Beautiful Geometry


Eli Maor and Eugen Jost
PRINCETON UNIVERSITY PRESS 2014, 208 PAGES
PRICE (HARDBACK) £19.95 ISBN 978-0-69115-099-4

Beautiful GeometryBeautiful Geometry by Maor and Jost, a mathematician and artist respectively, links art and mathematics through the common search for pattern. The wider perceived disconnect between the two is explored, of art expressing feelings and emotions and mathematics as rational and detached.  However the authors set out from the Renaissance tradition where both ‘not only were practised together, they were regarded as complementary aspects of the human mind’.

The colour plates are visually stunning and include many well-known images: the logarithmic spiral, the snowflake fractal, Esher style hexagonal staircase, and many more. The Fibonacci plate, resembles one of the artist Mondrian’s abstract paintings of coloured squares, rectangles and thick black lines. However artistic interpretation of the plates is minimal at best. The 1960’s psychedelic colour image of the series of circles in Steiner’s porism, evidence the outlook of a geometric construction as ‘a stark, black-and-white array of lines and circles. But add colour to it, and it can become an exquisite work of art’.

The book is split into fifty one sections, with the vast majority geometry, the rest centre on numbers and number progressions. Each section is explained in words and is accompanied by Jost’s colourful representations. The chapters largely stand alone, although there is linkage of a range of themes, including the history of geometry. The scene is set with the ancient Greeks, known for beginning the abstraction of geometry, Euclid’s Elements being the earliest existing record axiomatically defining plane geometry. The Greek approach is described using ‘Euclidean tools’ i.e. a straight edge and compass and any geometric representation must be derived from these alone. Euclid, ‘… insisted that a proof should be based strictly on geometric considerations’, whereas today it is preferred to be proved algebraically. The circle features strongly and for a simple geometric construct, ‘yields much’.

The influence of number in geometric construction is a strong element of the book. For example, the golden ratio appearing in art and architecture, with a proportionality fundamental to an artistic appreciation. However there was no evidence that the Greeks were aware of the effect of this proportion or deliberately used it to artistic effect. More difficult to explain perhaps is the appearance of the Fibonacci sequence in nature’s geometry. The discovery that √2 is irrational and not constructable by Euclidean tools brought about a serious intellectual crisis for the Greeks, believers in the rule of rational numbers.

The book raises fundamental questions, answered thousands of years later and evidencing the progress made. The Greeks knew that squaring the circle, i.e. constructing a square equal in area to that of a circle, was not possible with the tools at their disposal. In 1882 Carl von Lindemann proved that as pi is not a solution of a polynomial equation with integer coefficients, the circle cannot be squared. The development of analytical geometry, using algebraic methods not available to the Greeks, where a geometrical construct can be translated into a set of equations, allowed age old problems to be achievable with Euclidean tools. This is an engaging book of broad appeal and colourful approach to the history of geometry.

Wallace A Ferguson CMath CSci MIMA
Chatham and Clarendon Grammar School, Ramsgate

Book review published directly onto IMA website (April 2015)

Published