My way to non-Euclidean and fractal kaleidoscopes

Abstract

I have always been interested in geometry. When I was about 12 years old, my grandfather Oskar Stampfli, an excellent teacher and mathematician, showed me the tilings of the plane with regular polygons and that there are only five Platonic solids. The self-similar shapes of crystals and ferns were fascinating to me. Later, I discovered the geometric art of M. C. Escher as well as the concrete art of Max Bill and Verena Loewensberg. I admired their work. But then I was also disappointed, as the underlying geometrical ideas were too simple. On the other hand, I realized that these paintings and prints need a lot of work and talent. Yet, I was dreaming of creating images based on more sophisticated geometry and using less time-consuming manual labour. Meanwhile, Penrose and others discovered quasiperiodic tilings and Benoit Mandelbrotmade self-similar fractal structures popular. At the university, I studied non-Euclidean geometry. All these ideas are inspirations for mathematical art that goes beyond periodic ornaments. However, doing the images by hand takes a lot of time and is not accurate enough. Then came powerful personal computers. They can rapidly generate complicated images and allow extensive explorations. It is now possible to zoomnearly without end into details of an image. Thus, I can now realize my dreams. Usually, I beginwith some vague questions:Howcan I decorate a tilingwith fragments of a photo, such that they fit the symmetry of the tiling and that the resulting image appears to be continuous? Is there an iterative procedure tomake an image that resembles a snowflake? What happens to an image after multiple reflection in distorting mirrors? I prefer to map photos onto geometrical structures rather than doing abstract visualization. This makes more natural looking images and recognizable real world details make a surreal effect. For periodic and quasiperiodic tilings of the Euclidean plane, I use reflection at straight lines and thus I can directly make a collage of small pieces of a photo. Others are doing similar work. Frank Farris uses wave functions instead of mirrors for mapping photos, as you can see in his book ‘Creating Symmetry’ (Farris, 2015). Thus, he creates images for all wallpaper groups of the plane. With mirrors, I can only make a small subset. However, with distorting mirrors, such as inversion in a circle, it is much easier to create kaleidoscopes that make hyperbolic and fractal images. These kaleidoscope distort the pieces of the photo depending on their place in the resulting image and its geometry. This is determined by a computer program. I can choose the size, orientation and position of