Lectures on the Icosahedron

Some weeks ago, I was looking for examples giving me a quick overview on how to control transparency in the raytracer POV-Ray. This took me to the website of David Dumas who has beautiful illustrations of limit sets. (A very good and accessible introduction to the beauty of the Kleinian groups behind this is given by the book “Indra’s Pearls” [1].) What took me as even more beautiful were the pages from old mathematics books he is using as a background.

This again led me to look into the writings of Felix Klein and, in particular, it led me to start reading his wonderful book “Lectures on the Icosahedron” [2] (actually the German version [3]). This book is such an inspiration that I think that every sentient being should read it. In the first chapters, Klein is aiming at instilling “into comparatively elementary geometrical figures the ideas of the theory of groups, in such a form that the group-theory reflexions and the geometrical mode of illustration might henceforward supplement one another.” However, he is leaving the actual illustration, by images or models, of what he is describing to the reader. From a didactic point of view, this makes sense because the reader will learn a lot by producing the illustrations by himself. On the other hand, in this way the beauty is not immediately visible and the hasty reader will not grasp it.

I have started to make illustrations of what is described in Klein’s book and I will discuss them in upcoming posts. To start with, here is a taste of the illustrations. Go read the book!


[1] Mumford, David; Series, Caroline; Wright, David (2002), Indra’s pearls, Cambridge University Press, ISBN 978-0-521-35253-6, MR 1913879.

[2] Klein, Felix, Lectures on the ikosahedron and the solution of equations of the fifth degree, London: Trubner and Co, 1888.

[3] Klein, Felix, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, Teubner(Leipzig), 1884.

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