The associahedra form a family of polytopes whose two-dimensional member is a pentagon. The associahedron was discovered by Stasheff in 1963 and is also known as the Stasheff polytope. The vertices of the -dimensional associahedron correspond to ordered full binary trees with internal vertices, so their number is the Catalan number .

The associahedron is the basic tool in the study of homotopy associative Hopf spaces.

Loday (2004) constructs an associahedron using the set of ordered full binary trees with leaves. Number the internal vertices from left to right in inorder traversal. Let and count the leaves in the left and right subtrees of the th internal vertex, respectively. For in , define

The -associahedron is then defined as the convex hull of .

The associahedron can be obtained by removing facets from the permutohedron, and is related to the cyclohedron and permutohedron.

For each , the flip graph of triangulations of an -sided convex polygon is the 1-skeleton of the -dimensional associahedron. Two triangulations correspond to adjacent vertices when a single diagonal is replaced by the other diagonal of the quadrilateral formed by its two incident triangles. Tan (2026) proved that each graph in this family is 12-colorable.

See also

Cyclohedron, Pentagon, Permutohedron, Polytope

Portions of this entry contributed by Bryan Jacobs

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References

Baez, J. "The Associahedron and Little -Cubes Operads." https://math.ucr.edu/home/baez/hda/associahedron.html.Hemmi, Y. "Associahedron, Permutohedron." http://www.math.kochi-u.ac.jp/hemmi/kenkyuu/polytope.html.Hohlweg, C. and Lange, C. "Realizations of the Associahedron and Cyclohedron." 2 Dec 2005. https://arxiv.org/abs/math/0510614.Loday, J.-L. "Realization of the Stasheff Polytope." Arch. Math. 83, 267-278, 2004.Markl, M. "Simplex, Associahedron, and Cyclohedron." 9 Jul 1997. https://arxiv.org/abs/alg-geom/9707009.Postnikov, A. "Permutohedra, Associahedra, and Beyond." https://math.mit.edu/~apost/papers/permutohedron.pdf.Starck, M. "3D Representations." http://www.ac-noumea.nc/maths/amc/polyhedr/3D-img_.htm.Tan, J. "Associahedra Are 12-Colourable." 3 Nov 2026. https://arxiv.org/abs/2611.05323v1.

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Associahedron

Cite this as:

Weisstein, Eric W., with contributions by Bryan Jacobs. "Associahedron." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Associahedron.html

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