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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
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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.
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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.
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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!
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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 ...
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CHSUCB Seeing Mathematical ObjectsSeeing Mathematical Objects
Very big point
Large point
Small point
Tiny point
Mathematical point
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CHSUCB Geometrical View of Dimensions Geometrical View of Dimensions
Read my hands …(inspired by Scott Kim, ca 1977).
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CHSUCB
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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:
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CHSUCB Regular Polytopes in 3DRegular Polytopes in 3D
The Platonic Solids:
There are only 5. Why ? …
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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!
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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.
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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:
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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.
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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) ...
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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.
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CHSUCB Oblique ProjectionsOblique Projections
Cavalier Projection
3D Cube 2D 4D Cube 3D ( 2D )
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CHSUCB ProjectionsProjections: : VERTEXVERTEX / / EDGEEDGE / / FACEFACE // CELL CELL - First.- First.
3D Cube:
Paralell proj.
Persp. proj.
4D Cube:
Parallel proj.
Persp. proj.
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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.
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CHSUCB Edge TreatmentsEdge Treatments
Leonardo DaVinci – George Hart
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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°.
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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°).
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CHSUCB 5-Cell or Simplex in 4D5-Cell or Simplex in 4D
5 cells, 10 faces, 10 edges, 5 vertices. (self-dual).
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CHSUCB 4D Simplex4D Simplex
Using Polymorf TM Tiles
Additional tiles made on our FDM machine.
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CHSUCB 16-Cell or “Cross Polytope” in 4D16-Cell or “Cross Polytope” in 4D
16 cells, 32 faces, 24 edges, 8 vertices.
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CHSUCB 4D Cross Polytope4D Cross Polytope
Highlighting the eight tetrahedra from which it is composed.
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CHSUCB 4D Cross Polytope4D Cross Polytope
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CHSUCB Hypercube or Tessaract in 4DHypercube or Tessaract in 4D
8 cells, 24 faces, 32 edges, 16 vertices.
(Dual of 16-Cell).
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CHSUCB 4D Hypercube4D Hypercube
Using PolymorfTM Tiles
made byKiha Leeon FDM.
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CHSUCB Corpus HypercubusCorpus Hypercubus
Salvador Dali
“Unfolded”Hypercube
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CHSUCB 24-Cell in 4D24-Cell in 4D
24 cells, 96 faces, 96 edges, 24 vertices. (self-dual).
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CHSUCB
24-Cell, showing 3-fold symmetry24-Cell, showing 3-fold symmetry
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CHSUCB 24-Cell “Fold-out” in 3D24-Cell “Fold-out” in 3D
Andrew Weimholt
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CHSUCB 120-Cell in 4D120-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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CHSUCB 120 Cell120 Cell
Hands-on workshop with George Hart
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CHSUCB 120-Cell120-Cell
Thin face frames, Perspective projection.
Séquin(1982)
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CHSUCB 120-Cell120-Cell
Cell-first,extremeperspectiveprojection
Z-Corp. model
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CHSUCB (smallest ?) 120-Cell(smallest ?) 120-Cell
Wax model, made on Sanders machine
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CHSUCB Radial Projections of the 120-CellRadial Projections of the 120-Cell
Onto a sphere, and onto a dodecahedron:
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CHSUCB 120-Cell, “exploded”120-Cell, “exploded”
Russell Towle
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CHSUCB 120-Cell Soap Bubble120-Cell Soap Bubble
John Sullivan
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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.
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CHSUCB 600-Cell600-Cell
Cross-eye Stereo Picture by Tony Smith
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CHSUCB 600-Cell in 4D600-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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CHSUCB 600-Cell600-Cell
David Richter
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CHSUCB Slices through the 600-CellSlices through the 600-Cell
At each Vertex: 20 tetra-cells, 30 faces, 12 edges.
Gordon Kindlmann
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CHSUCB 600-Cell600-Cell
Cell-first, parallel projection,
Z-Corp. model
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CHSUCB Model FabricationModel Fabrication
Commercial Rapid Prototyping Machines:
Fused Deposition Modeling (Stratasys)
3D-Color Printing (Z-corporation)
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CHSUCB Fused Deposition ModelingFused Deposition Modeling
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CHSUCB Zooming into the FDM MachineZooming into the FDM Machine
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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
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CHSUCB 3D Printing:3D Printing: Z CorporationZ Corporation
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CHSUCB 3D Printing:3D Printing: Z CorporationZ Corporation
Cleaning up in the de-powdering station
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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
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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.
