The Family Tree of Fractal Curves
A taxonomy of plane-filling curves using complex integer lattices
by Jeffrey Ventrella
1 Introduction
2 Taxonomy
3 Iteration
4 Morphology
5 Family Resemblances
6 Undiscovered Curiosities
7 References




6. Undiscovered Curiosities


The invention of the telescope allowed astronomers to peer deeper into the cosmos. The invention of the microscope allowed biologists to peer deeper into the inner workings of cells. Although true fractals are Platonic entities—occupying an abstract mathematical realm—it is hard to avoid the feeling that these mathematical objects lay hidden, like stars and cells, unavailable to the human eye until the right tools come along to bring them out of darkness.

Here is a question: are fractals designed by humans or discovered by humans? Let us not rush to a binary answer, and instead savor the question. Amazing fractal patterns do appear in nature, untouched by the human hand. The information dynamic resulting from running a recursive function on a computer has similarities to many dynamical systems in nature. Thus, the computer can be considered as a tool to visualize aspects of nature.

Software and graphics technology have helped inspire new mathematical tools and ideas that made it possible to discover the fractal gems that we can now calculate at lightning speeds. This power of calculation never could have been imagined when Giuseppi Peano drew his nine-segment curve in 1890.

Conclusion
One goal of the taxonomy and associated techniques covered in this book is to provide a framework for describing several familiar plane-filling curves…under one system. It also provides a methodology for discovering more curves, including the infinity of beasts that lie far from home—far from their origins in the Gaussian and Eisenstein domains. Imagine exploring the far reaches of a family set—rich with genealogical self-similarity; we will find amazing new curves with complex variations on their ancestry.

The reason a taxonomy is possible may have something to do with the iterative nature of genesis, and the presence of certain constraints that encourage structures to emerge. Like biological organisms, these constraints come into play when genotypes guide the expression of phenotypes through recursion. For these curves, the lattice of complex integers corresponds to some of those constraints. This book lays the foundations of a framework for categorizing self-similar, plane-filling fractal curves. Mathematicians, geometers, designers, and anyone with a playful curious mind, will hopefully find it useful as a context for more discoveries—to reveal more amazing and beautiful fractal curves.





Figure 6.1. A self-avoiding curve of the E(2,1)2 family


The Family Tree of Fractal Curves
A taxonomy of plane-filling curves using complex integer lattices
by Jeffrey Ventrella
1 Introduction
2 Taxonomy
3 Iteration
4 Morphology
5 Family Resemblances
6 Undiscovered Curiosities
7 References