BRIDGES, July 2002

CHSUCB BRIDGES, July 2002BRIDGES, July 2002

3D Visualization Models of the Regular Polytopes

in Four and Higher Dimensions .

Carlo H. Séquin

University of California, Berkeley

CHSUCB Goals of This TalkGoals of This Talk

Expand your thinking.

Teach you “hyper-seeing,”seeing things that one cannot ordinarily see, in particular: Four- and higher-dimensional objects.

NOT an original math research paper !(facts have been known for >100 years)NOT a review paper on literature …(browse with “regular polyhedra” “120-Cell”)

Also: Use of Rapid Prototyping in math.

CHSUCB A Few Key References …A Few Key References …

Ludwig Schläfli: “Theorie der vielfachen Kontinuität,” Schweizer Naturforschende Gesellschaft, 1901.

H. S. M. Coxeter: “Regular Polytopes,” Methuen, London, 1948.

John Sullivan: “Generating and rendering four-dimensional polytopes,” The Mathematica Journal, 1(3): pp76-85, 1991.

Thanks to George Hart for data on 120-Cell, 600-Cell, inspiration.

CHSUCB What is the 4th Dimension ?What is the 4th Dimension ?

Some people think:

“it does not really exist,”

“it’s just a philosophical notion,”

“it is ‘TIME’ ,”

. . .

But, it is useful and quite real!

CHSUCB Higher-dimensional SpacesHigher-dimensional Spaces

Mathematicians Have No Problem:

A point P(x, y, z) in this room isdetermined by: x = 2m, y = 5m, z = 1.5m; has 3 dimensions.

Positions in other data sets P = P(d1, d2, d3, d4, ... dn).

Example #1: Telephone Numbersrepresent a 7- or 10-dimensional space.

Example #2: State Space: x, y, z, vx, vy, vz ...

CHSUCB Seeing Mathematical ObjectsSeeing Mathematical Objects

Very big point

Large point

Small point

Tiny point

Mathematical point

CHSUCB Geometrical View of Dimensions Geometrical View of Dimensions

Read my hands …(inspired by Scott Kim, ca 1977).

CHSUCB

CHSUCB What Is a Regular PolytopeWhat 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 indistinguishable form each another.

Examples in 2D: Regular n-gons:

CHSUCB Regular Polytopes in 3DRegular Polytopes in 3D

The Platonic Solids:

There are only 5. Why ? …

CHSUCB Why Only 5 Platonic Solids ?Why Only 5 Platonic Solids ?

Lets 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 (forming saddles) now the edges are no longer all alike!

CHSUCB Do All 5 Conceivable Objects Exist?Do All 5 Conceivable Objects Exist?

I.e., do they all close around the back ?

Tetra base of pyramid = equilateral triangle.

Octa two 4-sided pyramids.

Cube we all know it closes.

Icosahedron antiprism + 2 pyramids (are vertices at the sides the same as on top ?)Another way: make it from a cube with six lineson the faces split vertices symmetricallyuntil all are separated evenly.

Dodecahedron is the dual of the Icosahedron.

CHSUCB Constructing a Constructing a (d+1)(d+1)-D Polytope-D Polytope

Angle-deficit = 90°

creates a 3D corner creates a 4D corner

?

2D

3D 4D

3D

Forcing closure:

CHSUCB ““Seeing a Polytope”Seeing a Polytope”

I showed you the 3D Platonic Solids …But which ones have you actually seen ?

For some of them you have only seen projections. Did that bother you ??

Good projections are almost as good as the real thing. Our visual input after all is only 2D. -- 3D viewing is a mental reconstruction in your brain, -- that is where the real "seeing" is going on !

So you were able to see things that "didn't really exist" in physical 3-space, because you saw good enough “projections” into 2-space, yet you could still form a mental image ==> “Hyper-seeing.”

We will use this to see the 4D Polytopes.

CHSUCB ProjectionsProjections

How do we make “projections” ?

Simplest approach: set the coordinate values of all unwanted dimensions to zero, e.g., drop z, retain x,y, and you get a parallel projection along the z-axis. i.e., a 2D shadow.

Alternatively, use a perspective projection: back features are smaller depth queue. Can add other depth queues: width of beams, color, fuzziness, contrast (fog) ...

