Logo - springer
Slogan - springer

Mathematics - Geometry & Topology | Topological Crystallography - With a View Towards Discrete Geometric Analysis

Topological Crystallography

With a View Towards Discrete Geometric Analysis

Sunada, Toshikazu

2013, XII, 229 p. 94 illus., 40 illus. in color.

Available Formats:
eBook
Information

Springer eBooks may be purchased by end-customers only and are sold without copy protection (DRM free). Instead, all eBooks include personalized watermarks. This means you can read the Springer eBooks across numerous devices such as Laptops, eReaders, and tablets.

You can pay for Springer eBooks with Visa, Mastercard, American Express or Paypal.

After the purchase you can directly download the eBook file or read it online in our Springer eBook Reader. Furthermore your eBook will be stored in your MySpringer account. So you can always re-download your eBooks.

 
$39.95

(net) price for USA

ISBN 978-4-431-54177-6

digitally watermarked, no DRM

Included Format: PDF and EPUB

download immediately after purchase


learn more about Springer eBooks

add to marked items

Softcover
Information

Softcover (also known as softback) version.

You can pay for Springer Books with Visa, Mastercard, American Express or Paypal.

Standard shipping is free of charge for individual customers.

 
$49.95

(net) price for USA

ISBN 978-4-431-54176-9

free shipping for individuals worldwide

usually dispatched within 3 to 5 business days


add to marked items

  • 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.

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.

Content Level » Research

Keywords » Covering map - Discrete Abel--Jacobi map - Homology group - Standard realization - Topological crystal

Related subjects » Geometry & Topology

Table of contents 

​PrefaceList of SymbolsTopological crystals—Introduction—1 Quotient objects2 Generalities on graphs3 Homology groups of graphs4 Covering graphs5 Topological crystals6 Canonical placements7 Explicit construction8 MiscellanyAppendixBibliographyIndex

Popular Content within this publication 

 

Articles

Read this Book on Springerlink

Services for this book

New Book Alert

Get alerted on new Springer publications in the subject area of Algebraic Topology.