# Shaping Space. Exploring Polyhedra in Nature, Art, and the Geometrical Imagination

As the editor quotes in her introduction "plus que ça change, plus c'est la même chose" since indeed polyhedra are as new as they are old and given the recent evolution in graphs, discrete and computational geometry, combinatorial optimization, computer graphics, a new edition of the previous version (Birkhäuser, 1987) became unavoidable and it resulted in a complete makeover. The format is still the same (the first edition grew out of a 1984 conference), it consists of a collection of essays by different authors about many different aspects related to polyhedra.

The papers are ordered in such a way that they start with elementary, less formal definitions an properties, and suggestions and practical tips about how to actually organize hands-on sessions where children are encouraged to construct the three-dimensional objects. But polyhedra are also followed along their historical and cultural trail from the pyramids in old Egypt and the Platonic solids, till recent developments.

In a second part, appearance and use of polyhedra in art and nature is the the central theme. They lived in the minds of the architects of the pyramids but they also appear in futuristic constructions of modern architecture. Because their graphs have some optimality and stability properties also nature's architect is eager to make use of these structures. Crystals, chemical bindings, cell biology quite often follow the geometrical laws of polyhedral constellations. And of course many artists made 2 of 3-dimensional artwork inspired by these forms.

In part 3, called "polyhedra in geometrical imagination", the contributions become more mathematical. Here we find more general polyhedra, and discussions about molecular stability, dual graphs, Dirichlet tessellations and spider webs, diophantine equations, rigidity, decomposition of solids, etc. The final contribution is a set of 10 geometrical problems that are still (partly) open problems still waiting for a solution.

Although there are 22 papers by many different authors, there is an extensive global index that helps you to find the items you are looking for. The readability of the papers is kept as smooth as possible by collecting notes, remarks and references in a section at the end of the book. Of course the style cannot be uniform since there is a difference between an historical survey, an exposition of how to glue pieces of cardboard together, and a mathematical paper with theorems. However, by the ordering of the papers, the reader grows gradually into the mathematics as he of she is reading on towards the end of the book.

The book is amply illustrated and aiming at a public from 9 till 99. It will be of interest to a very broad public. Form a mathematical side children might be interested in geometrical puzzles and advanced mathematicians may be interested in solving the open problems, and the whole range in between will probably find something interesting of their own taste. But of cause also the non-mathematician will be attracted by these fascinating building blocks in nature, art, science and engineering.

**Submitted by Adhemar Bultheel |

**17 / Jun / 2013