How many "different" triangulations does $S^1$ have? Let us say that two triangulations $T, T'$ of a space $X$ are "the same" if there is a homeomorphism $f:X \to X$ such that, for every $k, f$ identifies each $k-$simplex of $T$ with another $k-$simplex of $T'$.
Classical triangulations of $S^1$ are the following: $S^1$ in itself, with $2$ vertices and $2$ edges, an equilateral triangle, a square and so on. More generally, the circle can be viewed as a "curved" regular $n-$gon, and this are triangulations of $S^1$ (where the $1-$gon is the circle itself with one edge and one vectex, and the $2-$gon the the circle with $2$ vertices and $2$ edges). Also, these triangulations are "different", since they have a different number of vertices ($0-$simplex) and edges ($1-$simplex). However, how do we prove there aren't any other triangulations?