ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.01
https://doi.org/10.26493/1855-3974.2712.6be
(Also available at http://amc-journal.eu)
The edge-transitive polytopes that are not
vertex-transitive*
Frank Göring
Faculty of Mathematics, University of Technology, 09107 Chemnitz, Germany
Martin Winter †
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Received 27 October 2021, accepted 19 March 2022, published online 28 October 2022
Abstract
In 3-dimensional Euclidean space there exist two exceptional polyhedra, the rhombic
dodecahedron and the rhombic triacontahedron, the only known polytopes (besides poly-
gons) that are edge-transitive without being vertex-transitive. We show that these poly-
hedra do not have higher-dimensional analogues, that is, that in dimension d ≥ 4, edge-
transitivity of convex polytopes implies vertex-transitivity.
More generally, we give a classification of all convex polytopes which at the same time
have all edges of the same length, an edge in-sphere and a bipartite edge-graph. We show
that any such polytope in dimension d ≥ 4 is vertex-transitive.
Keywords: Convex polytopes, symmetry of polytopes, vertex-transitive, edge-transitive.
Math. Subj. Class. (2020): 52B15, 52B11
*This article appears as Chapter 6 in the second author’s doctoral thesis from 2021. The authors thank the
anonymous referees for their careful reading and their many remarks that led to an improvement of the arti-
cle in several ways.
†Corresponding author.
E-mail addresses: frank.goering@mathematik.tu-chemnitz.de (Frank Göring),
martin.h.winter@warwick.ac.uk (Martin Winter)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 1855-3966 (tiskana izd.), ISSN 1855-3974 (elektronska izd.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.01
https://doi.org/10.26493/1855-3974.2712.6be
(Dostopno tudi na http://amc-journal.eu)
Povezavno tranzitivni politopi, ki niso točkovno
tranzitivni*
Frank Göring
Faculty of Mathematics, University of Technology, 09107 Chemnitz, Germany
Martin Winter †
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Prejeto 27. oktobra 2021, sprejeto 19. marca 2022, objavljeno na spletu 28. oktobra 2022
Povzetek
V 3-dimenzionalnem evklidskem prostoru obstajata dva izjemna poliedra, rombski do-
dekaeder in rombski triakontaeder, edina znana politopa (poleg poligonov), ki sta pove-
zavno tranzitvna, ne pa tudi točkovno tranzitivna. Pokažemo, da za ta dva poliedra ne
obstajajo nobene podobne strukture v višjih dimenzijah, to pomeni, da v dimenziji d ≥ 4
povezavna tranzitivnost konveksnega politopa implicira točkovno tranzitivnost.
Splošneje, podamo klasifikacijo vseh konveksnih politopov, ki imajo hkrati vse povezave
iste dolžine, včrtano sfero, ki se vsake povezave dotika v eni sami točki, ter dvodelen
povezavni graf. Pokažemo, da je vsak tak politop v dimenziji d ≥ 4 točkovno tranzitiven.
Ključne besede: Konveksni politopi, simetrija politopov, točkovno tranzitiven, povezavno tranzitiven.
Math. Subj. Class. (2020): 52B15, 52B11
*Ta članek predstavlja 6. poglavje doktorske disertacije drugega avtorja iz leta 2021. Avtorja se zahvaljujeta
neznanim recenzentom za skrbno branje in številne pripombe, ki so vodile k marsikaterim izboljšavam.
†Kontaktni avtor.
E-poštna naslova: frank.goering@mathematik.tu-chemnitz.de (Frank Göring),
martin.h.winter@warwick.ac.uk (Martin Winter)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/