Equivalence Between Four Models of Associahedra

We present a combinatorial isomorphism between Stasheff associahedra and an inductive cone construction of those complexes given by Loday. We give an alternate description of certain polytopes, known as multiplihedra, which arise in the study of $A_{\mathrm{\infty }}$ maps. We also prove a combinatorial isomorphism between Stasheff associahedra, collapsed multiplihedra and graph cubeahedra for path graphs.