I have been working on a taxonomy of space-filling curves in the square and triangular lattices. I came across Adam Goucher’s blog: **Complex Projective 4-Space**. I asked Adam what he thought of my proposed **fractal curve families **and he told me that I should study the **Gaussian** and **Eisenstein** integers. The image below shows Gaussian integers (red dots) and Eisenstein integers (blue dots).

So cool! Adam pointed me to a whole new world of discovery that I am still unraveling and unfolding to this day.

I will not get into the details of **Gaussian** and **Eisenstein** integers here, except to say that they are sets of **complex numbers** occupying square and triangular lattices respectively. (You can click on the links provided and follow the search to any level of detail you wish).

They can be seen as two-dimensional versions of the one-dimensional number line of integers that we are all so familiar with.

Like their familiar one-dimensional counterpart, the Gaussian and Eisenstein integers form an **integral domain**: the rules of addition and multiplication apply, and these operations always result in a new number which is within the domain. Gaussian and Eisenstein integers have their own unique prime number and composite number *personalities* – which make for a fascinating study – including how they relate to families of plane-filling curves.

The **Koch Curve** and it’s Squarish cousin occupy triangular and square grids, respectively. And each has a “spine” that traverses exactly three units.

In these two lattices, all distances of grid points from the origin (0+0i) are square roots of integers (** Euclidian norm**).

Squaring the Euclidian norm gives us the ** field norm** , the integer that we use to denote the fractal family. It is either prime or composite.

Here are some examples of plane-filling curves that occupy the square lattice:

Example 1 (also discovered by **Carbajo**)

Example 2 (Mandelbrot’s “Quartet”)

Example 3: (**a Root 13 self-avoiding curve**)

And here are some examples of plane-filling curves that occur in the triangular lattice:

Example 4: (**a curve I discovered**)

Example 5: (**a member of the Root 7 family**)

Example 6: (**a member of the triangle Root 9 family**)

Check out the generators of these curves (shown at the left of each diagram). You may have noticed that some of them consist of line segments of ** differing lengths**. These represent members of

*composite*fractal families: their spines traverse a distance that is equal to the square root of a composite number in the domain. (Examples 1, 4, and 6 are composite).

The members of “prime” families (examples 2, 3, and 5 above) can only have segments of length 1. Consequently, as they are iterated to create higher-level **teragons**, with more fractal detail, the texture becomes more even and smooth. The composite curves – on the other hand – have varying density within their flesh, which increases in variety as they are iterated.

This is awesome: it’s another way to show the recursive complexity of composite numbers. And if you ask me (and **Greg Chaitin**), the composite numbers are way more interesting than the primes!

Here’s a plane-filling curve of the root-8 family. Since 8 = 2^3, this family contains interesting variants of the famous **dragon curve**, which is a member of the root 2 family. This example has the corners rounded for aesthetics. It shows some of the variation in scale within its flesh.

I’ll be posting more as discoveries are made. Stay tuned!

-Jeffrey

## 2 thoughts on “Fractal Curves in Gaussian and Eisenstien Integer Lattices”