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The n -dimensional associahedron or Stasheff polytope is a convex n -polytope where each vertex corresponds to a binary tree of n + 2 leaves, equivalently the ways to bracket (or associate) 1 • 2 • ... • (n + 2) where is an infix binary operator. Two vertices are connected by an edge if their bracketings differ by only one bracket.

Associahedra were originally conceived as abstract polytopes, but they have many possible realizations as convex polytopes in Euclidean space. Jean-Louis Loday devised a particularly elegant one.[1]

The 2D associahedron is the regular pentagon.

The 3D associahedron, also known as the order-4 truncated triangular tegum, can be realized as a near-miss Johnson solid with 3 rhombi and 6 mirror-symmetric pentagons as faces. It is the result of taking a triangular bipyramid and truncating the three vertices where four triangles meet, and is the dual of the triaugmented triangular prism.

The 4D associahedron has the same symmetry as the 7-2 step prism. Its cells are 7 bilaterally symmetric associahedra and 7 bilaterally symmetric pentagonal prisms.

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