CHSUCB Wire Frame ProjectionsWire Frame Projections

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.

CHSUCB Oblique ProjectionsOblique Projections

Cavalier Projection

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

CHSUCB ProjectionsProjections: : VERTEXVERTEX / / EDGEEDGE / / FACEFACE // CELL CELL - First.- First.

3D Cube:

Paralell proj.

Persp. proj.

4D Cube:

Parallel proj.

Persp. proj.

CHSUCB 3D Models Need Physical Edges3D Models Need Physical Edges

Options:

Round dowels (balls and stick)

Profiled edges – edge flanges convey a sense of the attached face

Actual composition from flat tiles– with holes to make structure see-through.

CHSUCB Edge TreatmentsEdge Treatments

Leonardo DaVinci – George Hart

CHSUCB How Do We Find All 4D Polytopes?How Do We Find All 4D Polytopes?

Reasoning by analogy helps a lot:-- How did we find all the Platonic solids?

Use the Platonic solids as “tiles” and ask:

What can we build from tetrahedra?

From cubes?

From the other 3 Platonic solids?

Need to look at dihedral angles!

Tetrahedron: 70.5°, Octahedron: 109.5°, Cube: 90°, Dodecahedron: 116.5°, Icosahedron: 138.2°.

CHSUCB All Regular Polytopes in 4DAll Regular Polytopes in 4D

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) Hyper-octahedron

Using Dodecahedra (116.5°):

3 around an edge (349.5°) (120 cells)

Using Icosahedra (138.2°):

none: angle too large (414.6°).

CHSUCB 5-Cell or Simplex in 4D5-Cell or Simplex in 4D

5 cells, 10 faces, 10 edges, 5 vertices. (self-dual).

CHSUCB 4D Simplex4D Simplex

Using Polymorf TM Tiles

Additional tiles made on our FDM machine.

CHSUCB 16-Cell or “Cross Polytope” in 4D16-Cell or “Cross Polytope” in 4D

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

CHSUCB 4D Cross Polytope4D Cross Polytope

Highlighting the eight tetrahedra from which it is composed.


CHSUCB Hypercube or Tessaract in 4DHypercube or Tessaract in 4D


(Dual of 16-Cell).

CHSUCB 4D Hypercube4D Hypercube

Using PolymorfTM Tiles

made byKiha Leeon FDM.

CHSUCB Corpus HypercubusCorpus Hypercubus

Salvador Dali

“Unfolded”Hypercube

CHSUCB 24-Cell in 4D24-Cell in 4D

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

CHSUCB

24-Cell, showing 3-fold symmetry24-Cell, showing 3-fold symmetry

CHSUCB 24-Cell “Fold-out” in 3D24-Cell “Fold-out” in 3D

Andrew Weimholt



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

CHSUCB 120 Cell120 Cell

Hands-on workshop with George Hart

CHSUCB 120-Cell120-Cell

Thin face frames, Perspective projection.

Séquin(1982)


Cell-first,extremeperspectiveprojection

Z-Corp. model

CHSUCB (smallest ?) 120-Cell(smallest ?) 120-Cell

Wax model, made on Sanders machine

CHSUCB Radial Projections of the 120-CellRadial Projections of the 120-Cell

Onto a sphere, and onto a dodecahedron:

CHSUCB 120-Cell, “exploded”120-Cell, “exploded”

Russell Towle

CHSUCB 120-Cell Soap Bubble120-Cell Soap Bubble

John Sullivan

CHSUCB 600-Cell, A Classical Rendering600-Cell, A Classical Rendering

Oss, 1901

Frontispiece of Coxeter’s 1948 book “Regular Polytopes,”and John Sullivan’s Paper “The Story of the 120-Cell.”

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

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


Cross-eye Stereo Picture by Tony Smith


Dual of 120 cell.


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


David Richter

CHSUCB Slices through the 600-CellSlices through the 600-Cell

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

Gordon Kindlmann


Cell-first, parallel projection,

Z-Corp. model

CHSUCB Model FabricationModel Fabrication

Commercial Rapid Prototyping Machines:

Fused Deposition Modeling (Stratasys)

3D-Color Printing (Z-corporation)

CHSUCB Fused Deposition ModelingFused Deposition Modeling

CHSUCB Zooming into the FDM MachineZooming into the FDM Machine

CHSUCB SFF: 3D Printing -- PrincipleSFF: 3D Printing -- Principle

Selectively deposit binder droplets onto a bed of powder to form locally solid parts.

