Abstract
In the classical setting, a convex polytope is called semiregular if its facets are regular and its symmetry group is transitive on vertices. This paper continues our study of alternating abstract semiregular polytopes \(\mathcal {S}\). These structures have two kinds of abstract regular facets \(\mathcal {P}\) and \(\mathcal {Q}\), still with combinatorial automorphism group transitive on vertices. Furthermore, for some interlacing number \(k\geqslant 1\), k copies each of \(\mathcal {P}\) and \(\mathcal {Q}\) can be assembled in alternating fashion around each face of co-rank 2 in \(\mathcal {S}\). Here we focus on constructions involving interesting pairs of polytopes \(\mathcal {P}\) and \(\mathcal {Q}\). In some cases, \(\mathcal {S}\) can be constructed for general values of k. In other remarkable instances, interlacing with certain finite interlacing numbers proves impossible.
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Notes
Polytopes like this are often called quasi-regular in the literature. See [2, pp. 18, 69], for example, where the cuboctahedron is denoted by \(\bigl \{{}^{3}_{4}\bigr \}\). We generalize such notation later.
We thank the referee for suggesting this diagram, along with other improvements.
As the referee pointed out, this is the pullback of the canonical projections for \(N_\mathcal {P}\) and \(N_\mathcal {Q}\),
in the category of groups. We also thank the referee for simplifying the proof of Lemma 5.1.
We include chiral polytopes here for completeness but actually do not need to work with any of their special properties. A chiral polytope has two flag orbits under its full automorphism group, with adjacent flags always in different orbits. We refer the reader to [21] for more details.
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Egon Schulte: Supported by the Simons Foundation Award No. 420718.
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Monson, B., Schulte, E. The Assembly Problem for Alternating Semiregular Polytopes. Discrete Comput Geom 64, 453–482 (2020). https://doi.org/10.1007/s00454-019-00118-6
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DOI: https://doi.org/10.1007/s00454-019-00118-6