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Is it possible for an edge to connect two non-adjacent vertices of a polygonal face in a regular abstract polytope? Here “adjacent” means that the two vertices are connected by an edge that is a facet of the face.

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  • $\begingroup$ It is common to say that, if two vertices are connected by an edge, they are adjacent. So it seems your question might depend on the meaning of "edge" and "adjacent"? $\endgroup$ Apr 30, 2022 at 16:50
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    $\begingroup$ I clarified the question. $\endgroup$ Apr 30, 2022 at 18:13
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    $\begingroup$ I think this would violate "intersection property", as defined in the book by McMullen and Schulte. $\endgroup$ Apr 30, 2022 at 22:26
  • $\begingroup$ @Dima, can you elaborate? $\endgroup$ May 1, 2022 at 8:46
  • $\begingroup$ IIRC, it's the same intersection property as pops up in diagram geometries (in sense of Buekenhout-Tits). Flat geometries don't satisfy intersection property. $\endgroup$ May 2, 2022 at 19:59

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Yes, it is possible.

An example is the hemicube (Wikipedia, Atlas of small regular polytopes, Weddslist). It has four vertices, six edges (every pair of vertices is connected by an edge), and three square faces. The funny thing is that every face also contains all four vertices, but in different orders.

Here is a nice hemicube picture from Wikipedia (by user Apocheir, source). We can see, for example, that vertices $a$ and $b$ are connected by an edge ($1$), which lies on two faces (I and II), but not on the third face (III).

So edge $1$ connects vertices that, on another face (III), are nonadjacent.

Hemicube

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