Surveys and Tutorials in the Applied Mathematical Sciences

Topological Crystallography

With a View Towards Discrete Geometric Analysis

Authors: Sunada, Toshikazu

  • Useful for both mathematicians and practical scientists, who know how ideas developed in pure mathematics are applied to a practical problem
  • Designed to be as self-contained as possible, so that the reader can start from the scratch and reach the advanced level of the field
  • This monograph will give a surprise to the reader since he will discover that a problem in one area leads us into a quite different area of mathematics.
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  • ISBN 978-4-431-54177-6
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About this book

Geometry 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.

Reviews

From the reviews:

“The book’s main interest lies in the unexpected, pure mathematical thrust. The author’s ‘topological crystals’ emerge as prototypes--as cases simpler than the usual simplest cases­--of objects studied in deep theories such as algebraic number theory and algebraic geometry. The book is highly suitable for undergraduate capstone courses; all necessary algebraic topology is developed from scratch. Summing Up: Recommended. Upper-division undergraduates and above.” (D. V. Feldman, Choice, Vol. 51 (3), November, 2013)

Table of contents (10 chapters)

Buy this book

eBook $34.99
price for USA (gross)
  • ISBN 978-4-431-54177-6
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $49.95
price for USA
  • ISBN 978-4-431-54176-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the eBook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Topological Crystallography
Book Subtitle
With a View Towards Discrete Geometric Analysis
Authors
Series Title
Surveys and Tutorials in the Applied Mathematical Sciences
Series Volume
6
Copyright
2013
Publisher
Springer Japan
Copyright Holder
Springer Japan
eBook ISBN
978-4-431-54177-6
DOI
10.1007/978-4-431-54177-6
Softcover ISBN
978-4-431-54176-9
Series ISSN
2199-4765
Edition Number
1
Number of Pages
XII, 229
Number of Illustrations and Tables
54 b/w illustrations, 40 illustrations in colour
Topics