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001 978-4-431-54177-6
003 DE-He213
005 20140220082926.0
007 cr nn 008mamaa
008 121227s2013 ja | s |||| 0|eng d
020 _a9784431541776
_9978-4-431-54177-6
024 7 _a10.1007/978-4-431-54177-6
_2doi
050 4 _aQA612-612.8
072 7 _aPBPD
_2bicssc
072 7 _aMAT038000
_2bisacsh
082 0 4 _a514.2
_223
100 1 _aSunada, Toshikazu.
_eauthor.
245 1 0 _aTopological Crystallography
_h[electronic resource] :
_bWith a View Towards Discrete Geometric Analysis /
_cby Toshikazu Sunada.
264 1 _aTokyo :
_bSpringer Japan :
_bImprint: Springer,
_c2013.
300 _aXII, 229 p. 94 illus., 40 illus. in color.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aSurveys and Tutorials in the Applied Mathematical Sciences ;
_v6
520 _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.
650 0 _aMathematics.
650 0 _aGlobal differential geometry.
650 0 _aAlgebraic topology.
650 1 4 _aMathematics.
650 2 4 _aAlgebraic Topology.
650 2 4 _aDifferential Geometry.
650 2 4 _aGraph Theory.
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9784431541769
830 0 _aSurveys and Tutorials in the Applied Mathematical Sciences ;
_v6
856 4 0 _uhttp://dx.doi.org/10.1007/978-4-431-54177-6
912 _aZDB-2-SMA
999 _c99050
_d99050