08 March 2005

Hyperdimensional "Solids"

It's fun to imagine in 4 or more dimensions!




Poltyope visualization using "Peek". The Peek code is currently under (re)development and is apparently not available for download, but the screen shots are Very Cool. You can watch a hypersolid rotate and slide its way through our 3-space.

And, of course, today's new kernels of knowledge:

  • "Polychoron" is 4-space's equivalent of the concept "polyhedron".
  • "Polytope" is the generalized n-dimensional term for "polygon/polyhedron/polychoron..."
  • In 3-space, there are 5 regular polyhedra. Per Euclid and Plato, these are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
  • In 4-space, there are 6 regular polychora. Per Schlafli and Coxeter, these are the tesseract, '5-cell', '16-cell', '24-cell', '120-cell', and '600-cell'. There are many alternative namees for each of these.
  • In 5-space and above, there are always 3 (!) regular polytopes. These are the n-simplex, the n-cube, and the n-D cross-polytope.
See for example this site on the history of regular convex polytopes.

Much of the above taken from "MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com"