4D Polytopes and 3D Models of Them Florida 1999 4D Polytopes and 3D Models of Them George W. Hart Stony Brook University

Goals of This Talk Expand your thinking. Florida 1999 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

What is the 4th Dimension ? Florida 1999 What is the 4th Dimension ? Some people think: “it does not really exist” “it’s just a philosophical notion” “it is ‘TIME’ ” . . . But, a geometric fourth dimension is as useful and as real as 2D or 3D.

Higher-dimensional Spaces Florida 1999 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.

What Is a Regular Polytope? Florida 1999 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:

Regular Convex Polytopes in 3D Florida 1999 Regular Convex Polytopes in 3D The Platonic Solids: There are only 5. Why ? …

Why Only 5 Platonic Solids ? Florida 1999 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)

Constructing a (d+1)-D Polytope Florida 1999 Constructing a (d+1)-D Polytope Angle-deficit = 90° 2D 3D Forcing closure: ? 3D 4D creates a 3D corner creates a 4D corner

Florida 1999 “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.

Florida 1999 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) ...

Wire Frame Projections Florida 1999 Wire Frame Projections Shadow of a solid object is mostly a blob. Better to use wire frame, so we can see components.

Oblique Projections Cavalier Projection 3D Cube  2D Florida 1999 Oblique Projections Cavalier Projection 3D Cube  2D 4D Cube  3D ( 2D )

Projections: VERTEX / EDGE / FACE / CELL – centered Florida 1999 Projections: VERTEX / EDGE / FACE / CELL – centered 3D Cube: Paralell proj. Persp. proj. 4D Cube: Parallel proj.

3D Objects Need Physical Edges Florida 1999 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.

Florida 1999 Edge Treatments (Leonardo Da Vinci)

How Do We Find All 4D Polytopes? Florida 1999 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°.

All Regular Convex 4D Polytopes Florida 1999 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.

5-Cell or 4D Simplex 5 cells, 10 faces, 10 edges, 5 vertices. Florida 1999 5-Cell or 4D Simplex 5 cells, 10 faces, 10 edges, 5 vertices. Carlo Sequin Can make with Zometool also

16-Cell or “4D Cross Polytope” Florida 1999 16-Cell or “4D Cross Polytope” 16 cells, 32 faces, 24 edges, 8 vertices.

4D Hypercube or “Tessaract” Florida 1999 4D Hypercube or “Tessaract” 8 cells, 24 faces, 32 edges, 16 vertices.

Hypercube, Perspective Projections Florida 1999

Nets: 11 Unfoldings of Cube Florida 1999 Nets: 11 Unfoldings of Cube

Hypercube Unfolded -- “Net” Florida 1999 Hypercube Unfolded -- “Net” One of the 261 different unfoldings

Florida 1999 Corpus Hypercubus “Unfolded” Hypercube Salvador Dali

24-Cell 24 cells, 96 faces, 96 edges, 24 vertices. (self-dual). Florida 1999 24-Cell 24 cells, 96 faces, 96 edges, 24 vertices. (self-dual).

Florida 1999 24-Cell “Net” in 3D Andrew Weimholt

120-Cell 120 cells, 720 faces, 1200 edges, 600 vertices. Florida 1999 120-Cell 120 cells, 720 faces, 1200 edges, 600 vertices. Cell-first parallel projection, (shows less than half of the edges.)

Florida 1999 120-cell Model Marc Pelletier

120-Cell Thin face frames, Perspective projection. Carlo Séquin Florida 1999 120-Cell Carlo Séquin Thin face frames, Perspective projection.

120-Cell – perspective projection Florida 1999 120-Cell – perspective projection

Florida 1999 (smallest ?) 120-Cell Wax model, made on Sanders machine

120-Cell – perspective projection Florida 1999 120-Cell – perspective projection Selective laser sintering

3D Printing — Zcorp Florida 1999

Florida 1999 120-Cell, “exploded” Russell Towle

120-Cell Soap Bubble John Sullivan Florida 1999 120-Cell Soap Bubble John Sullivan Stereographic projection preserves 120 degree angles

120-Cell “Net” with stack of 10 dodecahedra George Olshevski Florida 1999 120-Cell “Net” with stack of 10 dodecahedra George Olshevski

600-Cell -- 2D projection Oss, 1901 Florida 1999 600-Cell -- 2D projection 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 book “Regular Polytopes,”

Cross-eye Stereo Picture by Tony Smith Florida 1999 600-Cell Cross-eye Stereo Picture by Tony Smith

Florida 1999 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

Florida 1999 600-Cell Straw model by David Richter

Slices through the 600-Cell Florida 1999 Slices through the 600-Cell Gordon Kindlmann At each Vertex: 20 tetra-cells, 30 faces, 12 edges.

