On Vietoris-Rips Complexes of the 2-Sphere

We provide statistical evidence of the homotopy types of Vietoris-Rips complexes for circles. We determine the homotopy type of the Vietoris-Rips complexes of the vertex sets of the platonic solids, with a focus on the dodecahedron. We also offer homology computations of some higher-dimensional polytopes, especially hypercubes and 4-dimensional polytopes.