{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:04:09Z","timestamp":1760241849320,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2018,9,27]],"date-time":"2018-09-27T00:00:00Z","timestamp":1538006400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002261","name":"Russian Foundation for Basic Research","doi-asserted-by":"publisher","award":["18-07-00170"],"award-info":[{"award-number":["18-07-00170"]}],"id":[{"id":"10.13039\/501100002261","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The work reports the finding and the study of transformation groups with two conditional elements (binary transformations of abstract structures of the finite numerical sets with broken symmetry). The term Broken Symmetry Group (BSG) is introduced. Transformation examples of relevant structures are studied with computer visualization and application in real structure study. A special type of BSG was discovered, which describes the subsets of \u201cevolutionary trees\u201d with convergent and divergent properties of the oriented graph (orgraph) with structure-development direction edges and \u201cgrowth spirals\u201d.<\/jats:p>","DOI":"10.3390\/sym10100440","type":"journal-article","created":{"date-parts":[[2018,9,28]],"date-time":"2018-09-28T02:54:54Z","timestamp":1538103294000},"page":"440","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Structure Transformations in Broken Symmetry Groups\u2014Abstraction and Visualization"],"prefix":"10.3390","volume":"10","author":[{"given":"Valery G.","family":"Rau","sequence":"first","affiliation":[{"name":"Russian Presidential Academy of National Economy and Public Administration, Vernadsky Prospekt, 119571 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Igor A.","family":"Togunov","sequence":"additional","affiliation":[{"name":"Russian Presidential Academy of National Economy and Public Administration, Vernadsky Prospekt, 119571 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tamara F.","family":"Rau","sequence":"additional","affiliation":[{"name":"General and Theoretical Physics Department, Vladimir State University, Gorkogo Street 87, 600000 Vladimir, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sergey V.","family":"Polyakov","sequence":"additional","affiliation":[{"name":"Russian Presidential Academy of National Economy and Public Administration, Vernadsky Prospekt, 119571 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,9,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"334","DOI":"10.2307\/1573781","article-title":"Symmetry in science and art","volume":"10","author":"Bornstein","year":"1977","journal-title":"Leonardo"},{"key":"ref_2","unstructured":"Veinstein, B.K. (1979). Modern Crystallography, Nauka."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1090\/pspum\/037\/604577","article-title":"Graphs, singularities and finite groups","volume":"37","author":"McKay","year":"1980","journal-title":"Proc. Symp. Pure Math."},{"key":"ref_4","unstructured":"Zorky, P.M. (1986). Symmetry of Molecules and Crystal Structures, Moscow University."},{"key":"ref_5","unstructured":"Chuprunov, E.V. (2015). Symmetry and Pseudosymmetry of Crystalls, Nizniy Novgorod State University."},{"key":"ref_6","unstructured":"Kostrikin, A.I. (1977). Introduction to Algebra, Nauka."},{"key":"ref_7","unstructured":"Hall, M.J. (1967). Combinatorial Theory, Blaisdell Publishing Company."},{"key":"ref_8","unstructured":"Nelipa, N.F. (1985). Physics of Elementary Particles. Calibration fields, Vischaya Shkola Publishers."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"54","DOI":"10.3390\/sym5010054","article-title":"Non-crystallographic symmetry in packing spaces","volume":"5","author":"Rau","year":"2013","journal-title":"Symmetry"},{"key":"ref_10","unstructured":"Rau, V.G., Lomtev, L.A., Rau, T.F., and Nikitin, O.R. (2016, January 21\u201326). Broken symmetry groups with non-classic substitution within a computer experiment. Congress Digest of the First Russian Crystallography Congress, Moscow, Russia."},{"key":"ref_11","unstructured":"Schulte, E. (2018). Polyhedral Structures, Symmetry, and Applications, MDPI."},{"key":"ref_12","unstructured":"(2018, April 15). The Cambridge Structural Database. Available online: webcsd.ccdc.cam.ac.uk."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1134\/S0022476618010171","article-title":"Broken symmetry group in the octahedral [Me(urea)6]2+,3+ cation with Chelate hydrogen bonds","volume":"59","author":"Rau","year":"2018","journal-title":"J. Struct. Chem."},{"key":"ref_14","unstructured":"Rau, V.G., Bondar, V.I., Rau, T.F., and Stepanov, S.V. (1990). Problems of Crystal Chemistry, Nauka."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Rau, V.G., Lomtev, L.A., Rau, T.F., Gorshkov, K.A., and Nikitin, O.R. (2017). Computer experiments in groups of permutations with broken symmetry. Mod. Prob. Sci. Educ., 43\u201349.","DOI":"10.17513\/snt.36614"},{"key":"ref_16","unstructured":"Koonin, E.V. (2011). The Logic of Chance: The Nature and Origin of Biological Evolution, FT Press."},{"key":"ref_17","unstructured":"Togunov, I.A. (2016). Structural Functional Model of Social Systems, RANEPA."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/10\/440\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:22:48Z","timestamp":1760196168000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/10\/440"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,9,27]]},"references-count":17,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2018,10]]}},"alternative-id":["sym10100440"],"URL":"https:\/\/doi.org\/10.3390\/sym10100440","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2018,9,27]]}}}