|Bytes||Contains:||Schläfli Symbol||Screen Shot||Download (USA)||Download (Japan)|
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).