Three non-equivalent realizations of the associahedron

TL;DR

This paper reviews three different geometric realizations of the associahedron, demonstrating that they are affinely non-equivalent regardless of parameter choices, highlighting the diversity of its geometric structures.

Contribution

It introduces and compares three realizations of the associahedron, establishing their affine non-equivalence under any parameter selection.

Findings

The three realizations are affinely non-equivalent.

Each realization arises from different mathematical constructions.

The associahedron's geometric diversity is confirmed.

Abstract

We review three realizations of the associahedron that arise as secondary polytopes, from cluster algebras, and as Minkowski sums of simplices, and show that under any choice of parameters, the resulting associahedra are affinely non-equivalent.