The following families of polytopes have received a lot of attention:
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?
Best Answer
Philosophical questions deserve philosophical answers, so I am afraid no amount of references and specific results will probably satisfy you. Let me try to explain it in a somewhat generic way.
Think about it this way - why care about sequences like $\{n!\}$, Fibonacci or Catalan numbers? The honest answer is "because they come up all the time". Now, once you know these sequences, you may want to understand the underlying structures (permutations, trees, Dyck paths, triangulations, etc.) You may then want to understand connections between structures (e.g. bijections), algebraic or geometric interpretations (e.g. group representations, volumes of polytopes), etc. Once you have developed some kind of structures you may want to understand the relations between different structures, whether your bijections are structure-preserving, etc. That's how you develop the theory starting with just numbers!
In general, basic objects in combinatorics tend to lack structure. Adding structures is always welcome as they present a deeper understanding of the underlying objects (and sometimes even just numbers). It's what allows to employ and further develop tools from other parts of Combinatorics and other fields. This is the setup in which one can understand results such as Kuperberg's proof of the number of ASMs or the Adiprasito-Huh-Katz theorem, but it doesn't have to be so spectacular. Sometimes even a weak structure can lead to unexpected connections and generalizations unforeseen otherwise.
In summary, "these polytopes are just further examples of polytopes" is a misunderstanding of the context in the same way as Fibonacci and Catalan numbers are not "just numbers". Viewed in context, permutahedra and associahedra exhibit structures of combinatorial objects invisible otherwise.