LEADER 03831nam 2200565 a 450 001 9910437863203321 005 20200520144314.0 010 $a1-283-94639-4 010 $a4-431-54177-2 024 7 $a10.1007/978-4-431-54177-6 035 $a(CKB)2670000000317956 035 $a(EBL)1083416 035 $a(OCoLC)823640937 035 $a(SSID)ssj0000870786 035 $a(PQKBManifestationID)11526928 035 $a(PQKBTitleCode)TC0000870786 035 $a(PQKBWorkID)10819330 035 $a(PQKB)10948312 035 $a(DE-He213)978-4-431-54177-6 035 $a(MiAaPQ)EBC1083416 035 $a(PPN)168332566 035 $a(EXLCZ)992670000000317956 100 $a20121101d2013 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aTopological crystallography $ewith a view towards discrete geometric analysis /$fToshikazu Sunada 205 $a1st ed. 2013. 210 $aNew York $cSpringer$d2013 215 $a1 online resource (235 p.) 225 1 $aSurveys and tutorials in the applied mathematical sciences ;$vv. 6 300 $aDescription based upon print version of record. 311 $a4-431-54176-4 320 $aIncludes bibliographical references and index. 327 $apt. I. Prerequisites for modern crystallography -- pt. II. Geometry of crystal structures -- pt. III. Advanced topics. 330 $aGeometry in ancient Greece is said to have originated in the curiosity of mathematicians about the shapes of crystals, with that curiosity culminating in the classification of regular convex polyhedra addressed in the final volume of Euclid?s Elements. Since then, geometry has taken its own path and the study of crystals has not been a central theme in mathematics, with the exception of Kepler?s work on snowflakes. Only in the nineteenth century did mathematics begin to play a role in crystallography as group theory came to be applied to the morphology of crystals. This monograph follows the Greek tradition in seeking beautiful shapes such as regular convex polyhedra. The primary aim is to convey to the reader how algebraic topology is effectively used to explore the rich world of crystal structures. Graph theory, homology theory, and the theory of covering maps are employed to introduce the notion of the topological crystal which retains, in the abstract, all the information on the connectivity of atoms in the crystal. For that reason the title Topological Crystallography has been chosen. Topological crystals can be described as ?living in the logical world, not in space,? leading to the question of how to place or realize them ?canonically? in space. Proposed here is the notion of standard realizations of topological crystals in space, including as typical examples the crystal structures of diamond and lonsdaleite. A mathematical view of the standard realizations is also provided by relating them to asymptotic behaviors of random walks and harmonic maps. Furthermore, it can be seen that a discrete analogue of algebraic geometry is linked to the standard realizations. Applications of the discussions in this volume include not only a systematic enumeration of crystal structures, an area of considerable scientific interest for many years, but also the architectural design of lightweight rigid structures. The reader therefore can see the agreement of theory and practice. 410 0$aSurveys and tutorials in the applied mathematical sciences ;$v6. 606 $aCrystallography 615 0$aCrystallography. 676 $a514 700 $aSunada$b Toshikazu$0300039 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910437863203321 996 $aTopological Crystallography$92510941 997 $aUNINA