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A 10-dimensional Jewel

EECS Computer Science DivisionEECS Computer Science DivisionUniversity of California, BerkeleyUniversity of California, Berkeley

Carlo H. Séquin

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 from each another.

Examples in 2D: Regular n-gons:

There are infinitely many of them!

In 3 Dimensions . . . In 3 Dimensions . . .

There are only 5 Platonic solids:

They are composed from the regular 2D polygons.

Only triangles, squares, and pentagons are useful: other n-gons are too ”round”;they cannot form nice 3D corners.

In 4D Space . . .In 4D Space . . .

The same constructive approach continues:

We can use the Platonic solids as building blocks to form the “crust” of regular 4D polychora.

Only 6 constructions are successful.

Only 4 of the 5 Platonic solids can be used; the icosahedron is too round (dihedral angle > 120°).

This is the result . . .

The 6 Regular Polychora in 4-D . . .The 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

In Higher-Dimensional Spaces . . . In Higher-Dimensional Spaces . . .

We can recursively construct new regular polytopes

Using the ones from one dimension lower spceas the boundary (“surface”) element.

But from dimension 5 onwards, there are just 3 each:

N-Simplices (like tetrahedron)

N-Cubes (hypercubes, measure-polytopes)

N-Orthoplexes (cross-polytopes = duals of n-cube)

Thinking Outside the Box . . .Thinking Outside the Box . . .

Allow polyhedron faces to intersect . . .

or even to be self-intersecting:

In 3D: In 3D: Kepler-Poinsot Solids Kepler-Poinsot Solids

Mutually intersecting faces: (all)

Faces in the form of pentagrams: (3,4)

+ 10 such objects in 4D space !

Gr. Dodeca, Gr. Icosa, Gr. Stell. Dodeca, Sm. Stell. Dodeca

1 2 3 4

Single-Sided Polychora in 4DSingle-Sided Polychora in 4D

Let’s allow single-sided polytope constructions like a Möbius band or a Klein bottle.

In 4D we can make objects that close on themselves;they have the topology of the projective plane.

The simplest one is the hemi-cube . . .

But we can do even wilder things ...

Hemi-CubeHemi-Cube Single-sided; not a solid any more!

Has the connectivity of the projective plane!

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

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 the BUILDING BLOCK for the 10D JEWEL !

connect oppositeperimeter points

connectivity: graph K6

5-D simplex;warped octahedron

The Complete Connectivity DiagramThe Complete Connectivity Diagram

From: Coxeter [2], colored by Tom Ruen

Combining Two CellsCombining Two Cells

A highly confusing, intersecting mess!

Add new cells on the inside !

All the edges of the first 5 cells.

Starter cell with4 tetra faces

Six More Cells !Six More Cells !

Regular Hendecachoron (11-Cell)Regular Hendecachoron (11-Cell)

11 vertices, 55 edges, 55 faces, 11 cells self dual

Solid faces Transparency

The Full 11-CellThe Full 11-Cell

The The 1010D Jewel D Jewel

– a building block of our universe ?

660 automorphisms

Hands-on Construction ProjectHands-on Construction Project

This afternoon we will build card-board models of the hemi-icosahedron.

Thanks to Chris Palmer (now at U.C. Berkeley):

for designing the parameterized template and for laser cutting the 30 colored parts.

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 may talk about it at G4G57 . . .