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## BRIDGES, July 2002

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**BRIDGES, July 2002**3D Visualization Models of the Regular Polytopesin Four and Higher Dimensions. Carlo H. Séquin University of California, Berkeley**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 >100 years)NOT a review paper on literature …(browse with “regular polyhedra” “120-Cell”) • Also: Use of Rapid Prototyping in math.**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.**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!**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 ...**Seeing Mathematical Objects**• Very big point • Large point • Small point • Tiny point • Mathematical point**Geometrical View of Dimensions**• Read my hands …(inspired by Scott Kim, ca 1977).**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 indistinguishable form each another. • Examples in 2D: Regular n-gons:**Regular Polytopes in 3D**• The Platonic Solids: There are only 5. Why ? …**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!**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.**Constructing a (d+1)-D Polytope**Angle-deficit = 90° 2D 3D Forcing closure: ? 3D 4D creates a 3D corner creates a 4D corner**“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.**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) ...**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.**Oblique Projections**• Cavalier Projection 3D Cube 2D 4D Cube 3D ( 2D )**Projections: VERTEX/ EDGE /FACE/CELL - First.**• 3D Cube: Paralell proj. Persp. proj. • 4D Cube: Parallel proj. Persp. proj.**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.**Edge Treatments**Leonardo DaVinci – George Hart**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°.**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°).**5-Cell or Simplex in 4D**• 5 cells, 10 faces, 10 edges, 5 vertices. • (self-dual).**4D Simplex**• Using Polymorf TM Tiles Additional tiles made on our FDM machine.**16-Cell or “Cross Polytope” in 4D**• 16 cells, 32 faces, 24 edges, 8 vertices.**4D Cross Polytope**• Highlighting the eight tetrahedra from which it is composed.**Hypercube or Tessaract in 4D**• 8 cells, 24 faces, 32 edges, 16 vertices. • (Dual of 16-Cell).**4D Hypercube**• Using PolymorfTM Tilesmade byKiha Leeon FDM.**Corpus Hypercubus**Salvador Dali “Unfolded”Hypercube**24-Cell in 4D**• 24 cells, 96 faces, 96 edges, 24 vertices. • (self-dual).**24-Cell “Fold-out” in 3D**Andrew Weimholt**120-Cell in 4D**• 120 cells, 720 faces, 1200 edges, 600 vertices.Cell-first parallel projection,(shows less than half of the edges.)**120 Cell**• Hands-on workshop with George Hart**120-Cell**Séquin(1982) Thin face frames, Perspective projection.**120-Cell**• Cell-first,extremeperspectiveprojection • Z-Corp. model**(smallest ?) 120-Cell**Wax model, made on Sanders machine**Radial Projections of the 120-Cell**• Onto a sphere, and onto a dodecahedron:**120-Cell, “exploded”**Russell Towle**120-Cell Soap Bubble**John Sullivan**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 Coxeter’s 1948 book “Regular Polytopes,” and John Sullivan’s Paper “The Story of the 120-Cell.”**600-Cell**Cross-eye Stereo Picture by Tony Smith**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.**600-Cell**• David Richter**Slices through the 600-Cell**At each Vertex: 20 tetra-cells, 30 faces, 12 edges. Gordon Kindlmann**600-Cell**• Cell-first, parallel projection, • Z-Corp. model**Model Fabrication**Commercial Rapid Prototyping Machines: • Fused Deposition Modeling (Stratasys) • 3D-Color Printing (Z-corporation)