The First Nine Families 



The comprehensive set
I have designed a comprehensive scheme for categorizing every possible planefilling curve (2D spacefilling curve) using edgereplacement (Koch construction). The technique is explained in the book, Brainfilling Curves. This method is used to both classify and generate planefilling curves, including the familiar Original Peano curve, the Dragon curve, the Gosper curve, the Snowflake sweep, and several others. These classic curves were explained within a general framework in Benoit Mandelbrot's book, The Fractal Geometry of Nature. My system picks up where Mandelbrot left off, and introduces hundreds of new curves  with a method for finding many more. Some planefilling curves can be generated using other techniques. Another commonly used technique is 'node replacement' which is used to create the Hilbert curve, and many others. Both nodereplacement and edgereplacement curves can be generated using Lsystems. Also, both are related to recursive tiling. What curves qualify for this collection? All planefilling curves using edgereplacement (Kochconstruction) are included in this list. Each curve has a fractal dimension of exactly 2. It can touch itself, but it can never cross itself or overlap with itself. Not all planefilling curves are topologically equivalent to a disc: some curves have holes or complex boundaries. However, all of them have regions that are infinitely filled  and this suffices to include them in this list. Some curves may have uneven density  and yet they are still planefilling. While this collection does not include curves that selfoverlap, a few classic selfoverlapping curves are included (e.g., Cesaro's Sweep and the Peano Sweep). Relation to Gaussian and Eisenstein integers Each family f of planefilling curves corresponds to a Gaussian integer (square lattice) or an Eisenstein integer (triangular lattice). A fractal generator can be seen as the visual expression of an ordered set of integers  each of which corresponds to one segment of the generator. The sum of these integers is the family number f. Generators of planefilling curves can only contain segments whose associated complex number is an integer root of the family number f. Another way of saying this is that the sum of the squared lengths (the norms) of each segment must equal f. For instance, the Eisenstein9 family is represented with generators that can have segments with norms 1 and 3. In contrast: the Eisenstein family 7 is a prime number, and so it can only have segments of length 1. 

The First Nine Families 



fractalcurves.com 