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.