Tilings of the Plane: From Escher via Moebius to Penrose

The book explores symmetries and tessellation, which have long fascinated artists and mathematicians. It describes three approaches: plane crystal groups, Harald Heesch's procedures, and the theory of groups of Möbius transformations. It also discusses the Penrose tessellation, which provides non-periodic parquetizations of the plane.

Format: Paperback / softback
Length: 283 pages
Publication date: 13 November 2022
Publisher: Springer

The captivating world of symmetries and tessellations has long fascinated artists and mathematicians alike. Renowned examples include the breathtaking creations of the Arabs in the Alhambra and the intricate paintings of the Dutch artist Maurits Escher. While mathematicians initially explored these concepts sparingly in the 19th century, the visual representation of mathematical relationships has yielded stunning imagery. This book delves into three distinct approaches to understanding and creating tessellations.

In Part I, we embark on a journey to explore the 17 fundamentally distinct possibilities of tessellation in the plane, known as plane crystal groups. Alongside this exploration, we delve into the ideas of Harald Heesch, a visionary mathematician who demonstrated how these theoretical findings can be translated into practical applications. Heesch provided a comprehensive catalog of 28 creative procedures, encouraging readers to follow in the footsteps of Escher and unlock their artistic potential.

In Part II, we shift our focus to the complex plane, replacing movements with bijective holomorphic mappings. This leads us into the realm of groups of Möbius transformations, including Kleinian groups and Schottky groups. We also uncover intriguing connections to hyperbolic geometry, adding another layer of depth to our understanding.

Finally, in Part III, we delve into the captivating world of Penrose tessellations. This section explores the groundbreaking results from the seventies, when easily describable and provably non-periodic parquetizations of the plane were first introduced. The Penrose tessellation represents a significant milestone in the field, showcasing the endless possibilities of mathematical beauty and creativity.

Through these three parts, this book offers a comprehensive and engaging exploration of symmetries and tessellations, providing both a theoretical foundation and practical guidance for artists and mathematicians alike. Whether you are a seasoned practitioner or a newcomer to the world of geometry, this book will captivate your imagination and inspire you to unlock the hidden wonders of symmetrical patterns and shapes.

Weight: 456g
Dimension: 235 x 155 (mm)
ISBN-13: 9783658388096
Edition number: 1st ed. 2022