On the enumeration of Archimedean polyhedra in the Lobachevsky space

V. S. Makarov ; P. V. Makarov

Informatics and Automation, Classical and modern mathematics in the wake of Boris Nikolaevich Delone, Tome 275 (2011), pp. 99-127.

Résumé

We describe the class of Archimedean polyhedra in the three-dimensional Lobachevsky space, which technically reduces to studying Archimedean tilings of the Lobachevsky plane. We analyze the possibility of obtaining Archimedean tilings by methods that are usually applied on the sphere and in the Euclidean plane. It is pointed out that such tilings can be constructed by using certain types of Fedorov groups in the Lobachevsky plane. We propose a general approach to the problem of classifying Archimedean tilings of the Lobachevsky plane.

Bibliographie

[1] Aleksandrov A.D., “O zapolnenii prostranstva mnogogrannikami”, Vestn. Leningr. un-ta. Matematika. Fizika. Khimiya, 1954, no. 2, 33–43

[2] Gutsul I.S., Makarov V.S., “Ob odnom svoistve fedorovskikh grupp prostranstva Lobachevskogo”, Tr. MIAN, 148, 1978, 106–108 | MR | Zbl

[3] Delone B.N., “Teoriya planigonov”, Izv. AN SSSR. Ser. mat., 23:3 (1959), 365–385 | MR

[4] Delone B.N., Dolbilin N.P., Shtogrin M.I., Galiulin R.V., “Lokalnyi kriterii pravilnosti sistem tochek”, DAN SSSR, 227:1 (1976), 19–21 | MR | Zbl

[5] Delone B.N., Dolbilin N.P., Shtogrin M.I., “Kombinatornaya i metricheskaya teoriya planigonov”, Tr. MIAN, 148, 1978, 109–140 | MR | Zbl

[6] Delone B.N., Galiulin R.V., Shtogrin M.I., “Sovremennaya teoriya pravilnykh razbienii evklidova prostranstva”: Fedorov E.S., Pravilnoe delenie ploskosti i prostranstva, Klassiki nauki, Nauka, L., 1979, 235–260 | MR

[7] Dolbilin N.P., “O lokalnykh svoistvakh diskretnykh pravilnykh sistem”, DAN SSSR, 230:3 (1976), 516–519 | MR | Zbl

[8] Zalgaller V.A., Vypuklye mnogogranniki s pravilnymi granyami, Zap. nauch. sem. LOMI, 2, Nauka, L., 1967 | MR | Zbl

[9] Zamorzaev A.M., “O pravilnykh mnogostoronnikakh i mnogogrannikakh v prostranstve Lobachevskogo”, Uchen. zap. Kishinev. gos. un-ta, 39 (1959), 195–207

[10] Kagan V.F., Osnovaniya geometrii: Uchenie ob osnovanii geometrii v khode ego istoricheskogo razvitiya, Ch. 1, Gostekhizdat, M.; L., 1949; Ч. 2, 1956

[11] Makarov V.S., “Polupravilnye mnogougolniki i mnogostoronniki na ploskostyakh Evklida i Lobachevskogo”, Uchen. zap. Kishinev. gos. un-ta, 54 (1960), 101–116

[12] Makarov V.S., “Ob odnom klasse dvumernykh fedorovskikh grupp”, Izv. AN SSSR. Ser. mat., 31:3 (1967), 531–542 | MR | Zbl

[13] Makarov V.S., “Geometricheskie metody postroeniya diskretnykh grupp dvizhenii prostranstva Lobachevskogo”, Problemy geometrii, T. 15, Itogi nauki i tekhniki, VINITI, M., 1983, 3–59 | MR | Zbl

[14] Makarov V.S., “O pravilnogrannykh mnogogrannikakh v prostranstve Lobachevskogo”, Izv. AN Resp. Moldova, 1992, no. 2, 3–6 | MR | Zbl

[15] Makarov V.S., Makarov P.V., “O vypuklykh mnogogrannikakh s pravilnymi granyami v prostranstve Lobachevskogo”, Diskretnaya matematika i ee prilozheniya, Mater. VIII Mezhdunar. sem., Izd. mekh.-mat. fak. MGU, M., 2004, 402–405

[16] Makarov V.S., Makarov P.V., “Pravilnye razbieniya prostranstv postoyannoi krivizny i ikh kristallograficheskie gruppy”, Matematicheskie issledovaniya v kristallografii, mineralogii i petrografii, Tr. II Vseros. nauch. shk., Apatity, 16–17 okt. 2006, K M, Apatity, 2006, 19–32

[17] Makarov V.S., Makarov P.V., “Pravilnogrannye mnogogranniki trekhmernogo prostranstva Lobachevskogo”, Metricheskaya geometriya poverkhnostei i mnogogrannikov, Tr. Mezhdunar. konf., posv. 100-letiyu N.V. Efimova, Moskva, 18–21 avg. 2010, MAKS Press, M. (to appear) | Zbl

[18] Makarov V.S., Pakhomii A.P., “O mnogogrannikakh s pravilnymi granyami v prostranstve Lobachevskogo”, VI Tirasp. simpoz. po obschei topologii i ee prilozheniyam, Shtiintsa, Kishinev, 1991, 159–160

[19] Makarov P.V., “O kelvinovskikh razbieniyakh trekhmernogo prostranstva Lobachevskogo pravilnymi mnogogrannikami”, UMN, 45:1 (1990), 179–180 | MR | Zbl

[20] Makarov P.V., “K voprosu o klassifikatsii A-razbienii ploskosti Lobachevskogo”, Matematicheskie issledovaniya v estestvennykh naukakh, Tr. V Vseros. nauch. shk., Apatity, 12–14 okt. 2009, K M, Apatity, 2009, 34–43

[21] Smogorzhevskii O.S., Osnovi geometriï, Radyanska shkola, Kiïv, 1954

[22] Feiesh Tot L., Raspolozheniya na ploskosti, sfere i v prostranstve, Fizmatgiz, M., 1958

[23] Boole Stott A., Geometrical deduction of semiregular from regular polytopes and space fillings, Verh. Kon. Akad. Wetensch. Amsterdam. 1910. Sect. 1, dl. 11, N 1.

[24] Coxeter H.S.M., “Regular honeycombs in hyperbolic space”, Proc. Intern. Congr. Math., Amsterdam, 1954, V. 3, North-Holland, Amsterdam, 1956, 155–169 | MR

[25] Fejes Tóth L., Regular figures, Pergamon Press, Oxford, 1964 | Zbl

[26] Poincaré H., “Théorie des groupes fuchsiens”, Acta math., 1 (1882), 1–62 | DOI | MR | Zbl