This seems like a very natural category of polyhedron to have, and would encompass a lot of things like the Platonic and Archimedean solids and more, but I haven't been able to find a name for it anywhere.
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1$\begingroup$ At numericana.com/answer/polyhedra.htm#equi they are called equilateral polyhedra. See also tum.dds.nl/polyh/heptagons/index.html where that term is used. $\endgroup$– Barry CipraSep 14, 2019 at 2:16
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There is a term called isotoxal, meaning that there exists a symmetry, which acts on all edges transitvely, cf. https://en.wikipedia.org/wiki/Isotoxal_figure.
The prefix iso- always unites into a single equivalence class. If you just wan to tell that those are alike (here: by size), then you could use the prefix equi- instead.
--- rk
answered Sep 15, 2019 at 18:42
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$\begingroup$ Symmetry is another story. With some effort, we may produce a polyhedron with edges of same length, but without any symmetry at all. $\endgroup$ Sep 16, 2019 at 14:52
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$\begingroup$ Yes, this is why I added equitoxal instead of the more restrictive isotoxal too. $\endgroup$ Sep 16, 2019 at 16:42
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$\begingroup$ If you'd like other further restrictions, like overall convexity plus regularity of the polygonal faces, then in 3D you would just encompass the well-known set of Johnson solids, cf. en.wikipedia.org/wiki/Johnson_solid. Within 4D this same restriction would result in what is understood by the acronym CRF (convex regular faced polychora), e.g. cf. bendwavy.org/klitzing/explain/johnson.htm#crf. $\endgroup$ Sep 16, 2019 at 16:46
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$\begingroup$ equilateral would be equally (sic) acceptable. $\endgroup$ Mar 19, 2020 at 13:24