# Vietoris-Rips Complexes of Regular Polygons

#### Main Theorem

Abstract. Persistent homology has emerged as a novel tool for data analysis in the past two decades. However, there are still very few shapes or even manifolds whose persistent homology barcodes (say of the Vietoris-Rips complex) are fully known. Towards this direction, we provide a near-complete characterization of the homotopy types of Vietoris-Rips complexes of any regular polygon in the plane. Indeed, for a regular polygon $$P_n$$ with $$n$$ sides, we describe the homotopy types and persistent homology of its Vietoris-Rips complexes up to scale $$r_n$$, where $$r_n$$ approaches the diameter of $$P_n$$ as $$n\to\infty.$$ Surprisingly, these homotopy types include spheres of all dimensions (as $$n\to\infty$$). Roughly speaking, the number of higher-dimensional spheres appearing is linked to the number of equilateral (but not necessarily equiangular) stars that can be inscribed into $$P_n$$; our main tool is the recently-developed theory of cyclic graphs and winding fractions. Furthermore, we show that the Vietoris-Rips complex of an arbitrarily dense subset of $$P_n$$ need not be homotopy equivalent to the Vietoris-Rips complex of $$P_n$$ itself.