The -dimensional cyclohedron and the correspondence between its vertices and edges with a cycle on three vertices
In
geometry, the cyclohedron is a -dimensional
polytope where can be any non-negative integer. It was first introduced as a combinatorial object by
Raoul Bott and
Clifford Taubes[1] and, for this reason, it is also sometimes called the Bott–Taubes polytope. It was later constructed as a polytope by Martin Markl[2] and by
Rodica Simion.[3] Rodica Simion describes this polytope as an
associahedron of type B.
Cyclohedra belong to several larger families of polytopes, each providing a general construction. For instance, the cyclohedron belongs to the generalized associahedra[5] that arise from
cluster algebra, and to the graph-associahedra,[6] a family of polytopes each corresponding to a
graph. In the latter family, the graph corresponding to the -dimensional cyclohedron is a cycle on vertices.
Just as the associahedron, the cyclohedron can be recovered by removing some of the
facets of the
permutohedron.[7]
Properties
The graph made up of the vertices and edges of the -dimensional cyclohedron is the
flip graph of the centrally symmetric
triangulations of a
convex polygon with vertices.[3] When goes to infinity, the asymptotic behavior of the diameter of that graph is given by