### Russell Towle's 4D Star Polytope Animations

You need the QuickTime player for these animations.

### Vertex-First Sections:

 Bytes Contains: Schläfli Symbol Screen Shot Download (USA) Download (Japan) 1,283,990 {3,3,52}vert.mov {3,3,5/2} (click) 3-3-52v.zip 3-3-52v.zip 1,766,046 {5,3,52}vert.mov {5,3,5/2} (click) 5-3-52v.zip 5-3-52v.zip 1,794,259 {5,52,3}vert.mov {5,5/2,3} (click) 5-52-3v.zip 5-52-3v.zip 1,459,341 {5,52,5}vert.mov {5,5/2,5} (click) 5-52-5v.zip 5-52-5v.zip 2,101,058 {52,3,3}vert.mov {5/2,3,3} (click) 52-3-3v.zip 52-3-3v.zip 886,771 {52,3,5}vert.mov {5/2,3,5} (click) 52-3-5v.zip 52-3-5v.zip

### Acknowledgement:

Japan web host space provided by Junichi Yananose (who also has interesting stuff, including polyhedra).

### Notes from Russell:

These may be the first animations ever made of the solid sections of four-dimensional star polytopes. To get a better idea of just what these "polytopes" are, one should read H.S.M. Coxeter's "Regular Polytopes" (Coxeter01). Briefly, plane polygons are two-dimensional polytopes, and polyhedra, three-dimensional polytopes. Where polygons are bounded by line segments, and polyhedra by polygons, a 4-polytope is bounded by polyhedra.

Just as we may have any number of planes in three dimensions, in 4-space we may have any number of 3-spaces. Two 3-spaces might be a millionth of an inch apart and yet have no common point (thus the popular idea of parallel universes). It follows that, given a fixed direction in the 4-space, we can take solid sections of objects in the 4-space, perpendicular to that direction.

If you find these concepts difficult, you are not alone. Even when a person is blessed with some extraordinary faculty for visualizing objects in higher space--as was Alicia Boole Stott, a century ago--it is a matter of years, and considerable patience, before much progress is made in the subject.

In these animations, a 3-space is passed from one vertex of each star polytope, to the opposite vertex, and sections taken at small intervals. The star polytopes were constructed, and the sections found, using Mathematica 4.0. The sections were rendered in POV-Ray (a freeware ray-tracer).

Contact: Russell Towle