For n >= 6, you can take a Császár polyhedron sharing the vertex-set (and indeed edge-set) with a regular 7-vertex simplex. The problems for n in {3, 4, 5} are certainly interesting! -- APG.
Sent: Saturday, May 23, 2020 at 4:48 PM From: "Dan Asimov" <dasimov@earthlink.net> To: "math-fun" <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Can an equilateral toroidal polyhedron
----- have fewer than 32 faces? —rwg -----
If it's not embedded anywhere, it can have just two faces.
If it's embedded in n-space for fixed n, this is a terrific question!
(Likewise for other surfaces, like the Klein bottle.)
—Dan
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