Adam Goucher, who writes a blog called **Complex Projective 4-Space**, got me motivated several years ago to work on a taxonomy of plane-filling curves, based on complex integers. I’m working on a new book on the subject. The current draft of the book is **parked on the Internet Archive**. Adam looked at an earlier version of the draft, and made an interesting observation, relating my scheme for generating plane-filling curves to the **pinwheel tiling** – which is a **non-periodic tiling**.

image above from this web site: http://www.quadibloc.com/math/til04.htm

Imagine having an unlimited number of tiles that are right triangles, with sides of length 1, 2, and √5. The tiles can be assembled in unlimited ways – as long as adjacent edges are the same length.

The pinwheel tiling can be represented in my plane-filling curve taxonomy as part of the G(25,0) family (**see page 73 of the book**) by way of a reflection, a rotation, and a scaling by √5. The third and fourth segments of the generator are overlapping.

In the picture below, the top row shows three iterations (progressive subdivision) of the triangle. Below that is the genetic code for the plane-filling curve, whose generator is shown in the box as a collection of Gaussian integers (line segments in the square lattice). All of the segments have norm 5 (Euclidean distance = √5). At the bottom of the picture is the result of iterating the generator.

It’s always exciting to see geometry ideas come together like this. So much interesting math, research, and design is happening right now with tilings and plane-filling curves (which are of course related!)

**A ramified curve**

Since the generator for this curve is self-overlapping (two of the five segments overlap), it is more “dense” than the self-avoiding variety of curves. Mandelbrot referred to these curves as “ramified”. One way to visualize self-overlapping and self-contacting curves is to render them with splines (rounding off the corners). This helps to reveal the sweep of the curve:

Aperiodic tilings (and curves) are stimulating to the eye and brain, perhaps because of the peculiar mix of symmetry and asymmetry.

Plane-filling curves, when related to tilings like these, provide a scheme for ordering the tiles. It opens up a new way to think about what may otherwise seem like a disordered mess.

Adam suggested that this tiling could be rendered using a node-replacement curve (in addition to an edge-replacement curve). Node replacement curves have a more obvious relation to plane-filling tilings in that they can be described as a way of connecting the centers of adjacent tiles. The ordering of connections is the same as that of edge-replacement.

I am sure that there is a lot more to say on this subject. If you are a fellow fractal geek, you might have something to add. Please chime in!

-Jeffrey