Finding out if a polychoron is orientable

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Finding out if a polychoron is orientable

Postby lllllllllwith10ls » Thu Nov 26, 2020 1:20 am

I was wondering if there was a good way to find out if a polychoron was orientable without needing to check all combinations of ridges you could move across. Do you guys know how?
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Re: Finding out if a polychoron is orientable

Postby Challenger007 » Fri Dec 11, 2020 2:04 pm

Have you tried looking for special software for analyzing the received data? I think it would be convenient.
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Re: Finding out if a polychoron is orientable

Postby mr_e_man » Thu May 20, 2021 3:30 am

Well, orientability is a global property, so you'll have to check all of something or other.

Take the omnitruncate's 1-skeleton, which is a (combinatorial) graph, representing the structure of flags in the polytope. (A flag is a sequence of (-1)-face (nulloid), 0-face (vertex), 1-face (edge), 2-face, ... , n-face (body), all incident with each other. The first and last are often omitted, since the nulloid and body are unique.) Two nodes in the graph being connected by an edge means that the flags are related by the dyadic property, thus sharing all but one element. It also means that the two flags should have opposite orientations. So start at one node, label it '+', and travel along the graph's edges, alternating '+' and '-' signs with each step, until all nodes have been labelled. Then check all the edges (in particular, the ones you didn't already travel along), to see whether the two nodes always have opposite signs. If so, the polytope is orientable. If not, it's non-orientable.
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