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Tilings of the Plane - (Mathematics Study Resources) by Ehrhard Behrends (Paperback)

Tilings of the Plane - (Mathematics Study Resources) by  Ehrhard Behrends (Paperback) - 1 of 1
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About this item

Highlights

  • The aim of the book is to study symmetries and tesselation, which have long interested artists and mathematicians.
  • About the Author: Prof. Dr. Ehrhard Behrends, Free University of Berlin, Department of Mathematics and Computer Science
  • 283 Pages
  • Mathematics, General
  • Series Name: Mathematics Study Resources

Description



Book Synopsis



The aim of the book is to study symmetries and tesselation, which have long interested artists and mathematicians. Famous examples are the works created by the Arabs in the Alhambra and the paintings of the Dutch painter Maurits Escher. Mathematicians did not take up the subject intensively until the 19th century. In the process, the visualisation of mathematical relationships leads to very appealing images. Three approaches are described in this book.

In Part I, it is shown that there are 17 principally different possibilities of tesselation of the plane, the so-called 'plane crystal groups'. Complementary to this, ideas of Harald Heesch are described, who showed how these theoretical results can be put into practice: He gave a catalogue of 28 procedures that one can use creatively oneself - following in the footsteps of Escher, so to speak - to create artistically sophisticated tesselation.

In the corresponding investigations forthe complex plane in Part II, movements are replaced by bijective holomorphic mappings. This leads into the theory of groups of Möbius transformations: Kleinian groups, Schottky groups, etc. There are also interesting connections to hyperbolic geometry.

Finally, in Part III, a third aspect of the subject is treated, the Penrose tesselation. This concerns results from the seventies, when easily describable and provably non-periodic parquetisations of the plane were given for the first time.






From the Back Cover



The aim of the book is to study symmetries and tesselation, which have long interested artists and mathematicians. Famous examples are the works created by the Arabs in the Alhambra and the paintings of the Dutch painter Maurits Escher. Mathematicians did not take up the subject intensively until the 19th century. In the process, the visualisation of mathematical relationships leads to very appealing images. Three approaches are described in this book.

In Part I, it is shown that there are 17 principally different possibilities of tesselation of the plane, the so-called "plane crystal groups". Complementary to this, ideas of Harald Heesch are described, who showed how these theoretical results can be put into practice: He gave a catalogue of 28 procedures that one can use creatively oneself - following in the footsteps of Escher, so to speak - to create artistically sophisticated tesselation.

In the corresponding investigations for the complex plane in Part II, movements are replaced by bijective holomorphic mappings. This leads into the theory of groups of Möbius transformations: Kleinian groups, Schottky groups, etc. There are also interesting connections to hyperbolic geometry.

Finally, in Part III, a third aspect of the subject is treated, the Penrose tesselation. This concerns results from the seventies, when easily describable and provably non-periodic parquetisations of the plane were given for the first time.


The ContentsPart I: Escher seen over the shoulders- Part II: Möbius transformations - Part III: Penrose tesselation

The AuthorProf. Dr. Ehrhard Behrends, Free University of Berlin, Department of Mathematics and Computer Science, is the author of numerous mathematical textbooks and popular science books.



Review Quotes




"Tilings is a fascinating and actively developing topic of modern mathematics. On the one hand, it is full of deep results with important physicalapplications. On the other hand, many results of tilings theory can be understood by non-professionals. ... I highly recommend to read this book to any who are interested in tilings." (Anton Shutov, zbMATH 1511.52001, 2023)



About the Author



Prof. Dr. Ehrhard Behrends, Free University of Berlin, Department of Mathematics and Computer Science
Dimensions (Overall): 9.21 Inches (H) x 6.14 Inches (W) x .62 Inches (D)
Weight: .92 Pounds
Suggested Age: 22 Years and Up
Number of Pages: 283
Genre: Mathematics
Sub-Genre: General
Series Title: Mathematics Study Resources
Publisher: Springer
Format: Paperback
Author: Ehrhard Behrends
Language: English
Street Date: November 13, 2022
TCIN: 1004136381
UPC: 9783658388096
Item Number (DPCI): 247-25-1389
Origin: Made in the USA or Imported
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Shipping details

Estimated ship dimensions: 0.62 inches length x 6.14 inches width x 9.21 inches height
Estimated ship weight: 0.92 pounds
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