CHS UCB BRIDGES, July D Visualization Models of the Regular Polytopes in Four and Higher Dimensions. Carlo H. Séquin University of California, Berkeley

CHS UCB Goals of This Talk u Expand your thinking. u Teach you “hyper-seeing,” seeing things that one cannot ordinarily see, in particular: Four- and higher-dimensional objects.

CHS UCB 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!

CHS UCB Higher-dimensional Spaces Mathematicians Have No Problem: u A point P(x, y, z) in this room is determined by: x = 2m, y = 5m, z = 1.5m; has 3 dimensions. u Positions in other data sets P = P(d1, d2, d3, d4,... dn). u Example #1: Telephone Numbers represent a 7- or 10-dimensional space. u Example #2: State Space: x, y, z, v x, v y, v z...

CHS UCB Seeing Mathematical Objects u Very big point u Large point u Small point u Tiny point u Mathematical point

CHS UCB Geometrical View of Dimensions u Read my hands … (inspired by Scott Kim, ca 1977).

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What Is a Regular Polytope u “Polytope” is the generalization of the terms “Polygon” (2D), “Polyhedron” (3D), … to arbitrary dimensions. u “Regular” means: All the vertices, edges, faces… are indistinguishable form each another. u Examples in 2D: Regular n-gons:

CHS UCB Regular Polytopes in 3D u The Platonic Solids: There are only 5. Why ? …

CHS UCB Why Only 5 Platonic Solids ? Lets try to build all possible ones: u from triangles: 3, 4, or 5 around a corner; u from squares: only 3 around a corner; u from pentagons: only 3 around a corner; u from hexagons:  floor tiling, does not close. u higher N-gons:  do not fit around vertex without undulations (forming saddles)  now the edges are no longer all alike!

CHS UCB Do All 5 Conceivable Objects Exist? I.e., do they all close around the back ? u Tetra  base of pyramid = equilateral triangle. u Octa  two 4-sided pyramids. u Cube  we all know it closes. u Icosahedron  antiprism + 2 pyramids (are vertices at the sides the same as on top ?) Another way: make it from a cube with six lines on the faces  split vertices symmetrically until all are separated evenly. u Dodecahedron  is the dual of the Icosahedron.

CHS UCB Constructing a (d+1)-D Polytope Angle-deficit = 90° creates a 3D cornercreates a 4D corner ? 2D 3D4D 3D Forcing closure:

CHS UCB “Seeing a Polytope” u I showed you the 3D Platonic Solids … But which ones have you actually seen ? u For some of them you have only seen projections. Did that bother you ?? u 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 ! u 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.” u We will use this to see the 4D Polytopes.

CHS UCB Projections How do we make “projections” ? u 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. u Alternatively, use a perspective projection: back features are smaller  depth queue. Can add other depth queues: width of beams, color, fuzziness, contrast (fog)...

CHS UCB Wire Frame Projections u Shadow of a solid object is mostly a blob. u Better to use wire frame, so we can also see what is going on on the back side.

CHS UCB Oblique Projections u Cavalier Projection 3D Cube  2D4D Cube  3D (  2D )

CHS UCB Projections : VERTEX / EDGE / FACE / CELL - First. u 3D Cube: Paralell proj. Persp. proj. u 4D Cube: Parallel proj. Persp. proj.

CHS UCB 3D Models Need Physical Edges Options: u Round dowels (balls and stick) u Profiled edges – edge flanges convey a sense of the attached face u Actual composition from flat tiles – with holes to make structure see-through.

CHS UCB Edge Treatments Leonardo DaVinci – George Hart

CHS UCB How Do We Find All 4D Polytopes? u Reasoning by analogy helps a lot: -- How did we find all the Platonic solids? u Use the Platonic solids as “tiles” and ask: l What can we build from tetrahedra? l From cubes? l 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°.

CHS UCB 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°).

CHS UCB 5-Cell or Simplex in 4D u 5 cells, 10 faces, 10 edges, 5 vertices. u (self-dual).

CHS UCB 4D Simplex u Using Polymorf TM Tiles Additional tiles made on our FDM machine.

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

CHS UCB 4D Cross Polytope u Highlighting the eight tetrahedra from which it is composed.

CHS UCB 4D Cross Polytope

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

CHS UCB 4D Hypercube u Using Polymorf TM Tiles made by Kiha Lee on FDM.

CHS UCB Corpus Hypercubus Salvador Dali “Unfolded” Hypercube

CHS UCB 24-Cell in 4D u 24 cells, 96 faces, 96 edges, 24 vertices. u (self-dual).

CHS UCB 24-Cell, showing 3-fold symmetry

CHS UCB 24-Cell “Fold-out” in 3D Andrew Weimholt

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

CHS UCB 120 Cell u Hands-on workshop with George Hart

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

CHS UCB 120-Cell u Cell-first, extreme perspective projection u Z-Corp. model

CHS UCB (smallest ?) 120-Cell Wax model, made on Sanders machine

CHS UCB Radial Projections of the 120-Cell u Onto a sphere, and onto a dodecahedron:

CHS UCB 120-Cell, “exploded” Russell Towle

CHS UCB 120-Cell Soap Bubble John Sullivan

CHS UCB 600-Cell, A Classical Rendering u Oss, 1901 Frontispiece of Coxeter’s 1948 book “Regular Polytopes,” and John Sullivan’s Paper “The Story of the 120-Cell.” u Total: 600 tetra-cells, 1200 faces, 720 edges, 120 vertices. u At each Vertex: 20 tetra-cells, 30 faces, 12 edges.

