4D Polytopesand 3D Models of Them George W. Hart Stony Brook University
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 ? 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 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? • “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 The Platonic Solids: There are only 5. Why ? …
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 Angle-deficit = 90° 2D 3D Forcing closure: ? 3D 4D creates a 3D corner creates a 4D corner
“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.
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 • 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 4D Cube 3D ( 2D )
Projections:VERTEX/ EDGE /FACE/CELL – centered • 3D Cube: Paralell proj. Persp. proj. • 4D Cube: Parallel proj. Persp. proj.
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.
Edge Treatments (Leonardo Da Vinci)
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 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. Carlo Sequin Can make with Zometool also
16-Cell or “4D Cross Polytope” • 16 cells, 32 faces, 24 edges, 8 vertices.
4D Hypercube or “Tessaract” • 8 cells, 24 faces, 32 edges, 16 vertices.
Hypercube Unfolded -- “Net” One of the 261 different unfoldings
Corpus Hypercubus Salvador Dali “Unfolded”Hypercube
24-Cell • 24 cells, 96 faces, 96 edges, 24 vertices. • (self-dual).
24-Cell “Net” in 3D Andrew Weimholt
120-Cell • 120 cells, 720 faces, 1200 edges, 600 vertices. • Cell-first parallel projection,(shows less than half of the edges.)
120-cell Model Marc Pelletier
120-Cell Carlo Séquin Thin face frames, Perspective projection.
(smallest ?) 120-Cell Wax model, made on Sanders machine
120-Cell – perspective projection Selective laser sintering
120-Cell, “exploded” Russell Towle
120-Cell Soap Bubble John Sullivan Stereographic projection preserves 120 degree angles
120-Cell “Net” with stack of 10 dodecahedra George Olshevski
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,”
600-Cell Cross-eye Stereo Picture by Tony Smith
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
600-Cell Straw model by David Richter
Slices through the 600-Cell At each Vertex: 20 tetra-cells, 30 faces, 12 edges. Gordon Kindlmann
History3D 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 • His paper shows cardboard models of layers 3 layers of 120-cell (45 dodecahedra)
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 “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 From Walther Dyck’s 1892 Math and Physics Catalog
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
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.
120 Cell • Zome Model • Orthogonal projection
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