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.
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Saleh et al. (2024) studied this question.
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