What is geometry about?

Geometry ‘measuring the world’ attempts to describe and understand space around us and all that is in it. It is the central activity and main driving force in many branches of math and physics, and offers a whole range of views on the nature and meaning of the universe. This book treats geometry in a wide context, including a wealth of relations with surrounding areas of math and other aspects of human experience.

Any discussion of geometry involves tension between the twin ideals of intuition and precision. Descriptive or synthetic geometry takes as its starting point our ideas and experience of the observed world, and treats geometric objects such as lines and shapes as objects in their own right. For example, a line could be the path of a light ray in space; you can envisage comparing line segments or angles by ‘moving’ one over another, thus giving rise to notions of ‘congruent’ figures, equal lengths, or equal angles that are independent of any quantitative measurement. If A, B, C are points along a line segment, what it means for B to be between A and C is an idea hard-wired into our consciousness. While descriptive geometry is intuitive and natural, and can be made mathematically rigorous (and, of course, Euclidean geometry was studied in these terms for more than two millennia, compare 9.1), this is not my main approach in this book.