Depends on how you define "polytope", doesn't it --- which isn't straightforward in general dimension. But I imagine that people who think about these things would normally expect a 4-D polytope to possess 3-D elements, meeting in pairs at the faces. https://en.wikipedia.org/wiki/Polytope But there are plenty of well-established examples atypical in some respect, such as mathworld.wolfram.com/11-Cell.html Fred Lunnon On 3/21/15, Tom Karzes <karzes@sonic.net> wrote:
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
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