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The recently posted arxiv paper Spectrum of the Laplacian on Regular Polyhedra by Evan Greif, Daniel Kaplan, Robert S. Strichartz, and Samuel C. Wiese, collects numerical evidence for conjectured asymptotic estimates of the eigenvalue counting function (found in the paper here of Strichartz).

For higher dimensional domains (like the regular convex polytopes), the problem of estimating eigenvalues becomes much more involving numerically, which makes it valuable to have polytopes for which the Laplace-Beltrami eigenvalues are known exactly to compare numerics against. How many such examples are known?

In three dimensions the regular tetrahedron is such an example. Are there other examples in higher dimensions?

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  • $\begingroup$ Note their computation for the regular tetrahedron relies on using the fact that it's covered by torus. If a polytope surface is not covered by a torus then I am skeptical that its eigenvalues can be explicitly computed. $\endgroup$ – Neal Dec 17 '18 at 21:15

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