the 57-Cell (2)Bottom-up Assembly of the 57-Cell (2)

10 cells around a common (central) vertex

Vertex Vertex ClusterCluster

(v0)(v0)

10 cells with one corner at v0

Edge ClusterEdge Clusteraround v1-v0around v1-v0

+ vertex clusters at both ends.

Connectivity Graph of the 57-CellConnectivity 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)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)Connectivity Graph of the 57-Cell (3)

Every possible combination of 2 primary edges is used in a pentagonal face

Graphprojected into plane

Connectivity in shell 2 : truncated hemi-icosahedron

Connectivity Graph of the 57-Cell (4)Connectivity Graph of the 57-Cell (4)

Connectivity Graph of the 57-Cell (5)Connectivity Graph of the 57-Cell (5)

The 3 “shells” around a vertex Diameter of graph is 3

20 vertices

30 vertices

6 vertices

1 vertex

57 vertices total

Connectivity Graph of the 57-Cell (6)Connectivity Graph of the 57-Cell (6)

The 20 vertices in the outermost shellare connected as in a dodecahedron.

An “Aerial Shot” of the 57-CellAn “Aerial Shot” of the 57-Cell

A “Deconstruction” of the 57-CellA “Deconstruction” of the 57-Cell

E X T R AE X T R A

Hemi-cube Hemi-cube ((single-sidedsingle-sided, , not a solidnot a solid any more!) any more!)

Simplest object with the connectivity of the projective plane,

(But too simple to form 4-D polychora)

3 faces only vertex graph K4 3 saddle faces

Physical Model of a Hemi-cubePhysical Model of a Hemi-cube

Made on a Fused-Deposition Modeling Machine