ISSN 2590-9770
The Art of Discrete and Applied Mathematics 4 (2021) #P3.07
https://doi.org/10.26493/2590-9770.1354.b40
(Also available at http://adam-journal.eu)
Locally spherical hypertopes from
generalised cubes*
Antonio Montero† , Asia Ivić Weiss
Department of Mathematics and Statistics, York University,
Toronto, Ontario M3J 1P3, Canada
Received 28 January 2020, accepted 31 July 2020, published online 23 August 2021
Abstract
We show that every non-degenerate regular polytope can be used to construct a thin,
residually-connected, chamber-transitive incidence geometry, i.e. a regular hypertope.
These hypertopes are related to the semi-regular polyotopes with a tail-triangle Coxeter
diagram constructed by Monson and Schulte. We discuss several interesting examples de-
rived when this construction is applied to generalised cubes. In particular, we produce an
example of a rank 5 finite locally spherical proper hypertope of hyperbolic type. No such
examples were previously known.
Keywords: Regularity, thin geometries, hypermaps, hypertopes, abstract polytopes.
Math. Subj. Class.: 52B15, 51E24, 51G05
*Supported by NSERC. The authors wish to thank the anonymous referee for their useful comments. Their
suggestions helped to improve the manuscript.
†Corresponding author.
E-mail addresses: amontero@yorku.ca (Antonio Montero), weiss@mathstat.yorku.ca (Asia Ivić Weiss)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 4 (2021) #P3.07
https://doi.org/10.26493/2590-9770.1354.b40
(Dostopno tudi na http://adam-journal.eu)
Lokalno sferični hipertopi, dobljeni iz
posplošenih kock*
Antonio Montero† , Asia Ivić Weiss
Department of Mathematics and Statistics, York University,
Toronto, Ontario M3J 1P3, Canada
Prejeto 28. januarja 2020, sprejeto 31. julija 2020, objavljeno na spletu 23. avgusta 2021
Povzetek
Pokažemo, da lahko iz vsakega nedegeneriranega pravilnega politopa konstruiramo
tanko, residualno povezano, komorno tranzitivno incidenčno geometrijo oz. pravilni hiper-
top. Ti hipertopi so povezani s polpravilnimi politopi z repno trikotnim Coxeterjevim dia-
gramom, ki sta ga konstruirala Monson in Schulte. Obravnavamo več zanimivih primerov,
ki jih dobimo, ko to konstrukcijo uporabimo na posplošenih kockah. Še posebej, pred-
stavimo primer končnega lokalno sferičnega pravilnega hipertopa hiperboličnega tipa ranga
5. Noben tak primer ni bil znan doslej.
Ključne besede: Pravilnost, tanke geometrije, hiperzemljevidi, hipertopi, abstraktni politopi.
Math. Subj. Class.: 52B15, 51E24, 51G05
*Podprto s strani NSERC. Avtorja se želita zahvaliti anonimnim recenzentom za njihove koristne pripombe.
Njihovi predlogi so pomagali izboljšati članek.
†Kontaktni avtor.
E-poštni naslovi: amontero@yorku.ca (Antonio Montero), weiss@mathstat.yorku.ca (Asia Ivić Weiss)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/