The Symmetries of Things

This book is divided into three parts with an increasing level of difficulty. The first part, which needs almost no prerequisites, is devoted to symmetries of repeating planar and spherical patterns and their classification. The authors have chosen an original approach based on the concepts of orbifold and orbifold signature. The key ingredients in the classification process is the so-called magic theorem, together with the classification theorem for topological surfaces (Conway's zip proof is included). The result is the well-known fact that there are exactly seventeen distinct possible types of repeating planar patterns. The second part requires a basic understanding of group theory (somewhat surprisingly, there is no group theory in the first part). Planar and spherical symmetries are now described in a group-theoretic language (using their presentations, i.e. generators and relations). This part also provides an enumeration of coloured planar and spherical patterns. The third part, which is difficult reading even for a professional mathematician, is devoted to repeating patterns in other spaces. The topics discussed in this part include patterns in the hyperbolic plane, Archimedean tilings, generalised Schäfli symbols, three-dimensional crystallographic groups, flat universes and higher dimensional groups. The book contains many new results. Moreover, it is printed on glossy pages with a large number of beautiful full-colour illustrations, which can be enjoyed even by non-mathematicians.

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