BRIDGES, July 2002

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BRIDGES, July 2002

• 3D Visualization Models of the Regular
  Polytopes in Four and Higher Dimensions .
• Carlo H. Séquin
• University of California, Berkeley

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Goals 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 gt100 years)NOT a review paper on
  literature (browse with regular polyhedra
  120-Cell)
• Also Use of Rapid Prototyping in math.

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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.

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What is the 4th Dimension ?

• Some people think it does not really exist,
  its just a philosophical notion,it is
  TIME , . . .
• But, it is useful and quite real!

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Higher-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 ...

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Seeing Mathematical Objects

• Very big point
• Large point
• Small point
• Tiny point
• Mathematical point

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Geometrical View of Dimensions

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

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What Is a Regular Polytope

• Polytope is the generalization of the terms
  Polygon (2D), Polyhedron (3D), to
  arbitrary dimensions.
• Regularmeans All the vertices, edges,
  facesare indistinguishable form each another.
• Examples in 2D Regular n-gons

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Regular Polytopes in 3D

• The Platonic Solids

There are only 5. Why ?
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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!

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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.

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Constructing a (d1)-D Polytope
Angle-deficit 90
2D
3D
Forcing closure
?
3D
4D
creates a 3D corner
creates a 4D corner
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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 gt
  Hyper-seeing.
• We will use this to see the 4D Polytopes.

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Projections

• 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) ...

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Wire 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.

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Oblique Projections

• Cavalier Projection

3D Cube ? 2D
4D Cube ? 3D (? 2D )
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Projections VERTEX / EDGE / FACE /
CELL - First.

• 3D Cube
• Paralell proj.
• Persp. proj.
• 4D Cube
• Parallel proj.
• Persp. proj.

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3D 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.

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Edge Treatments

• Leonardo DaVinci George Hart

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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.

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All 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).

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5-Cell or Simplex in 4D

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

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4D Simplex
Additional tiles made on our FDM machine.

• Using Polymorf TM Tiles

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16-Cell or Cross Polytope in 4D

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

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4D Cross Polytope

• Highlighting the eight tetrahedra from which it
  is composed.

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4D Cross Polytope
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Hypercube or Tessaract in 4D

• 8 cells, 24 faces, 32 edges, 16 vertices.
• (Dual of 16-Cell).

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4D Hypercube

• Using PolymorfTM Tilesmade byKiha Leeon FDM.

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Corpus Hypercubus
UnfoldedHypercube

• Salvador Dali

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24-Cell in 4D

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

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24-Cell, showing 3-fold symmetry
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24-Cell Fold-out in 3D

• Andrew Weimholt

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120-Cell in 4D

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

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120 Cell

• Hands-on workshop with George Hart

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120-Cell
Séquin(1982)
Thin face frames, Perspective projection.
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120-Cell

• Cell-first,extremeperspectiveprojection
• Z-Corp. model

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(smallest ?) 120-Cell

• Wax model, made on Sanders machine

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Radial Projections of the 120-Cell

• Onto a sphere, and onto a dodecahedron

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120-Cell, exploded

• Russell Towle

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120-Cell Soap Bubble

• John Sullivan

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600-Cell, A Classical Rendering

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

• Oss, 1901

Frontispiece of Coxeters 1948 book Regular
Polytopes, and John Sullivans Paper The Story
of the 120-Cell.
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600-Cell

• Cross-eye Stereo Picture by Tony Smith

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600-Cell in 4D

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

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600-Cell

• David Richter

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Slices through the 600-Cell
Gordon Kindlmann

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

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600-Cell

• Cell-first, parallel projection,
• Z-Corp. model

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Model Fabrication

• Commercial Rapid Prototyping Machines
• Fused Deposition Modeling (Stratasys)
• 3D-Color Printing (Z-corporation)

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Fused Deposition Modeling
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Zooming into the FDM Machine
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SFF 3D Printing -- Principle

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

Head
Powder Spreading
Printing
Powder
Feeder
Build
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3D Printing Z Corporation
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3D Printing Z Corporation

• Cleaning up in the de-powdering station

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Designing 3D Edge Models

• Is not totally trivial because of shortcomings
  of CAD tools
• Limited Rotations weird angles
• Poor Booleans need water tight shells

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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.

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Beyond 4 Dimensions

• What happens in higher dimensions ?
• How many regular polytopes are therein 5, 6, 7,
  dimensions ?

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Polytopes 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.

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Hypercube Series

• Measure Polytope Series(introduced in the
  pantomime)
• Consecutive perpendicular sweeps

1D 2D 3D
4D
This series extents to arbitrary dimensions!
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Simplex Series

• Connect all the dots among n1 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.
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Cross 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
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5D 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!

Duals !
Dim.
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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.

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Constructing 4D Regular Polytopes

• Let's construct all 4D regular polytopes-- or
  rather, good projections of them.
• What is a goodprojection ?
• 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 ???

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Parade of Projections

• 1. HYPERCUBES

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Hypercube, Perspective Projections
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Tiled Models of 4D Hypercube

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

U.C. Berkeley, CS 285, Spring 2002,
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4D Hypercube

• Vertex-first Projection

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Preferred Hypercube Projections

• Use Cavalier Projections to maintain sense of
  parallel sweeps

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6D Hypercube

• Oblique Projection

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6D Zonohedron

• Sweep symmetrically in 6 directions (in 3D)

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Modular Zonohedron Construction

• Injection Molded Tiles

Kiha Lee, CS 285, Spring 2002
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4D Hypercube squished

• to serve as basis for the 6D Hypercube

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Composed of 3D Zonohedra Cells

• The flat and the pointy cell

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5D Zonohedron

• Extrude by an extra story

• Extrusion

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5D Zonohedron ? 6D Zonohedron

• Another extrusion

Triacontrahedral Shell
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Parade of Projections (cont.)

• 2. SIMPLICES

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3D 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
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4D Simplex Projection 5 Vertices

• Edge-first parallel projection V5 in center
  of tetrahedron

V5
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5D Simplex 6 Vertices
Based on Octahedron

• Two methods

Avoid central intersection Offset edges from
middle.
Based on Tetrahedron(plus 2 vertices inside).
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5D Simplex with 3 Internal Tetras

• With 3 internal tetrahedra the 12 outer ones
  assumed to be transparent.

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6D Simplex 7 Vertices (Method A)

• Start from 5D arrangement that
• avoids central edge intersection,
• Then add point in center

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6D Simplex (Method A)

• skewed octahedron with center vertex

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6D Simplex 7 Vertices (Method B)

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

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7D and 8D Simplices

• Use a warped cube to avoid intersecting diagonals

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Parade of Projections (cont.)

• 3. CROSS POLYTOPES

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4D Cross Polytope

• Profiled edges, indicating attached faces.

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5D Cross Polytope

• FDM --- SLIDE

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5D Cross Polytope with Symmetry

• Octahedron Tetrahedron (10 vertices)

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6D Cross Polytope
12 vertices ? icosahedral symmetry
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7D Cross Polytope

• 14 vertices ? cube octahedron

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New Work in progress

• other ways to color these edges

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Coloring 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!

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4D Simplex 2 Hamiltonian Paths

• Two identical paths, complementing each other

C2
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4D Cross Polytopes 3 Paths

• All vertices have valence 6 !

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Hypercube 2 Hamiltonian Paths
C4 (C2)

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

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24-Cell 4 Hamiltonian Paths

• Aligned

? 4-fold symmetry
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The Big Ones ?

• . . . to be done !

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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.

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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.

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