© 2013

Topological Crystallography

With a View Towards Discrete Geometric Analysis


Part of the Surveys and Tutorials in the Applied Mathematical Sciences book series (STAMS, volume 6)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Toshikazu Sunada
    Pages 1-8
  3. Prerequisites for Modern Crystallography

    1. Front Matter
      Pages 9-9
    2. Toshikazu Sunada
      Pages 11-20
    3. Toshikazu Sunada
      Pages 21-35
    4. Toshikazu Sunada
      Pages 37-51
    5. Toshikazu Sunada
      Pages 53-70
  4. Geometry of Crystal Structures

    1. Front Matter
      Pages 71-71
    2. Toshikazu Sunada
      Pages 73-92
    3. Toshikazu Sunada
      Pages 93-124
    4. Toshikazu Sunada
      Pages 125-152
  5. Advanced Topics

    1. Front Matter
      Pages 153-153
    2. Toshikazu Sunada
      Pages 155-180
    3. Toshikazu Sunada
      Pages 181-200
  6. Back Matter
    Pages 201-229

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.


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

Authors and affiliations

  1. 1.School of Science and TechnologyMeiji UniversityKawasakiJapan

Bibliographic information

  • Book Title Topological Crystallography
  • Book Subtitle With a View Towards Discrete Geometric Analysis
  • Authors Toshikazu Sunada
  • Series Title Surveys and Tutorials in the Applied Mathematical Sciences
  • Series Abbreviated Title Surveys,Tutorials Math.Sci.
  • DOI
  • Copyright Information Springer Japan 2013
  • Publisher Name Springer, Tokyo
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-4-431-54176-9
  • eBook ISBN 978-4-431-54177-6
  • Edition Number 1
  • Number of Pages XII, 229
  • Number of Illustrations 54 b/w illustrations, 40 illustrations in colour
  • Topics Algebraic Topology
    Differential Geometry
    Graph Theory
  • Buy this book on publisher's site


“This book is unique in its coverage, and is therefore a necessary addition to the library of any mathematician working on mathematical or geometric crystallography particularly with respect to crystal design and engineering.” (G. L. McColm, MAA Reviews, April 05, 2019)


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)