I’ve been interested in the subject of non-Euclidean tessellations for many years now, although the reality is that I’ve only managed to work on them infrequently. My most recent spurt was in early 2009, when I created this image (click on it to open a larger version in a new window).
If you look closely at it, you’ll see that it consists of regions coloured blue, purple, and gray. Regions of the same colour are congruent, but not in the usual Euclidean sense of being rotations and translations of one another. It turns out that the class of congruency transforms is the more general set of Moebius transformations. If you’re not familiar with these, a visually intuitive introduction is given in this popular video.
Moebius transformations don’t preserve “straightness” and distance, but they map circles to circles and preserve angles. As such, they still preserve a lot of information, and they give rise to a vast number of beautiful tessellations.
Unlike the familiar Euclidean tessellations that consist of tiles with straight edges, the “tiles” in these non-Euclidean tessellations have edges that are circular arcs. Using the appropriate circle-preserving Moebius transformations, they can transformed so that copies match perfectly along their edges, leaving no spaces, and fill the plane.
For the tessellation in question, the basic tile is shown in Figure 2 below (click on it to see a larger version). It has been transformed by the Moebius transformation 1/z to make it a finite region.
It consists of three regions – purple, gray and blue. Shapewise, the tile is radially symmetric. Although the purple/blue regions appear to have 4 sides, they actually have 6 sides each – the small sides are easier to see on the full-sized image. Similarly, the gray region has 8 sides – two are quite small and are truncations on what at first look like corners.
Another view of the basic tile – known in the parlance as a “fundamental domain”, or “Ford domain” – is given below.
Look for the labels “A”, “B”, “C” in the above image. They label, respectively, purple, blue and gray regions. The gray region is outside of the other regions in the picture, and has infinite extent (contains the point at infinity). That’s why we transformed it by 1/z for the previous picture.
If you look carefully at the full sized version of the first picture in this post, you can see how the tiles fit together. I find it remarkable that such an unlikely tile shape could tessellate the plane, let alone do it so beautifully. I like to contemplate a Moebius soup in which swim untold multitudes of such tiles, gradually assembling according to edge pairing rules (self-deforming as they mate along edges) into an infinitely detailed quasifuchsian crystal.
Anybody looking for more technical details should take a look at the book Indra’s Pearls. The tessellation here corresponds to the limit set in Figure 8.13 of that book. And here is a movie that zooms out between Figure 1 and Figure 3: