Jan Aarts apparently set out to create a book that covers Euclidean Geometry in one, two and three dimensions. He realized early in the endeavor that the usual approach of building from abstract geometrical axioms was a huge task, and instead conceived of writing a book on Euclidean Geometry in which he could de-emphasize the usual axiom-theorem-proof approach. In particular he chose to use concepts of metric space, groups, transformations and symmetry to construct a rigorous Euclidean Geometry but do so in a rapid, rigorous, and powerful fashion.
Having loved Euclidean Geometry in the traditional approach of axiom and theorem, I was somewhat uncertain what to expect of the book, and whether I would like this new approach. To my delight, I found that Euclidean geometry is still a wonderful field, and this new approach allows one to reach not only the standard results quickly but also many more advanced results.
In the process of reading through the book, I discovered that there are fair number of Euclidean results on circles, triangles, conics and other topics that I’ve been using for years, but never really thought about how to prove. There was certain freshness and fun in following the proofs given in the text and in creating my own proofs patterned after those in the book. This approach to Euclidean Geometry has a wonderful way of tying different mathematical concepts together, and going beyond what might be normally part of Euclidean Geometry course material, with topics such as Voronoi diagrams, Voronoi cells, periodic tilings, inner and outer products, fractals, and frieze diagrams.
The book contains excellent illustrative diagrams, a reasonable selection of exercises, clear exposition, and fair number of worked examples and proofs, and a superb index. Altogether I found the book delightful. I still have a certain fondness for the old Euclidean Axioms, but for a unifying vision of mathematics, and economy of effort to get results, Aarts’s approach to Euclidean Geometry is hard to beat. If you love Euclidean geometry you will certainly appreciate this book as part of your collection. The book would also make an excellent text for those in physics, chemistry that deal with crystallography, and other practical aspects of Euclidean geometry.
Collin Carbno is a specialist in process improvement and methodology. He holds a Master’s of Science Degree in theoretical physics and completed course work for Ph.D. in theoretical physics (relativistic rotating stars) in 1979 at the University of Regina. He has been employed for nearly 30 years in various IT and process work at Saskatchewan Telecommunications and currently holds a Professional Physics Designation from the Canadian Association of Physicists, and the Information System Professional designation from the Canadian Information Process Society.