History 3D Models of 4D Polytopes Florida 1999 History 3D Models of 4D Polytopes Ludwig Schlafli discovered them in 1852. Worked algebraically, no pictures in his paper. Partly published in 1858 and 1862 (translation by Cayley) but not appreciated. Many independent rediscoveries and models.

Stringham (1880) First to rediscover all six Florida 1999 Stringham (1880) First to rediscover all six His paper shows cardboard models of layers 3 layers of 120-cell (45 dodecahedra)

Victor Schlegel (1880’s) Five regular polytopes Florida 1999 Victor Schlegel (1880’s) 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

Sommerville’s Description of Models Florida 1999 Sommerville’s 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.”

Cardboard Models of 120-Cell Florida 1999 Cardboard Models of 120-Cell From Walther Dyck’s 1892 Math and Physics Catalog

Paul S. Donchian’s Wire Models Florida 1999 Paul S. Donchian’s Wire Models 1930’s Rug Salesman with “visions” Wires doubled to show how front overlays back Widely displayed Currently on view at the Franklin Institute

Florida 1999 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.

Florida 1999 120 Cell Zome Model Orthogonal projection

Uniform 4D Polytopes Analogous to the 13 Archimedean Solids Florida 1999 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

Truncated 120-Cell Florida 1999

Truncated 120-Cell - Stereolithography Florida 1999

Zometool Truncated 120-Cell Florida 1999 Zometool Truncated 120-Cell MathCamp 2000

Florida 1999 Ambo 600-Cell Bridges Conference, 2001

Ambo 120-Cell Orthogonal projection Stereolithography Can do with Zome Florida 1999 Ambo 120-Cell Orthogonal projection Stereolithography Can do with Zome

Florida 1999 Expanded 120-Cell Mira Bernstein, Vin de Silva, et al.

Expanded Truncated 120-Cell Florida 1999

Florida 1999 Big Polytope “Net” George Olshevski

Big Polytope Zome Model Florida 1999 Big Polytope Zome Model Steve Rogers

Florida 1999 48 Truncated Cubes Poorly designed FDM model

Prism on a Snub Cube – “Net” Florida 1999 Prism on a Snub Cube – “Net” George Olshevski

Duo-Prisms - “Nets” Andrew Weimholt George Olshevski Andrew Weimholt Florida 1999 Duo-Prisms - “Nets” Andrew Weimholt George Olshevski Andrew Weimholt Robert Webb

Grand Antiprism “Net” with stack of 10 pentagonal antiprisms Florida 1999 Grand Antiprism “Net” with stack of 10 pentagonal antiprisms George Olshevski

Non-Convex Polytopes Components may pass through each other Florida 1999 Non-Convex Polytopes Components may pass through each other Slices may be useful for visualization Slices may be disconnected Jonathan Bowers

Beyond 4 Dimensions … What happens in higher dimensions ? Florida 1999 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

Hypercubes A.k.a. “Measure Polytope” Florida 1999 Hypercubes A.k.a. “Measure Polytope” Perpendicular extrusion in nth direction: 1D 2D 3D 4D

Orthographic Projections Florida 1999 Orthographic Projections Parallel lines remain parallel

Florida 1999 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.

7D Simplex A warped cube avoids intersecting diagonals. Florida 1999 7D Simplex A warped cube avoids intersecting diagonals. Up to 6D can be constructed with Zometool. Open problem: 7D constructible with Zometool?

A square frame for every pair of axes Florida 1999 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

6D Cross Polytope 12 vertices suggests using icosahedron Florida 1999 6D Cross Polytope 12 vertices suggests using icosahedron Can do with Zometool.

Florida 1999 6D Cross Polytope Chris Kling

Florida 1999 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, 1990. 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.

Florida 1999 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