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4D Polytopes and 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°
creates a 3D corner creates a 4D corner
?
2D
3D 4D
3D
Forcing closure:
“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 PolytopesUsing Tetrahedra (70.5°):
3 around an edge (211.5°) (5 cells) Simplex 4 around an edge (282.0°) (16 cells) Cross polytope5 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, Perspective Projections
Nets: 11 Unfoldings of Cube
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
Thin face frames, Perspective projection.
Carlo Séquin
120-Cell – perspective projection
(smallest ?) 120-Cell
Wax model, made on Sanders machine
120-Cell – perspective projection
Selective laser sintering
3D Printing — Zcorp
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
• Oss, 1901
Frontispiece of Coxeter’s book “Regular Polytopes,”
• Total: 600 tetra-cells, 1200 faces, 720 edges, 120 vertices.
• At each Vertex: 20 tetra-cells, 30 faces, 12 edges.
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 3Dprojection 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
Truncated 120-Cell
Truncated 120-Cell - Stereolithography
Zometool Truncated 120-Cell
MathCamp 2000
Ambo 600-Cell
Bridges Conference, 2001
Ambo 120-Cell
Orthogonal projection
Stereolithography
Can do with Zome
Expanded 120-Cell
Mira Bernstein,Vin de Silva, et al.
Expanded Truncated 120-Cell
Big Polytope
“Net”
George Olshevski
Big Polytope Zome Model
Steve Rogers
48 Truncated Cubes
Poorly designed FDM model
Prism on a Snub Cube – “Net”
George Olshevski
Duo-Prisms - “Nets”
Robert WebbAndrew Weimholt
Andrew WeimholtGeorge Olshevski
Grand Antiprism “Net”
with stack of 10 pentagonal antiprisms
George Olshevski
Non-Convex Polytopes
Jonathan Bowers
• Components may pass
through each other
• Slices may be useful for visualization
• Slices may be disconnected
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”• Perpendicular extrusion in nth direction:
1D 2D 3D 4D
Orthographic Projections
Parallel lines remain parallel
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.
Up to 6D can be constructed with Zometool.
Open problem: 7D constructible with Zometool?
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 Polytope12 vertices suggests using icosahedron
Can do with Zometool.
6D Cross Polytope
Chris Kling
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.
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