Powder Spreading Printing

Build

Feeder

Powder

Head

CHSUCB 3D Printing:3D Printing: Z CorporationZ Corporation

CHSUCB 3D Printing:3D Printing: Z CorporationZ Corporation

Cleaning up in the de-powdering station

CHSUCB Designing 3D Edge ModelsDesigning 3D Edge Models

Is not totally trivial …

because of shortcomings of CAD tools:

Limited Rotations – weird angles

Poor Booleans – need water tight shells

CHSUCB How We Did It …How We Did It …

SLIDE (Jordan Smith, U.C.Berkeley)

Some “cheating” …

Exploiting the strength and weaknesses of the specific programs that drive the various rapid prototyping machines.

CHSUCB Beyond 4 Dimensions …Beyond 4 Dimensions …

What happens in higher dimensions ?

How many regular polytopes are therein 5, 6, 7, … dimensions ?

CHSUCB Polytopes in Higher DimensionsPolytopes in Higher Dimensions

Use 4D tiles, look at “dihedral” angles between cells:

5-Cell: 75.5°, Tessaract: 90°, 16-Cell: 120°, 24-Cell: 120°, 120-Cell: 144°, 600-Cell: 164.5°.

Most 4D polytopes are too round …

But we can use 3 or 4 5-Cells, and 3 Tessaracts.

There are three methods by which we can generate regular polytopes for 5D and all higher dimensions.

CHSUCB Hypercube SeriesHypercube Series

“Measure Polytope” Series(introduced in the pantomime)

Consecutive perpendicular sweeps:

1D 2D 3D 4D

This series extents to arbitrary dimensions!

CHSUCB

Simplex SeriesSimplex Series

Connect all the dots among n+1 equally spaced vertices:(Find next one above COG).

1D 2D 3D

This series also goes on indefinitely!The issue is how to make “nice” projections.

CHSUCB Cross Polytope SeriesCross Polytope Series

Place vertices on all coordinate half-axes,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

CHSUCB 5D and Beyond5D and Beyond

The three polytopes that result from the

Simplex series,

Cross polytope series,

Measure polytope series,

. . . is all there is in 5D and beyond!

2D 3D 4D 5D 6D 7D 8D 9D … 5 6 3 3 3 3 3 3

Luckily, we live in one of the interesting dimensions!

Dim.

#

Duals !

CHSUCB ““Dihedral Angles in Higher Dim.”Dihedral Angles in Higher Dim.”

Consider the angle through which one cell has to be rotated to be brought on top of an adjoining neighbor cell.

Space 2D 3D 4D 5D 6D SimplexSeries

60° 70.5° 75.5° 78.5° 80.4° 90°

Cross Polytopes

90° 109.5° 120° 126.9° 131.8° 180°

MeasurePolytopes

90° 90° 90° 90° 90° 90°

CHSUCB Constructing 4D Regular PolytopesConstructing 4D Regular Polytopes

Let's construct all 4D regular polytopes-- or rather, “good” projections of them.

What is a “good”projection ?

Maintain as much of the symmetry as possible;

Get a good feel for the structure of the polytope.

What are our options ? A parade of various projections

CHSUCB Parade of Projections …Parade of Projections …

1. HYPERCUBES

CHSUCB Hypercube, Perspective ProjectionsHypercube, Perspective Projections

CHSUCB Tiled Models of 4D HypercubeTiled Models of 4D Hypercube

Cell-first - - - - - - - - - Vertex-first

U.C. Berkeley, CS 285, Spring 2002,


Vertex-first Projection

CHSUCB Preferred Hypercube ProjectionsPreferred Hypercube Projections

Use Cavalier Projections to maintain sense of parallel sweeps:


Oblique Projection

CHSUCB 6D Zonohedron6D Zonohedron

Sweep symmetrically in 6 directions (in 3D)

CHSUCB Modular Zonohedron ConstructionModular Zonohedron Construction

Injection Molded Tiles:

Kiha Lee, CS 285, Spring 2002

CHSUCB 4D Hypercube – “squished”…4D Hypercube – “squished”…

… to serve as basis for the 6D Hypercube

CHSUCB Composed of 3D Zonohedra CellsComposed of 3D Zonohedra Cells

The “flat” and the “pointy” cell:

CHSUCB 5D Zonohedron 5D Zonohedron

Extrude by an extra story …

Extrusion

CHSUCB 5D Zonohedron 5D Zonohedron 6D Zonohedron 6D Zonohedron

Another extrusion

Triacontrahedral Shell

CHSUCB Parade of Projections (cont.)Parade of Projections (cont.)

