Wikipedia:Reference desk/Archives/Mathematics/2024 June 1

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June 1

Antiprisms in Higher dimensions.

Antiprism talks about higher dimensions, but only in the context of four dimensional Antiprisms created from a Polyhedron and its Polar dual. Is there any reason not to extend this to, for example, being able to make an n+1 dimensional Antiprism out of the n dimensional cube and the n dimensional orthoplex or the 24-cell with itself? Also would the 24-cell anti-prism defined this way be a uniform 5-polytope or is the fact that all but two of the 4-dimensional facets are octohedral pyramids make it non-uniform?Naraht (talk) 19:51, 1 June 2024 (UTC)

That section also mentions five-dimensional antiprisms:

"However, there exist four-dimensional polyhedra that cannot be combined with their duals to form five-dimensional antiprisms.^[8]"

Apparently, the generalization to higher dimensions is not straightforward.  --Lambiam 04:07, 2 June 2024 (UTC)

However, the fact that it needs to be constructed *probably* means that it doesn't apply to any of the six regular 4-polytopes. Looked at the paper on www.semanticscholar.org . Interesting.Naraht (talk) 16:02, 3 June 2024 (UTC)

A 24-cell antiprism exists. However, the usual definition of a uniform polytope doesn't just require congruent edges and transitivity of the symmetry group on the vertices, but also that all facets are uniform. The 24-cell antiprism satisfies the first two requirements but not the last one, indeed because octahedral pyramids are not uniform. Such polytopes, satisfying all requirements of uniformity except for the uniformity of all facets, have been called "scaliform".

The n-simplex antiprism always exists, and is (if you pick the appropriate height, to give congruent edges) the (n+1)-orthoplex. However, the tesseract pyramid has zero height if you try to make it with congruent edges, because the tesseract's circumradius equals its edge length. It follows therefore that the penteract antiprism cannot be made with congruent edges, because it would need to have tesseract pyramids among its facets; and for similar reasons, there are no higher hypercube antiprisms with congruent edges. Likewise, you can't make a congruent-edged 120-cell antiprism because the dodecahedral pyramid cannot be made with congruent edges (since the circumradius of the dodecahedron is greater than its edge length). These can all nonetheless be made if you relax the assumption of congruent edges. Double sharp (talk) 09:01, 14 June 2024 (UTC)

@Naraht: Answered your question (forgot to ping you, sorry). Double sharp (talk) 09:02, 14 June 2024 (UTC)