I was initially reading about the Apeirion concept in philosophy when I came across this mathematical concept of apeirogons and its higher-dimensional analogues. Being a layman in mathematics, the definition of an apeirogon doesn't seem of much use. I imagine that they must have some applications in higher-level mathematics, but I couldn't come up with any ideas.
What applications of these infinite-sided polygons, polyhedra, and/or n-polytopes do you know of?
They're one case of the classification of abstract polygons (or polyhedra, or polytopes, for the higher-dimensional versions). It's not a very interesting case, but we include it so that the classification is complete.
In hyperbolic space, apeirogons can be more interesting, because they can be closer to our usual definition of a polygon: sides of the same length with the same (nontrivial!) angle at every corner. For example, here is a tiling of the hyperbolic plane by infinitely many apeirogons, which is no different in spirit from tiling the Euclidean plane by infinitely many hexagons.