by 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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