Illustrating Euclid inspired by the Axioms of Kandinsky

Abstract

Before I studied mathematics, I had already finished my studies in design with a focus on Illustration and 3D-Animation. My teacher in art philosophy used to say ‘Art is always contradictory, if you are confronted with a piece of art and you can decipher it completely, you can be pretty sure that you are looking at kitsch’ (Engelmann, 2003). One should not take this statement as a general rating, since it does not distinguish between good and bad art. It also includes that kitsch could be work of high artistic quality (although one should pause for a moment to think about what this implies for math-art in general, since mathematics is not very suitable to capture contradictions). But it shows a huge difference between art and illustration: while art is about asking questions, illustration is about giving answers. When you are looking at assembly instructions for an IKEA shelf or a mathematical proof, you want the illustrations to be as clear as possible. Children’s book illustrations normally give us answers about the characters and the surrounding world while an illustration of a poem is supposed to capture the mood and rhythm of the poem. Of course these boundaries are very blurry, so in the following I want to present some illustrations that concentrate on the ‘poetic’ side of mathematical proofs. Inspired by musical compositions, Wassily Kandinsky developed a (very flexible) axiomatic system that enabled him to construct his abstract paintings. In his bookPoint and Line to Plane (Kandinsky, 1926/1955) from 1926 Kandinsky claims that points are the primal element of every painting. A line is the trace of a moving point, and the characteristics of a line or the resulting shapes are defined by the movement of the points. The combination of points, lines, and shapes on the canvas creates tension that we perceive intuitively when we study an artwork, but which in principle could be measured mathematically, if one understands the underlying grammar of the art-language. His approach to not take nature as a model for his paintings, but to instead construct his compositions out of simple geometrical forms was a radical break with the predominant traditions. He claimed to be the first, whoever painted an abstract painting. But there are other contenders who created abstract paintings around the same time, like Robert Delaunay, Piet Mondrian and Hilma af Klint, who could also be regarded as the first abstract painter, depending on your definition of abstract art. To some degree his approach resembles the work of Euclid, who a few thousand years before also developed a (very rigid) axiomatic system based on simple geometrical forms.