# Vietoris-Rips complexes of Platonic solids

## TL;DR

This paper investigates the topological properties of Vietoris-Rips complexes constructed from the vertex sets of Platonic solids, revealing complex homotopy types, notably for the dodecahedron.

## Contribution

It determines the homotopy types of Vietoris-Rips complexes for Platonic solids, including the novel result that the dodecahedron's complex is a wedge of nine 3-spheres.

## Key findings

- Vietoris-Rips complex of the dodecahedron is a wedge of nine 3-spheres between distances 3 and 4.
- Homotopy types of complexes for other Platonic solids are characterized.
- Provides new insights into the topology of complexes derived from symmetric polyhedra.

## Abstract

We determine the homotopy type of the Vietoris-Rips complexes of the (vertex sets of the) platonic solids. The most interesting case is that the Vietoris-Rips complex of the dodecahedron is a wedge of nine 3-spheres when the parameter is between combinatorial distance 3 and 4.