2. SIMPLICES

CHSUCB 3D Simplex Projections3D Simplex Projections

Look for symmetrical projections from 3D to 2D, or …

How to put 4 vertices symmetrically in 2Dand so that edges do not intersect.

Similarly for 4D and higher…

CHSUCB 4D Simplex Projection: 5 Vertices4D Simplex Projection: 5 Vertices

“Edge-first” parallel projection: V5 in center of tetrahedron

V5

CHSUCB 5D Simplex: 6 Vertices5D Simplex: 6 Vertices

Two methods:

Avoid central intersection:Offset edges from middle.

Based on Tetrahedron(plus 2 vertices inside).

Based on Octahedron

CHSUCB 5D Simplex with 3 Internal Tetras5D Simplex with 3 Internal Tetras

With 3 internal tetrahedra;

the 12 outer ones assumed to be transparent.

CHSUCB 6D Simplex: 7 Vertices (Method A)6D Simplex: 7 Vertices (Method A)

Start from 5D arrangement that avoids central edge intersection,

Then add point in center:

CHSUCB 6D Simplex (Method A)6D Simplex (Method A)

= skewed octahedron with center vertex

CHSUCB 6D Simplex: 7 Vertices (Method B)6D Simplex: 7 Vertices (Method B)

Skinny Tetrahedron plusthree vertices around girth,(all vertices on same sphere):

CHSUCB 7D and 8D Simplices7D and 8D Simplices

Use a warped cube to avoid intersecting diagonals

CHSUCB Parade of Projections (cont.)Parade of Projections (cont.)

3. CROSS POLYTOPES


Profiled edges, indicating attached faces.


FDM --- SLIDE

CHSUCB 5D Cross Polytope with Symmetry5D Cross Polytope with Symmetry

Octahedron + Tetrahedron (10 vertices)


12 vertices icosahedral symmetry


14 vertices cube + octahedron

CHSUCB New Work – in progressNew Work – in progress

other ways to color these edges …

CHSUCB Coloring with Hamiltonian PathsColoring with Hamiltonian Paths

Graph Colorings:

Euler Path: visiting all edges

Hamiltonian Paths: visiting all vertices

Hamiltonian Cycles: closed paths

Can we visit all edges with multiple Hamiltonian paths ?

Exploit symmetry of the edge graphs of the regular polytopes!

CHSUCB 4D Simplex: 2 Hamiltonian Paths4D Simplex: 2 Hamiltonian Paths

Two identical paths, complementing each other

C2

CHSUCB 4D Cross Polytopes: 3 Paths4D Cross Polytopes: 3 Paths

All vertices have valence 6 !

CHSUCB Hypercube: 2 Hamiltonian PathsHypercube: 2 Hamiltonian Paths

4-fold (2-fold) rotational symmetry around z-axis.

C4 (C2)

CHSUCB 24-Cell: 4 Hamiltonian Paths24-Cell: 4 Hamiltonian Paths

Aligned:

4-fold symmetry

CHSUCB The Big Ones … ?The Big Ones … ?

. . . to be done !

CHSUCB Conclusions -- Questions ?Conclusions -- Questions ?

Hopefully, I was able to make you see some of these fascinating objectsin higher dimensions, and to make them appear somewhat less “alien.”

CHSUCB

CHSUCB

CHSUCB What is a Regular Polytope?

How do we know that we have a completely regular polytope ? I show you a vertex ( or edge or face) and then spin the object -- can you still identify which one it was ? -- demo with irregular object -- demo with symmetrical object.

Notion of a symmetry group -- all the transformations rotations (mirroring) that bring object back into cover with itself.

CHSUCB

Documents

BRIDGES, July 2002