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4D Polytopes and 3D Models of Them George W. Hart Stony Brook University

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Goals of This Talk Expand your thinking. Visualization of four- and higher-dimensional objects. Show Rapid Prototyping of complex structures. Note: Some Material and images adapted from Carlo Sequin

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What is the 4th Dimension ? Some people think: it does not really exist its just a philosophical notion it is TIME... But, a geometric fourth dimension is as useful and as real as 2D or 3D.

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Higher-dimensional Spaces Coordinate Approach: A point (x, y, z) has 3 dimensions. n-dimensional point: (d1, d2, d3, d4,..., dn). Axiomatic Approach: Definition, theorem, proof... Descriptive Geometry Approach: Compass, straightedge, two sheets of paper.

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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 equivalent. Assume convexity for now. Examples in 2D: Regular n-gons:

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

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Why Only 5 Platonic Solids ? 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 (not convex)

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

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Seeing a Polytope Real planes, lines, points, spheres, …, do not exist physically. We understand their properties and relationships as ideal mental models. Good projections are very useful. Our visual input is only 2D, but we understand as 3D via mental construction in the brain. You are able to see things that don't really exist in physical 3-space, because you understand projections into 2-space, and you form a mental model. We will use this to visualize 4D Polytopes.

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Projections Set the coordinate values of all unwanted dimensions to zero, e.g., drop z, retain x,y, and you get a orthogonal projection along the z-axis. i.e., a 2D shadow. Linear algebra allows arbitrary direction. Alternatively, use a perspective projection: rays of light form cone to eye. Can add other depth queues: width of lines, color, fuzziness, contrast (fog)...

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Wire Frame Projections Shadow of a solid object is mostly a blob. Better to use wire frame, so we can see components.

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Oblique Projections Cavalier Projection 3D Cube 2D 4D Cube 3D ( 2D )

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Projections : VERTEX / EDGE / FACE / CELL – centered 3D Cube: Paralell proj. Persp. proj. 4D Cube: Parallel proj. Persp. proj.

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3D Objects Need Physical Edges Options: Round dowels (balls and stick) Profiled edges – edge flanges convey a sense of the attached face Flat tiles for faces – with holes to make structure see- through.

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Edge Treatments (Leonardo Da Vinci)

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How Do We Find All 4D Polytopes? Sum of dihedral angles around each edge must be less than 360 degrees. Use the Platonic solids as cells Tetrahedron: 70.5° Octahedron: 109.5° Cube: 90° Dodecahedron: 116.5° Icosahedron: 138.2°.

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All Regular Convex 4D Polytopes 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) Using Dodecahedra (116.5°): 3 around an edge (349.5°) (120 cells) Using Icosahedra (138.2°): none: angle too large.

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5-Cell or 4D Simplex 5 cells, 10 faces, 10 edges, 5 vertices. Carlo Sequin Can make with Zometool also

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

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4D Hypercube or Tessaract 8 cells, 24 faces, 32 edges, 16 vertices.

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Hypercube, Perspective Projections

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Nets: 11 Unfoldings of Cube

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Hypercube Unfolded -- Net One of the 261 different unfoldings

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Corpus Hypercubus Salvador Dali Unfolded Hypercube

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24-Cell 24 cells, 96 faces, 96 edges, 24 vertices. (self-dual).

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24-Cell Net in 3D Andrew Weimholt

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

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120-cell Model Marc Pelletier

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120-Cell Thin face frames, Perspective projection. Carlo Séquin

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120-Cell – perspective projection

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

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120-Cell – perspective projection Selective laser sintering

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3D Printing Zcorp

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120-Cell, exploded Russell Towle

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120-Cell Soap Bubble John Sullivan Stereographic projection preserves 120 degree angles

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120-Cell Net with stack of 10 dodecahedra George Olshevski

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600-Cell -- 2D projection Oss, 1901 Frontispiece of Coxeters book Regular Polytopes, Total: 600 tetra-cells, 1200 faces, 720 edges, 120 vertices. At each Vertex: 20 tetra-cells, 30 faces, 12 edges.

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600-Cell Cross-eye Stereo Picture by Tony Smith

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600-Cell Dual of 120 cell. 600 cells, 1200 faces, 720 edges, 120 vertices. Cell-first parallel projection, shows less than half of the edges. Can make with Zometool

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600-Cell Straw model by David Richter

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

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History 3D Models of 4D Polytopes Ludwig Schlafli discovered them in Worked algebraically, no pictures in his paper. Partly published in 1858 and 1862 (translation by Cayley) but not appreciated. Many independent rediscoveries and models.