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CHSUCB Beyond 4 Dimensions …Beyond 4 Dimensions …
What happens in higher dimensions ?
How many regular polytopes are therein 5, 6, 7, … dimensions ?
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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.
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CHSUCB Hypercube SeriesHypercube 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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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.
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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
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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 !
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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°
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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
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CHSUCB Parade of Projections …Parade of Projections …
1. HYPERCUBES
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CHSUCB Hypercube, Perspective ProjectionsHypercube, Perspective Projections
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CHSUCB Tiled Models of 4D HypercubeTiled Models of 4D Hypercube
Cell-first - - - - - - - - - Vertex-first
U.C. Berkeley, CS 285, Spring 2002,
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CHSUCB 4D Hypercube4D Hypercube
Vertex-first Projection
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CHSUCB Preferred Hypercube ProjectionsPreferred Hypercube Projections
Use Cavalier Projections to maintain sense of parallel sweeps:
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CHSUCB 6D Hypercube6D Hypercube
Oblique Projection
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CHSUCB 6D Zonohedron6D Zonohedron
Sweep symmetrically in 6 directions (in 3D)
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CHSUCB Modular Zonohedron ConstructionModular Zonohedron Construction
Injection Molded Tiles:
Kiha Lee, CS 285, Spring 2002
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CHSUCB 4D Hypercube – “squished”…4D Hypercube – “squished”…
… to serve as basis for the 6D Hypercube
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CHSUCB Composed of 3D Zonohedra CellsComposed of 3D Zonohedra Cells
The “flat” and the “pointy” cell:
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CHSUCB 5D Zonohedron 5D Zonohedron
Extrude by an extra story …
Extrusion
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CHSUCB 5D Zonohedron 5D Zonohedron 6D Zonohedron 6D Zonohedron
Another extrusion
Triacontrahedral Shell
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CHSUCB Parade of Projections (cont.)Parade of Projections (cont.)
2. SIMPLICES
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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…
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CHSUCB 4D Simplex Projection: 5 Vertices4D Simplex Projection: 5 Vertices
“Edge-first” parallel projection: V5 in center of tetrahedron
V5
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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
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CHSUCB 5D Simplex with 3 Internal Tetras5D Simplex with 3 Internal Tetras
With 3 internal tetrahedra;
the 12 outer ones assumed to be transparent.
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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:
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CHSUCB 6D Simplex (Method A)6D Simplex (Method A)
= skewed octahedron with center vertex
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CHSUCB 6D Simplex: 7 Vertices (Method B)6D Simplex: 7 Vertices (Method B)
Skinny Tetrahedron plusthree vertices around girth,(all vertices on same sphere):
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CHSUCB 7D and 8D Simplices7D and 8D Simplices
Use a warped cube to avoid intersecting diagonals
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CHSUCB Parade of Projections (cont.)Parade of Projections (cont.)
3. CROSS POLYTOPES
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CHSUCB 4D Cross Polytope4D Cross Polytope
Profiled edges, indicating attached faces.
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CHSUCB 5D Cross Polytope5D Cross Polytope
FDM --- SLIDE
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CHSUCB 5D Cross Polytope with Symmetry5D Cross Polytope with Symmetry
Octahedron + Tetrahedron (10 vertices)
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CHSUCB 6D Cross Polytope6D Cross Polytope
12 vertices icosahedral symmetry
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CHSUCB 7D Cross Polytope7D Cross Polytope
14 vertices cube + octahedron
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CHSUCB New Work – in progressNew Work – in progress
other ways to color these edges …
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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!
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CHSUCB 4D Simplex: 2 Hamiltonian Paths4D Simplex: 2 Hamiltonian Paths
Two identical paths, complementing each other
C2
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CHSUCB 4D Cross Polytopes: 3 Paths4D Cross Polytopes: 3 Paths
All vertices have valence 6 !
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CHSUCB Hypercube: 2 Hamiltonian PathsHypercube: 2 Hamiltonian Paths
4-fold (2-fold) rotational symmetry around z-axis.
C4 (C2)
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CHSUCB 24-Cell: 4 Hamiltonian Paths24-Cell: 4 Hamiltonian Paths
Aligned:
4-fold symmetry
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CHSUCB The Big Ones … ?The Big Ones … ?
. . . to be done !
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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.”
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CHSUCB
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CHSUCB
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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.
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CHSUCB