From: Tom Karzes <karzes@sonic.net> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] genus-1 polytopes in 4-space Message-ID: <21773.64616.883302.948124@gargle.gargle.HOWL> Content-Type: text/plain; charset=us-ascii
I was thinking about genus-1 polytopes in 4-space, and I was wondering if some of them would be considered "regular". For example, the cartesian product of two identical regular polygons is a two-dimensional surface (like the surface of a torus), which can be distorted to fit in 3-space but which is much more symmetrical in 4-space.
As a specific example, here's a torus surface in 4-space whose cross sections are squares:
max(|w|, |x|) = 1 max(|y|, |z|) = 1
Unless I'm mistaken, all of its faces are squares (16 of them, the cartesian products of the edges of the squares), and it is face-transitive, edge-transitive, and vertex-transitive. Does that make it regular?
Tom
--By that defn, any cartesian product of any set of regular polytopes in any dimensions, would be "regular," right? I think you need to cut them down a bit, otherwise too much boring stuff is generated by your defn.