CHS UCB 600-Cell Cross-eye Stereo Picture by Tony Smith

CHS UCB 600-Cell in 4D u Dual of 120 cell. u 600 cells, 1200 faces, 720 edges, 120 vertices. u Cell-first parallel projection, shows less than half of the edges.

CHS UCB 600-Cell u David Richter

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

CHS UCB 600-Cell u Cell-first, parallel projection, u Z-Corp. model

CHS UCB Model Fabrication Commercial Rapid Prototyping Machines: u Fused Deposition Modeling (Stratasys) u 3D-Color Printing (Z-corporation)

CHS UCB Fused Deposition Modeling

CHS UCB Zooming into the FDM Machine

CHS UCB SFF: 3D Printing -- Principle u Selectively deposit binder droplets onto a bed of powder to form locally solid parts. Powder SpreadingPrinting Build Feeder Powder Head

CHS UCB 3D Printing: Z Corporation

CHS UCB 3D Printing: Z Corporation Cleaning up in the de-powdering station

CHS UCB Beyond 4 Dimensions … u What happens in higher dimensions ? u How many regular polytopes are there in 5, 6, 7, … dimensions ?

CHS UCB Polytopes in Higher Dimensions u 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°. u Most 4D polytopes are too round … But we can use 3 or 4 5-Cells, and 3 Tessaracts. u There are three methods by which we can generate regular polytopes for 5D and all higher dimensions.

CHS UCB Hypercube Series u “Measure Polytope” Series (introduced in the pantomime) u Consecutive perpendicular sweeps: 1D 2D 3D 4D This series extents to arbitrary dimensions!

CHS UCB Simplex Series u 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.

CHS UCB Cross Polytope Series u Place vertices on all coordinate half-axes, a unit-distance away from origin. u 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

CHS UCB 5D and Beyond The three polytopes that result from the l Simplex series, l Cross polytope series, l Measure polytope series,... is all there is in 5D and beyond! 2D 3D 4D 5D 6D 7D 8D 9D …  Luckily, we live in one of the interesting dimensions! Dim. # Duals !

CHS UCB “Dihedral Angles in Higher Dim.” u Consider the angle through which one cell has to be rotated to be brought on top of an adjoining neighbor cell. Space2D3D4D5D6D  Simplex Series 60°70.5°75.5°78.5°80.4°90° Cross Polytopes 90°109.5°120°126.9°131.8°180° Measure Polytopes 90°

CHS UCB Constructing 4D Regular Polytopes u Let's construct all 4D regular polytopes -- or rather, “good” projections of them. u What is a “good”projection ? l Maintain as much of the symmetry as possible; l Get a good feel for the structure of the polytope. u What are our options ? A parade of various projections 

CHS UCB Parade of Projections … 1. HYPERCUBES

CHS UCB Hypercube, Perspective Projections

CHS UCB Tiled Models of 4D Hypercube Cell-first Vertex-first U.C. Berkeley, CS 285, Spring 2002,

CHS UCB 4D Hypercube Vertex-first Projection

CHS UCB Preferred Hypercube Projections u Use Cavalier Projections to maintain sense of parallel sweeps:

CHS UCB 6D Hypercube u Oblique Projection

CHS UCB 6D Zonohedron u Sweep symmetrically in 6 directions (in 3D)

CHS UCB Parade of Projections (cont.) 2. SIMPLICES

CHS UCB 3D Simplex Projections u Look for symmetrical projections from 3D to 2D, or … u How to put 4 vertices symmetrically in 2D and so that edges do not intersect. Similarly for 4D and higher…

CHS UCB 4D Simplex Projection: 5 Vertices u “Edge-first” parallel projection: V5 in center of tetrahedron V5

CHS UCB 5D Simplex: 6 Vertices u Two methods: Avoid central intersection: Offset edges from middle. Based on Tetrahedron (plus 2 vertices inside). Based on Octahedron

CHS UCB 5D Simplex with 3 Internal Tetras u With 3 internal tetrahedra; the 12 outer ones assumed to be transparent.

CHS UCB 6D Simplex: 7 Vertices (Method A) Start from 5D arrangement that avoids central edge intersection, Then add point in center:

CHS UCB 6D Simplex (Method A) = skewed octahedron with center vertex

CHS UCB 6D Simplex: 7 Vertices (Method B) u Skinny Tetrahedron plus three vertices around girth, (all vertices on same sphere):

CHS UCB 7D and 8D Simplices Use a warped cube to avoid intersecting diagonals

CHS UCB Parade of Projections (cont.) 3. CROSS POLYTOPES

CHS UCB 4D Cross Polytope Profiled edges, indicating attached faces.

CHS UCB 5D Cross Polytope u FDM --- SLIDE

CHS UCB 5D Cross Polytope with Symmetry Octahedron + Tetrahedron (10 vertices)

CHS UCB 6D Cross Polytope 12 vertices  icosahedral symmetry

CHS UCB 7D Cross Polytope 14 vertices  cube + octahedron

CHS UCB Conclusions -- Questions ? u Hopefully, I was able to make you see some of these fascinating objects in higher dimensions, and to make them appear somewhat less “alien.”

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