Geodesic
Symmetries of 3-polytopes with fixed edge lengths
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1580-1587

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We consider an interesting class of combinatorial symmetries of polytopes which we call edge-length preserving combinatorial symmetries. These symmetries not only preserve the combinatorial structure of a polytope but also map each edge of the polytope to an edge of the same length. We prove a simple sufficient condition for a polytope to realize all edge-length preserving combinatorial symmetries by isometries of ambient space. The proof of this condition uses Cauchy's rigidity theorem in an unusual way.
Keywords: polytope, isometry, edge-length preserving combinatorial symmetry, circle pattern. 
@article{SEMR_2020_17_a57,
     author = {E. A. Morozov},
     title = {Symmetries of 3-polytopes with fixed edge lengths},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1580--1587},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a57/}
}
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E. A. Morozov. Symmetries of 3-polytopes with fixed edge lengths. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1580-1587. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a57/