Cyclohedra belong to several larger families of polytopes, each providing a general construction. For instance, the cyclohedron belongs to the generalized associahedra that arise from
cluster algebra, and to the graph-associahedra, a family of polytopes each corresponding to a
graph. In the latter family, the graph corresponding to the d-dimensional cyclohedron is a cycle on d+1 vertices. In topological terms, the
configuration space of d+1 distinct points on the circle S^1 is a (d+1)-dimensional
manifold, which can be
compactified into a
manifold with corners by allowing the points to approach each other. This
compactification can be factored as S^1 \times W_{d+1}, where W_{d+1} is the d-dimensional cyclohedron. Just as the associahedron, the cyclohedron can be recovered by removing some of the
facets of the
permutohedron. ==Properties==