Below are Quicktime .movs depicting the slicing of four-dimensional stellated polytopes, created by Russell Towle. These are hypnotic, complex patterns, strangely organic for being so hard-edged and geometric. Towle explains: "These may be the first animations ever made of the solid sections of four-dimensional star polytopes. [...] 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. [...] 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)." The screenshot above is from the last .mov, "{5/2,3,5}."
[.mov files removed -- they no longer work in current (2017) browsers -- thanks, Apple!]
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Below are Quicktime .movs depicting the slicing of four-dimensional stellated polytopes, created by Russell Towle. These are hypnotic, complex patterns, strangely organic for being so hard-edged and geometric. Towle explains: "These may be the first animations ever made of the solid sections of four-dimensional star polytopes. [...] 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. [...] 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)." The screenshot above is from the last .mov, "{5/2,3,5}."
[.mov files removed -- they no longer work in current (2017) browsers -- thanks, Apple!]
[via]
- tom moody 9-18-2006 10:07 am