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.
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George W. Hart
Stony Brook University
Note: Some Material and images adapted from Carlo Sequin
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.
The Platonic Solids:
There are only 5. Why ? …
Try to build all possible ones:
Angle-deficit = 90°
creates a 3D corner
creates a 4D corner
3D Cube 2D
4D Cube 3D ( 2D )
(Leonardo Da Vinci)
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.
Can make with Zometool also
One of the 261 different unfoldings
Thin face frames, Perspective projection.
Wax model, made on Sanders machine
Selective laser sintering
Stereographic projection preserves 120 degree angles
with stack of 10 dodecahedra
Frontispiece of Coxeter’s book “Regular Polytopes,”
Cross-eye Stereo Picture by Tony Smith
Straw model by David Richter
At each Vertex: 20 tetra-cells, 30 faces, 12 edges.
3 layers of 120-cell
Invented “Schlegel Diagram”
3D 2D perspective transf.
Used analogous 4D 3D
projection in educational
Built wire and thread models.
Advertised and sold models via commercial catalogs: Dyck (1892) and Schilling (1911).
Some stored at Smithsonian.
Five regular polytopes
“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.”
From Walther Dyck’s 1892 Math and Physics Catalog
Bridges Conference, 2001
Can do with Zome
Vin de Silva, et al.
Poorly designed FDM model
with stack of 10 pentagonal antiprisms
1D 2D 3D 4D
Parallel lines remain parallel
This series also goes on indefinitely.
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
6 square frames= 24 edges
12 vertices suggestsusing icosahedron
Can do with Zometool.