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Hyper-Seeing 4D Polytopes: Mathematical Visualization Beyond Dimensions

Discover the world of regular polytopes in four and higher dimensions through "hyper-seeing." Learn about the geometric view of dimensions, regular polytope construction, and visualization techniques using projections and wireframe models. Explore the links between 3D and 4D representations, including edge treatments and physical edges. Join this talk to expand your thinking and understand the beauty of higher-dimensional spaces.

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Hyper-Seeing 4D Polytopes: Mathematical Visualization Beyond Dimensions

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  1. BRIDGES, July 2002 3D Visualization Models of the Regular Polytopesin Four and Higher Dimensions. Carlo H. Séquin University of California, Berkeley

  2. 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.

  3. 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.

  4. 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!

  5. 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 ...

  6. Seeing Mathematical Objects • Very big point • Large point • Small point • Tiny point • Mathematical point

  7. Geometrical View of Dimensions • Read my hands …(inspired by Scott Kim, ca 1977).

  8. 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:

  9. Regular Polytopes in 3D • The Platonic Solids: There are only 5. Why ? …

  10. 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!

  11. 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.

  12. Constructing a (d+1)-D Polytope Angle-deficit = 90° 2D 3D Forcing closure: ? 3D 4D creates a 3D corner creates a 4D corner

  13. “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.

  14. 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) ...

  15. 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.

  16. Oblique Projections • Cavalier Projection 3D Cube  2D 4D Cube  3D ( 2D )

  17. Projections: VERTEX/ EDGE /FACE/CELL - First. • 3D Cube: Paralell proj. Persp. proj. • 4D Cube: Parallel proj. Persp. proj.

  18. 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.

  19. Edge Treatments Leonardo DaVinci – George Hart

  20. 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°.

  21. 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°).

  22. 5-Cell or Simplex in 4D • 5 cells, 10 faces, 10 edges, 5 vertices. • (self-dual).

  23. 4D Simplex • Using Polymorf TM Tiles Additional tiles made on our FDM machine.

  24. 16-Cell or “Cross Polytope” in 4D • 16 cells, 32 faces, 24 edges, 8 vertices.

  25. 4D Cross Polytope • Highlighting the eight tetrahedra from which it is composed.

  26. 4D Cross Polytope

  27. Hypercube or Tessaract in 4D • 8 cells, 24 faces, 32 edges, 16 vertices. • (Dual of 16-Cell).

  28. 4D Hypercube • Using PolymorfTM Tilesmade byKiha Leeon FDM.

  29. Corpus Hypercubus Salvador Dali “Unfolded”Hypercube

  30. 24-Cell in 4D • 24 cells, 96 faces, 96 edges, 24 vertices. • (self-dual).

  31. 24-Cell, showing 3-fold symmetry

  32. 24-Cell “Fold-out” in 3D Andrew Weimholt

  33. 120-Cell in 4D • 120 cells, 720 faces, 1200 edges, 600 vertices.Cell-first parallel projection,(shows less than half of the edges.)

  34. 120 Cell • Hands-on workshop with George Hart

  35. 120-Cell Séquin(1982) Thin face frames, Perspective projection.

  36. 120-Cell • Cell-first,extremeperspectiveprojection • Z-Corp. model

  37. (smallest ?) 120-Cell Wax model, made on Sanders machine

  38. Radial Projections of the 120-Cell • Onto a sphere, and onto a dodecahedron:

  39. 120-Cell, “exploded” Russell Towle

  40. 120-Cell Soap Bubble John Sullivan

  41. 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.”

  42. 600-Cell Cross-eye Stereo Picture by Tony Smith

  43. 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.

  44. 600-Cell • David Richter

  45. Slices through the 600-Cell At each Vertex: 20 tetra-cells, 30 faces, 12 edges. Gordon Kindlmann

  46. 600-Cell • Cell-first, parallel projection, • Z-Corp. model

  47. Model Fabrication Commercial Rapid Prototyping Machines: • Fused Deposition Modeling (Stratasys) • 3D-Color Printing (Z-corporation)

  48. Fused Deposition Modeling

  49. Zooming into the FDM Machine

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