Associahedron
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. JeanLouis Loday devised a particularly elegant one.^{[1]}
The 2D associahedron is the regular pentagon.
The 3D associahedron, also known as the order4 truncated triangular tegum, can be realized as a nearmiss Johnson solid with 3 rhombi and 6 mirrorsymmetric 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 72 step prism. Its cells are 7 bilaterally symmetric associahedra and 7 bilaterally symmetric pentagonal prisms.
Gallery[edit  edit source]

3D associahedron

Net of 4D associahedron
See also[edit  edit source]
External links[edit  edit source]
 Wikipedia contributors. "Associahedron".
 Weisstein, Eric W. "Associahedron" at MathWorld.
References[edit  edit source]
 ↑ JeanLouis Loday. "Realization of the Stasheff polytope."