## Cardboard associahedron

After the dodecahedron comes the cardboard associahedron : it represents the combinatorics of moving parentheses to calculate an associative product of five things step by step. The vertices of the associahedron are thus the 14 different binary trees with 5 leaves.

Given four factors, there are exactly five ways of multiplying them using a binary operation: by moving parentheses according to the associativity rule, you go through five different trees in a cyclic way. This is known as MacLane’s pentagon coherence rule, which states that in not too weak notions of monoid, checking coherence for pentagon diagrams ensures that the definition of the product is well-behaved.

This associahedron can also be used to define $A_5$-algebras: such an algebra is a differential graded vector space (or abelian group), with a binary multiplication, and ternary operation $\mu_3(a,b,c)$ whose boundary is the difference between $(ab)c$ and $a(bc)$. Trees of operations containing a ternary vertex are represented in the associahedron are edges (there are 21 of them) between the corresponding binary trees. There is also a quaternary operation whose boundary is the sum of the five operations represented by the edges of the associated pentagonal diagram, and a quinary operation whose boundary is the sum of trees containing either a quaternary vertex (representing by 6 pentagons) or two ternary vertices (represented by 3 squares).

Associahedra exist in all dimensions, they have $\frac 1{n+1} \binom{2n}{n}$ vertices corresponding to the Catalan number of possible binary trees. In low dimensions, they look like a point, an interval, a pentagon, and the cardboard associahedron. Composition of trees defines an operad structure on the associahedra (maybe a cellular operad, or a topological operad).

The whole theory of $\mathcal A_\infty$-algebras is encoded by a differential graded operad, which is the chain complex of the operad of associahedra. It was used by Stasheff to define homotopy-associative monoids (thus the associahedra are also called Stasheff polytopes).