4D Polytopes and 3D Models of Them

George W. Hart

Stony Brook University

Goals of This Talk

• Expand your thinking.• Visualization of four- and higher-dimensional

objects.• Show Rapid Prototyping of complex structures.

Note: Some Material and images adapted from Carlo Sequin

What is the 4th Dimension ?

Some people think:

“it does not really exist”

“it’s just a philosophical notion”

“it is ‘TIME’ ”

. . .

But, a geometric fourth dimension is as useful and as real as 2D or 3D.

Higher-dimensional Spaces

Coordinate Approach:• A point (x, y, z) has 3 dimensions.• n-dimensional point: (d1, d2, d3, d4, ..., dn).

Axiomatic Approach:

• Definition, theorem, proof...

Descriptive Geometry Approach:• Compass, straightedge, two sheets of paper.

What Is a Regular Polytope?• “Polytope” is the generalization of the

terms “Polygon” (2D), “Polyhedron” (3D), … to arbitrary dimensions.

• “Regular” means: All the vertices, edges, faces… are equivalent.

• Assume convexity for now.• Examples in 2D: Regular n-gons:

Regular Convex Polytopes in 3D

The Platonic Solids:

There are only 5. Why ? …

Why Only 5 Platonic Solids ?Try to build all possible ones:• from triangles:

3, 4, or 5 around a corner;• from squares:

only 3 around a corner;• from pentagons:

only 3 around a corner;• from hexagons: floor tiling, does not close.

• higher n-gons: do not fit around vertex without undulations (not convex)

Constructing a (d+1)-D Polytope

Angle-deficit = 90°

creates a 3D corner creates a 4D corner

?

2D

3D 4D

3D

Forcing closure:

“Seeing a Polytope”• Real “planes”, “lines”, “points”, “spheres”, …, do

not exist physically.• We understand their properties and relationships

as ideal mental models. • Good projections are very useful. Our visual input

is only 2D, but we understand as 3D via mental construction in the brain.

• You are able to “see” things that don't really exist in physical 3-space, because you “understand” projections into 2-space, and you form a mental model.

• We will use this to visualize 4D Polytopes.

Projections• Set the coordinate values of all unwanted

dimensions to zero, e.g., drop z, retain x,y, and you get a orthogonal projection along the z-axis. i.e., a 2D shadow.

• Linear algebra allows arbitrary direction.• Alternatively, use a perspective projection:

rays of light form cone to eye. • Can add other depth queues: width of

lines, color, fuzziness, contrast (fog) ...

Wire Frame Projections

• Shadow of a solid object is mostly a blob. • Better to use wire frame, so we can see

components.

Oblique Projections• Cavalier Projection

3D Cube 2D 4D Cube 3D ( 2D )

Projections: VERTEX / EDGE / FACE / CELL – centered

• 3D Cube:

Paralell proj.

Persp. proj.

• 4D Cube:

Parallel proj.

Persp. proj.

3D Objects Need Physical Edges

Options:• Round dowels (balls and stick)• Profiled edges – edge flanges convey a

sense of the attached face• Flat tiles for faces

– with holes to make structure see-through.

Edge Treatments

(Leonardo Da Vinci)

How Do We Find All 4D Polytopes?

• Sum of dihedral angles around each edge must be less than 360 degrees.

• Use the Platonic solids as “cells” Tetrahedron: 70.5°

Octahedron: 109.5°

Cube: 90°

Dodecahedron: 116.5°

Icosahedron: 138.2°.

All Regular Convex 4D PolytopesUsing Tetrahedra (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)

Using Cubes (90°): 3 around an edge (270.0°) (8 cells) Hypercube

Using Octahedra (109.5°): 3 around an edge (328.5°) (24 cells)

Using Dodecahedra (116.5°): 3 around an edge (349.5°) (120 cells)

Using Icosahedra (138.2°): none: angle too large.

5-Cell or 4D Simplex• 5 cells, 10 faces, 10 edges, 5 vertices.

Carlo Sequin

Can make with Zometool also

16-Cell or “4D Cross Polytope”• 16 cells, 32 faces, 24 edges, 8 vertices.

4D Hypercube or “Tessaract”• 8 cells, 24 faces, 32 edges, 16 vertices.

Hypercube, Perspective Projections

Nets: 11 Unfoldings of Cube

Hypercube Unfolded -- “Net”

One of the 261 different unfoldings

Corpus Hypercubus

Salvador Dali

“Unfolded”Hypercube

24-Cell• 24 cells, 96 faces, 96 edges, 24 vertices.• (self-dual).

24-Cell “Net” in 3D

Andrew Weimholt

120-Cell

• 120 cells, 720 faces, 1200 edges, 600 vertices.

• Cell-first parallel projection,(shows less than half of the edges.)

120-cell Model

Marc Pelletier

120-Cell

Thin face frames, Perspective projection.

Carlo Séquin

120-Cell – perspective projection

(smallest ?) 120-Cell

Wax model, made on Sanders machine

120-Cell – perspective projection

Selective laser sintering

3D Printing — Zcorp

120-Cell, “exploded”

Russell Towle

120-Cell Soap Bubble

John Sullivan

Stereographic projection preserves 120 degree angles

120-Cell “Net”

with stack of 10 dodecahedra

George Olshevski

600-Cell -- 2D projection

• Oss, 1901

Frontispiece of Coxeter’s book “Regular Polytopes,”

• Total: 600 tetra-cells, 1200 faces, 720 edges, 120 vertices.

