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)