Realizations of the associahedron and cyclohedron
The associahedron first arose purely as the graph corresponding to the bistellar operations on triangulations of a convex polygon which only use the vertices of the polygon. Later, it was shown how to realize this graph as the edge-graph of a suitable simple convex polytope. Indeed, this polytope can be taken to be the secondary polytope (a special case of a fibre polytope) of the projection of an appropriate simplex on the polygon. In a similar way, the cyclohedron corresponds to centrally symmetric triangulations of a centrally symmetric polygon. There is also a connexion with the permutahedra arising from the Coxeter groups of types \(A\) and \(B\), respectively. In this paper, the authors discuss realizations of these polytopes with integer coordinates for their vertices, comparing them with these permutahedra; these coordinates are obtained through an algorithm which uses the corresponding oriented Coxeter graphs of types \(A\) and \(B\) as the only input data.
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