• At each Vertex: 20 tetra-cells, 30 faces, 12 edges.

600-Cell

Cross-eye Stereo Picture by Tony Smith

600-Cell

• Dual of 120 cell.• 600 cells,

1200 faces, 720 edges, 120 vertices.

• Cell-first parallel projection,shows less than half of the edges.

• Can make with Zometool

600-Cell

Straw model by David Richter

Slices through the 600-Cell

At each Vertex: 20 tetra-cells, 30 faces, 12 edges.

Gordon Kindlmann

History3D Models of 4D Polytopes

• Ludwig Schlafli discovered them in 1852. Worked algebraically, no pictures in his paper. Partly published in 1858 and 1862 (translation by Cayley) but not appreciated.

• Many independent rediscoveries and models.

Stringham (1880)

• First to rediscover all six• His paper shows cardboard models of layers

3 layers of 120-cell(45 dodecahedra)

Victor Schlegel (1880’s)Invented “Schlegel Diagram”3D 2D perspective transf.

Used analogous 4D 3Dprojection in educational models.

Built wire and thread models.

Advertised and sold models via commercial catalogs: Dyck (1892) and Schilling (1911).

Some stored at Smithsonian.

Five regular polytopes

Sommerville’s Description of Models

“In each case, the external boundary of the projection represents one of the solid boundaries of the figure. Thus the 600-cell, which is the figure bounded by 600 congruent regular tetrahedra, is represented by a tetrahedron divided into 599 other tetrahedra … At the center of the model there is a tetrahedron, and surrounding this are successive zones of tetrahedra. The boundaries of these zones are more or less complicated polyhedral forms, cardboard models of which, constructed after Schlegel's drawings, are also to be obtained by the same firm.”

Cardboard Models of 120-Cell

From Walther Dyck’s 1892 Math and Physics Catalog

Paul S. Donchian’s Wire Models• 1930’s

• Rug Salesman with “visions”

• Wires doubled to show how front overlays back

• Widely displayed

• Currently on view at the Franklin Institute

Zometool

• 1970 Steve Baer designed and produced "Zometool" for architectural modeling

• Marc Pelletier discovered (when 17 years old) that the lengths and directions allowed by this kit permit the construction of accurate models of the 120-cell and related polytopes.

• The kit went out of production however, until redesigned in plastic in 1992.

120 Cell

• Zome Model• Orthogonal

projection

Uniform 4D Polytopes

• Analogous to the 13 Archimedean Solids • Allow more than one type of cell• All vertices equivalent• Alicia Boole Stott listed many in 1910• Now over 8000 known • Cataloged by George Olshevski and

Jonathan Bowers

Truncated 120-Cell

Truncated 120-Cell - Stereolithography

Zometool Truncated 120-Cell

MathCamp 2000

Ambo 600-Cell

Bridges Conference, 2001

Ambo 120-Cell

Orthogonal projection

Stereolithography

Can do with Zome

Expanded 120-Cell

Mira Bernstein,Vin de Silva, et al.

Expanded Truncated 120-Cell

Big Polytope

“Net”

George Olshevski

Big Polytope Zome Model

Steve Rogers

48 Truncated Cubes

Poorly designed FDM model

Prism on a Snub Cube – “Net”

George Olshevski

Duo-Prisms - “Nets”

Robert WebbAndrew Weimholt

Andrew WeimholtGeorge Olshevski

Grand Antiprism “Net”

with stack of 10 pentagonal antiprisms

George Olshevski

Non-Convex Polytopes

Jonathan Bowers

• Components may pass

through each other

• Slices may be useful for visualization

• Slices may be disconnected

Beyond 4 Dimensions …

• What happens in higher dimensions ?• How many regular polytopes are there

in 5, 6, 7, … dimensions ?• Only three regular types:

– Hypercubes — e.g., cube– Simplexes — e.g., tetrahedron– Cross polytope — e.g., octahedron

Hypercubes• A.k.a. “Measure Polytope”• Perpendicular extrusion in nth direction:

1D 2D 3D 4D

Orthographic Projections

Parallel lines remain parallel

Simplex Series• Connect all the dots among n+1 equally

spaced vertices:(Put next one “above” center of gravity).

1D 2D 3D

This series also goes on indefinitely.

7D Simplex

A warped cube avoids intersecting diagonals.

Up to 6D can be constructed with Zometool.

Open problem: 7D constructible with Zometool?

Cross Polytope Series• Place vertex in + and – direction on each axis,

a unit-distance away from origin. • Connect all vertex pairs that lie on different axes.

1D 2D 3D 4D

A square frame for every pair of axes

6 square frames= 24 edges

6D Cross Polytope12 vertices suggests using icosahedron

Can do with Zometool.

6D Cross Polytope

Chris Kling

Some References• Ludwig Schläfli: “Theorie der vielfachen

Kontinuität,” 1858, (published in 1901).• H. S. M. Coxeter: “Regular Polytopes,” 1963,

(Dover reprint).• Tom Banchoff, Beyond the Third Dimension,

1990.• G.W. Hart, “4D Polytope Projection Models by

3D Printing” to appear in Hyperspace. • Carlo Sequin, “3D Visualization Models of the

Regular Polytopes…”, Bridges 2002.

Puzzle• Which of these shapes can / cannot

be folded into a 4D hypercube? • Hint: Hold the red cube still and fold

the others around it.

Scott Kim