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4D Polytopes and 3D Models of Them

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4D Polytopesand 3D Models of Them

George W. Hart

Stony Brook University

- 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

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.

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.

- “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:

The Platonic Solids:

There are only 5. Why ? …

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)

Angle-deficit = 90°

2D

3D

Forcing closure:

?

3D

4D

creates a 3D corner

creates a 4D corner

- 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.

- 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) ...

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

- Cavalier Projection

3D Cube 2D

4D Cube 3D ( 2D )

- 3D Cube:
Paralell proj.

Persp. proj.

- 4D Cube:
Parallel proj.

Persp. proj.

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.

(Leonardo Da Vinci)

- 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°.

Using Tetrahedra (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)

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 cells, 10 faces, 10 edges, 5 vertices.

Carlo Sequin

Can make with Zometool also

- 16 cells, 32 faces, 24 edges, 8 vertices.

- 8 cells, 24 faces, 32 edges, 16 vertices.

Hypercube, Perspective Projections

One of the 261 different unfoldings

Salvador Dali

“Unfolded”Hypercube

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

Andrew Weimholt

- 120 cells, 720 faces, 1200 edges, 600 vertices.
- Cell-first parallel projection,(shows less than half of the edges.)

Marc Pelletier

Carlo Séquin

Thin face frames, Perspective projection.

Wax model, made on Sanders machine

Selective laser sintering

Russell Towle

John Sullivan

Stereographic projection preserves 120 degree angles

with stack of 10 dodecahedra

George Olshevski

- Total: 600 tetra-cells, 1200 faces, 720 edges, 120 vertices.
- At each Vertex: 20 tetra-cells, 30 faces, 12 edges.

- Oss, 1901

Frontispiece of Coxeter’s book “Regular Polytopes,”

Cross-eye Stereo Picture by Tony Smith

- 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

Straw model by David Richter

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

Gordon Kindlmann

- 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.

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

3 layers of 120-cell

(45 dodecahedra)

Invented “Schlegel Diagram”

3D 2D perspective transf.

Used analogous 4D 3D

projection 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

“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.”

From Walther Dyck’s 1892 Math and Physics Catalog

- 1930’s
- Rug Salesman with
- “visions”
- Wires doubled to show how front overlays back
- Widely displayed
- Currently on view at the Franklin Institute

- 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.

- Zome Model
- Orthogonal projection

- 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

MathCamp 2000

Ambo 600-Cell

Bridges Conference, 2001

Orthogonal projection

Stereolithography

Can do with Zome

Mira Bernstein,

Vin de Silva, et al.

George Olshevski

Steve Rogers

Poorly designed FDM model

George Olshevski

Andrew Weimholt

George Olshevski

Andrew Weimholt

Robert Webb

with stack of 10 pentagonal antiprisms

George Olshevski

Non-Convex Polytopes

- Components may pass
- through each other
- Slices may be useful
- for visualization
- Slices may be
- disconnected

Jonathan Bowers

- What happens in higher dimensions ?
- How many regular polytopes are therein 5, 6, 7, … dimensions ?
- Only three regular types:
- Hypercubes — e.g., cube
- Simplexes — e.g., tetrahedron
- Cross polytope — e.g., octahedron

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

1D 2D 3D 4D

Parallel lines remain parallel

- 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.

A warped cube avoids intersecting diagonals.

Up to 6D can be constructed with Zometool.

Open problem: 7D constructible with Zometool?

- 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

12 vertices suggestsusing icosahedron

Can do with Zometool.

Chris Kling

- 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.

- 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