Thales series: all the rectangles of the world

Abstract

In January 2019, I turnedmy interests in art and geometry frommy work with incommensurable ratios and the grids they make to a theorem by Thales of Miletus. Thales observed that in a semi-circle, as AC in Figure 1a, any point, B, on the circumference of that semicircle, AC, when drawn to the ends of the diameter, AC, will always make a ninety-degree angle at that point, B. Because there are an infinite number of points on the circumference of the semi-circle, an infinite number of right triangles can be generated, as in Figure 1b. In Figure 1c, it follows then that in a complete circle, any right triangle, AKM, by rotating it a half-turn about the centre, O, of this circle, will produce a rectangle, AKMZ. This diagram shows one easy way to achieve this rotation a halfturn: draw a line from point K through the centre of the circle, O, to R. Because of the infinite number of points on the circle, an infinite number of rectangles – all the rectangles of the world in fact – can be generated. The result of my studies is a new series of drawings and watercolours entitled, ‘Thales Series: All the Rectangles of the World’ (ATROTW for convenience). When I began my series, I realized that any and all rectangles I drew using thismethod have three common features: (a) They share a common diagonal; (b) This diagonal is equal to the diameter of the generating circle; and, (c) This diagonal is also the hypotenuse of a right triangle. While other construction methods can produce axially-aligned rectangles as well as radial, rotational, and reflection symmetries in the circle, my interests so far have centred on generating specific ratios and families of rectangles into the circle using these three features of Thales’ theorem. I continue to work with the diagonal/diameter/hypotenuse relationship because I like the dynamic and unique appearance of the artworks. I also like the challenges and aesthetic considerations presented in the Thales construction.