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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 4 th dimension exists ! and it is NOT “time” !

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siggraph 2007 san diego
SIGGRAPH 2007, San Diego

Carlo H. Séquin & James F. Hamlin

University of California, Berkeley

The Regular 4-Dimensional

11-Cell & 57-Cell

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 francisco
San Francisco
  • Cannot be understood from one single shot !
to get to know san francisco
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 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
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
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
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
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
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
Oblique Projections
  • Cavalier Projection

3-D Cube  2-D

4-D Cube  3-D ( 2-D )

projections of a hypercube to 3 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
120-Cell ( 600V, 1200E, 720F )
  • Cell-first,extremeperspectiveprojection
  • Z-Corp. model
kepler poinsot solids in 3 d
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
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
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
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
Gluing Two Steiner-Cells Together

Hemi-icosahedron

  • Two cells share one triangle face
  • Together they use 9 vertices
adding cells sequentially

2 cells

inner faces

3rd cell

4th cell

1 cell

5th cell

Adding Cells Sequentially
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 set
Start: Highly Symmetrical Vertex-Set

Center Vertex+Tetrahedron+Octahedron

1 + 4 + 6 vertices

all 55 edges shown

the complete connectivity diagram
The Complete Connectivity Diagram

7 6 2

  • Based on [ Coxeter 1984, Ann. Disc. Math 20 ]
views of the 11 cell
Views of the 11-Cell

Solid faces Transparency

the full 11 cell
The Full 11-Cell

660 automorphisms

– a building block of our universe ?

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 dodecahedron
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
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 cluster v0
Vertex Cluster(v0)
  • 10 cells with one corner at v0
edge cluster around v1 v0
Edge Clusteraround v1-v0

+ vertex clusters at both ends.

connectivity graph of the 57 cell
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
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)

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 Graph of the 57-Cell (4)

Connectivity in shell 2 :  truncated hemi-icosahedron

connectivity graph of the 57 cell 5
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
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
Hemi-cube (single-sided, not a solid any more!)

3 faces only

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
Physical Model of a Hemi-cube

Made on a Fused-Deposition Modeling Machine