Rainbows form attractive, luminous patterns, whether they occur in the sky or on a light table in the laboratory. Here crisscrossing rainbow bands have been interwoven to form Celtic knot patterns. The challenge was to create only one asymmetrical tile that could be connected in arbitrary ways with itself, always leading to a seamless fit around its border, while overall creating many different irregular, but pleasing tiling patterns.
The tile geometry is based on the Poincaré disk for the regular hyperbolic {4,6} tiling. The geometry in the region between the two quarter disks has been obtained by a conformal distortion of the basic hyperbolic tiling pattern. There is only one tile type, but it comes in two different complementary colorings. This allows one to form ever new undulating patterns. It also raises the puzzling question, which color is the foreground and which one is the background?
This work was inspired by many artists, but especially by M.C. Escher, Oscar Reutersvärd, and Tamas Farkas. Inside the one hexagonal tile, I kept the parallel projection of the pseudo-3D corners consistent. Thus half of all the possible ways of joining two abutting tiles will maintain that consistent orientation in space globally. However, if some of the tiles are rotated by an odd multiple of 60 degrees, a Necker-cube like inversion is forced onto the viewer's perspective when the gaze travels from one tile to the next.
A collection of these tiles form a depiction of a luscious green jungle in the style of Henri Rousseau, celebrating a healthy green Earth. Again, I tried to use only a single tile to create a jungle scene as varied as possible; thus asymmetry and irregularity were important attributes. After some experimentation, I found a structure that keeps the snake body as well as the vines seamlessly connected across tile boundaries, and which creates many different leaf bundles by connection half-bundles across the tile boundaries. With this one tile, jungle scenes of arbitrary sizes can be composed.