The following families of polytopes have received a lot of attention:

- permutahedra,
- associahedra,
- cyclohedra,
- ...

My question is simple:

Why?

As I understand, at least the latter two were initially constructed by their face lattice representing certain combinatorial objects (e.g. ways to insert parentheses into a string). So I assumed that representing these structures as a face lattice was of some use.

But then people got interested in realizing these objects geometrically, and it turns out that, e.g. the associahedron can be realized in many ways. Was this surprising? Is there something to be learned from that fact? On the other hand, for the permutahedron the realization came probably first, so is there anything deep to learn from its combinatorial structure?

Further, there seem to exist connections to algebra, e.g. homotopy theory. I cannot wrap my head around these connections. For me, these polytopes are just further examples of polytopes, nothing else.

So what's up? Do they have some extremal properties? Are they especially symmetric (i.e. are they interesting for their symmetries)? Does the geometric point of view make apparent some hidden combinatorial properties of the underlying structures (e.g. the cyclohedron is said to be "useful in studying knot invariants")? What justifies this interest?