For centuries Euclidean geometry, fuelled by the inexorable logic of its five axioms, was the mathematical system par excellence. Violation of its postulates – especially the fifth, enigmatic, one – led to the awareness that the remarkable system was not unique: it is possible to tackle the infinite in other ways, via non-Euclidean geometries, which are equally beautiful and coherent and based on multidimensional spaces. Intellectual efforts have since ventured even further, to the point of imagining an entire city made up of many geometric constructions, having strange and marvellous shapes, some of which have been used by physicists, such as Einstein, to measure astronomical space. This book tells an engaging story of ideas, endeavours, and achievements, involving major thinkers such as Euler, Saccheri, Gauss, Beltrami, and Riemann.

Laura Catastini used to teach mathematics and physics in Italian upper secondary schools. She currently conducts research on the teaching of maths at the Tor Vergata University in Rome.

Franco Ghione used to be a professor of Geometry at the Tor Vegata University in Rome.