Some people believe that there exist links between the natural numbers and the physical world. This rather strange book looks for such links in what the authors call “the periodicity of matter.”

Readers with an interest in mathematics, science, or natural philosophy will find the book useful but perhaps hard to read. This is not a textbook; it is aimed at researchers working on various branches of physics who might find it interesting to find related ideas in number theory.

The first chapter discusses evidence of the existence periodic structures in matter and finds some links with the periodic structures of natural numbers, especially those appearing in modular arithmetic. Then, in chapter two, the authors sketch out some basic tools in number theory. They include some interesting topics, such as the “prime spiral.” Chapters three, four and five review some basic topics in physics, discussing the periodic table of elements, the structure of atomic nuclei, and elements of cosmology. In each of these chapters the authors start with historical background and then lead the reader to modern ideas.

To continue the exploration of periodic laws in the structure of matter, chapter six includes a number of tables and diagrams that display some of these laws. In the next chapter, using concepts like the Golden Ratio and Farey sequences, the authors describe periodic structures in matter that are connected to number theory, as the reader expects from the title of book. Chapters eight, nine and ten are similar to the seventh, focusing this time on nuclear structure. The authors conclude by studying chemical periodicity.

We believe that this book can be of interest to mathematicians, physicists and chemists. It will at least allow them to sit together and talk about matters of joint interest, which is a property shared by few other books.

Soheila Emamyari is M.S. student at the Institute for Advanced Studies in Basic Science in Zanjan, Iran, under the supervision of Dr. H. Fazli.

Mehdi Hassani is a Co-tutelle Ph.D. student in IASBS and the Université de Bordeaux I, under supervision of professors M.M. Shahshahani and J-M. Deshouillers.