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Stringham (1880) First to rediscover all six His paper shows cardboard models of layers 3 layers of 120-cell (45 dodecahedra)

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Victor Schlegel (1880s) Invented Schlegel Diagram 3D 2D perspective transf. Used analogous 4D 3D projection in educational models. Built wire and thread models. Advertised and sold models via commercial catalogs: Dyck (1892) and Schilling (1911). Some stored at Smithsonian. Five regular polytopes

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Sommervilles Description of Models In each case, the external boundary of the projection represents one of the solid boundaries of the figure. Thus the 600-cell, which is the figure bounded by 600 congruent regular tetrahedra, is represented by a tetrahedron divided into 599 other tetrahedra … At the center of the model there is a tetrahedron, and surrounding this are successive zones of tetrahedra. The boundaries of these zones are more or less complicated polyhedral forms, cardboard models of which, constructed after Schlegel's drawings, are also to be obtained by the same firm.

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Cardboard Models of 120-Cell From Walther Dycks 1892 Math and Physics Catalog

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Paul S. Donchians Wire Models 1930s Rug Salesman with visions Wires doubled to show how front overlays back Widely displayed Currently on view at the Franklin Institute

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Zometool 1970 Steve Baer designed and produced "Zometool" for architectural modeling Marc Pelletier discovered (when 17 years old) that the lengths and directions allowed by this kit permit the construction of accurate models of the 120-cell and related polytopes. The kit went out of production however, until redesigned in plastic in 1992.

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120 Cell Zome Model Orthogonal projection

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Uniform 4D Polytopes Analogous to the 13 Archimedean Solids Allow more than one type of cell All vertices equivalent Alicia Boole Stott listed many in 1910 Now over 8000 known Cataloged by George Olshevski and Jonathan Bowers

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Truncated 120-Cell

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Truncated 120-Cell - Stereolithography

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Zometool Truncated 120-Cell MathCamp 2000

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Ambo 600-Cell Bridges Conference, 2001

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Ambo 120-Cell Orthogonal projection Stereolithography Can do with Zome

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Expanded 120-Cell Mira Bernstein, Vin de Silva, et al.

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Expanded Truncated 120-Cell

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Big Polytope Net George Olshevski

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Big Polytope Zome Model Steve Rogers

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48 Truncated Cubes Poorly designed FDM model

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Prism on a Snub Cube – Net George Olshevski

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Duo-Prisms - Nets Robert Webb Andrew Weimholt George Olshevski

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Grand Antiprism Net with stack of 10 pentagonal antiprisms George Olshevski

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Non-Convex Polytopes Jonathan Bowers Components may pass through each other Slices may be useful for visualization Slices may be disconnected

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Beyond 4 Dimensions … What happens in higher dimensions ? How many regular polytopes are there in 5, 6, 7, … dimensions ? Only three regular types: –Hypercubes e.g., cube –Simplexes e.g., tetrahedron –Cross polytope e.g., octahedron

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Hypercubes A.k.a. Measure Polytope Perpendicular extrusion in n th direction: 1D 2D 3D 4D

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Orthographic Projections Parallel lines remain parallel

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Simplex Series Connect all the dots among n+1 equally spaced vertices: (Put next one above center of gravity). 1D 2D 3D This series also goes on indefinitely.

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7D Simplex A warped cube avoids intersecting diagonals. Up to 6D can be constructed with Zometool. Open problem: 7D constructible with Zometool?

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Cross Polytope Series Place vertex in + and – direction on each axis, 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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6D Cross Polytope 12 vertices suggests using icosahedron Can do with Zometool.

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6D Cross Polytope Chris Kling

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Some References Ludwig Schläfli: Theorie der vielfachen Kontinuität, 1858, (published in 1901). H. S. M. Coxeter: Regular Polytopes, 1963, (Dover reprint). Tom Banchoff, Beyond the Third Dimension, G.W. Hart, 4D Polytope Projection Models by 3D Printing to appear in Hyperspace. Carlo Sequin, 3D Visualization Models of the Regular Polytopes…, Bridges 2002.

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Puzzle Which of these shapes can / cannot be folded into a 4D hypercube? Hint: Hold the red cube still and fold the others around it. Scott Kim

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