The Assembly Problem for Alternating Semiregular Polytopes

  • Barry Monson
  • Egon SchulteEmail author
Branko Grünbaum Memorial Issue


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.


Abstract semiregular polytopes Tail-triangle C-groups 

Mathematics Subject Classification

Primary 51M20 Secondary 52B15 



Egon Schulte: Supported by the Simons Foundation Award No. 420718.


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Authors and Affiliations

  1. 1.University of New BrunswickFrederictonCanada
  2. 2.Northeastern UniversityBostonUSA

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