maisМоделирование и анализ информационных системModeling and Analysis of Information Systems1818-10152313-5417Yaroslavl State University10.18255/1818-1015-2013-6-103-110mais-161Research ArticleОригинальные статьиArticlesПравильные полигональные комплексы высших рангов в E³Regular Polygonal Complexes of Higher Ranks in E³ЭйгонШультеSchulteEgon

факультет математики, профессор,

Хаттингтон авеню, 360, Бостон, MA 02115, США

профессор, Department of Mathematics, 360 Huntington Avenue, Boston, MA 02115, USA

schulte@neu.eduСеверо-Восточный университет СШАСоединённые Штаты АмерикиNortheastern UniversityUnited States201320122013206103110Copyright © Эйгон Ш., 20132013Эйгон Ш.Schulte E.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.mais-journal.ru/jour/article/view/161

В статье рассматриваются ранги правильных полигональных комплексов и устанавливается, что ранги таких комплексов в E³ не могут превышать 4 и в E³ имеется только восемь правильных полигональных комплексов ранга 4. Статья публикуется в авторской редакции.

The paper establishes that the rank of a regular polygonal complex in E³ cannot exceed 4, and that the only regular polygonal complexes of rank 4 in E³ are the eight regular 4-apeirotopes in E³ . The article is published in the author’s wording.

полигональные комплексыабстрактные многогранникирегулярностьpolygonal complexabstract polytopesregularityReferencesH.S.M. Coxeter. Regular Polytopes (3rd edition), Dover.H.S.M. Coxeter. Regular Polytopes (3rd edition), Dover.H.S.M. Coxeter. Regular skew polyhedra in 3 and 4 dimensions and their topological analogues, Proc. London Math. Soc. (2) 43 (1937), 33–62. (Reprinted with amendments in Twelve Geometric Essays, Southern Illinois University Press (Carbondale, 1968), 76–105.)H.S.M. Coxeter. Regular skew polyhedra in 3 and 4 dimensions and their topological analogues, Proc. London Math. Soc. (2) 43 (1937), 33–62. (Reprinted with amendments in Twelve Geometric Essays, Southern Illinois University Press (Carbondale, 1968), 76–105.)L. Danzer, E. Schulte. Regul¨are Inzidenzkomplexe, I, Geom. Dedicata 13 (1982), 295–308.L. Danzer, E. Schulte. Regul¨are Inzidenzkomplexe, I, Geom. Dedicata 13 (1982), 295–308.A.W.M. Dress. A combinatorial theory of Gr¨unbaum’s new regular polyhedra, I: Gr¨unbaum’s new regular polyhedra and their automorphism group, Aequationes Math. 23 (1981), 252–265.A.W.M. Dress. A combinatorial theory of Gr¨unbaum’s new regular polyhedra, I: Gr¨unbaum’s new regular polyhedra and their automorphism group, Aequationes Math. 23 (1981), 252–265.A.W.M. Dress. A combinatorial theory of Gr¨unbaum’s new regular polyhedra, II: complete enumeration, Aequationes Math. 29 (1985), 222–243.A.W.M. Dress. A combinatorial theory of Gr¨unbaum’s new regular polyhedra, II: complete enumeration, Aequationes Math. 29 (1985), 222–243.B. Gr¨unbaum. Regular polyhedra — old and new, Aequat. Math. 16 (1977), 1–20.B. Gr¨unbaum. Regular polyhedra — old and new, Aequat. Math. 16 (1977), 1–20.P. McMullen. Realizations of regular polytopes, Aequationes Math. 37 (1989), 38–56.P. McMullen. Realizations of regular polytopes, Aequationes Math. 37 (1989), 38–56.P. McMullen. Geometric regular polytopes, monograph in preparation.P. McMullen. Geometric regular polytopes, monograph in preparation.P. McMullen, E. Schulte. Regular polytopes in ordinary space, Discrete Comput. Geom. 17 (1997), 449–478.P. McMullen, E. Schulte. Regular polytopes in ordinary space, Discrete Comput. Geom. 17 (1997), 449–478.P. McMullen, E. Schulte, Abstract regular polytopes, Encyclopedia of Mathematics and its Applications, Vol. 92, Cambridge University Press, Cambridge, UK, 2002.P. McMullen, E. Schulte, Abstract regular polytopes, Encyclopedia of Mathematics and its Applications, Vol. 92, Cambridge University Press, Cambridge, UK, 2002.D. Pellicer, E. Schulte. Regular polygonal complexes in space, I, Trans. Amer. Math. Soc. 362 (2010), 6679–6714.D. Pellicer, E. Schulte. Regular polygonal complexes in space, I, Trans. Amer. Math. Soc. 362 (2010), 6679–6714.D. Pellicer, E. Schulte. Regular polygonal complexes in space, II, Trans. Amer. Math. Soc. 365 (2013), 2031–2061.D. Pellicer, E. Schulte. Regular polygonal complexes in space, II, Trans. Amer. Math. Soc. 365 (2013), 2031–2061.E. Schulte. Regul¨are Inzidenzkomplexe, II, Geom. Dedicata 14 (1983), 33–56.E. Schulte. Regul¨are Inzidenzkomplexe, II, Geom. Dedicata 14 (1983), 33–56.

The authors declare that there are no conflicts of interest present.