  Consider $2n$ points on the unit circle and a reference dissection
$\mathrm{D}_\circ$ of the convex hull of the odd points. The accordion complex
of $\mathrm{D}_\circ$ is the simplicial complex of non-crossing subsets of the
diagonals with even endpoints that cross a connected subset of diagonals of
$\mathrm{D}_\circ$. In particular, this complex is an associahedron when
$\mathrm{D}_\circ$ is a triangulation and a Stokes complex when
$\mathrm{D}_\circ$ is a quadrangulation. In this paper, we provide geometric
realizations (by polytopes and fans) of the accordion complex of any reference
dissection $\mathrm{D}_\circ$, generalizing known constructions arising from
cluster algebras.

Combinatorics