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 Zh. Vychisl. Mat. Mat. Fiz., 2005, Volume 45, Number 8, Pages 1331–1344 (Mi zvmmf4626)

Two-dimensional discrete groups with finite fundamental regions and their physical and humanitarian interpretations

R. V. Galiulin

Institute of Cristallography RAS

Abstract: The crystallographic model is the global model of the universe that has been most intensively developed in recent years. Its best-known authors are R.J. Hauy, E.S. Fedorov, H. Poincaré, B.N. Delaunay, N.V. Belov, and D.D. Ivanenko. The latest astronomical observations suggest that the universe is a compact locally Euclidean manifold constructed on a Platonic dodecahedron. Each atom in the periodic table (which is represented as a regular or semiregular isogon) is also represented as a compact locally Euclidean manifold; i.e., the atoms and the universe are topologically identical. The periodic table itself is divided into four blocks ($s$, $p$, $d$, and $f$). The atoms in the $d$ and $f$ blocks have five-fold and sevenfold symmetry, respectively. These constructions are underlain by discrete groups with a finite fundamental region (crystallographic groups), which first appeared in Islamic ornaments and only several centuries later were discovered by scientists. Thus, science and religion have come to the modern (crystallographic) picture of the world. After an appropriate redesign, all the major principles of this picture can be represented in the two-dimensional case, which is done in this paper.

Key words: isohedron, isogon, tilings, periodic table, Delaunay systems, hyperbolic geometry, regular graphs, nanocrystals, Rydberg atoms.

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Bibliographic databases:
UDC: 519.63

Citation: R. V. Galiulin, “Two-dimensional discrete groups with finite fundamental regions and their physical and humanitarian interpretations”, Zh. Vychisl. Mat. Mat. Fiz., 45:8 (2005), 1331–1344

Citation in format AMSBIB
\Bibitem{Gal05} \by R.~V.~Galiulin \paper Two-dimensional discrete groups with finite fundamental regions and their physical and humanitarian interpretations \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2005 \vol 45 \issue 8 \pages 1331--1344 \mathnet{http://mi.mathnet.ru/zvmmf4626} \zmath{https://zbmath.org/?q=an:1087.51009} \elib{http://elibrary.ru/item.asp?id=9142409}