The Dragon of Eve
A selfavoiding planefilling fractal curve by Jeffrey Ventrella 
I discovered the
Dragon of Eve
while living in the basement
of the house of Eve Peters, in the early 1980's. Eve was my mother's art
teacher when my mother was young. I had just started working on my
Master's degree at Syracuse University. My thesis was on fractal art.
I had already spent about a year with
Mandelbrot's book,
studying techniques for creating planefilling fractal curves.
One technique is called "edge replacement". It is analogous to the more general technique of Lsystems. The technique I have devised encodes the geometrical operation of edge replacement using complex integer math. 
The Koch curve provides an easy and familiar example of how edge replacement works.
The generator is copied four times, with each copy scaled to onethird size, making
four baby generators. These are placed
onto the line segments (edges) of the original generator, thus causing the
generator (the mother) to give birth to four bumps that look like small versions
of the original bump.
Repeating this process results in a progression of bumps on bumps. Now, look at the example at right for how the Dragon of Eve is created: it requires a "flip" in one of the steps (the last baby is flipped), causing the sharp baby bump to aim downward instead of upward. Notice how each flipped baby "stays out of the way" as the curve grows more bumps. Every generation of mother bumps backs away and leaves room for the next generation of baby bumps. This generational dance continues to infinity. 
I designed many fractal generators using flipped segments
because I had observed a similar technique in Mandelbrot's amazing
Snowflake Sweep
(shown at right), and other fractal curves described in the book.
The boundary of the snowflake sweep is the same as the Koch Snowflake. It is topologically equal to a disk (the boundary never makes contact with itself). Now consider the classic HarterHeighway Dragon Curve: this curve fills a planar region that has a series of "waspwaists"  touchpoints that separate body parts, each topologically equal to a disk. There are infinetely many such body parts. Harter Heighway Dragon Curve similar body parts 
Snowflake Sweep 
The same is true for the Dragon of Eve: the sweep that fills it is divided into infinitely
many selfsimilar body parts. However, they are not all exactly selfsimilar! (look closely).
One way that the Dragon of Eve is different than the classic Dragon Curve is that it has varying lengths in its segments, owing to the fact that the middle segment in the generator has length equal to the square root of 2, while the other segments have length 1. The textural effect of having varying lengths can be seen in the highresolution rendering below: 
Dragon of Eve body parts 
As a continual unbroken curve (as a connected series of line segments), how long is the
Dragon of Eve? Well, it depends on how many times the process of iteration is
performed. By definition, a planefilling curve is infinitely long, since it has
to be iterated an infinite number of times in order to completely fill the entire
region of the plane. Below is an illustration showing how the length of the Dragon
of Eve increases exponentially each time the process of iteration is performed.
(In this illustration the length starts as 1  consistent with Mandelbrot's convention
of normalizing all generators to the unit interval).

Here's a
video
I made describing the Dragon of Eve, and some of its interesting properties:
My scheme for generating the Dragon of Eve is described on this page and illustrated below. This scheme uses Gaussian and Eisenstein integers, which allows the entire definition of an edgereplacement fractal curve to be represented in a compact way. 
Check out my
Fractal Curves
page, where many more strange spacefilling fractal beasts can be seen.

Finally, here is a reference to the
Great Wave of Kanagawa:

BrainFillingCurves.com  Brain Filling video  FractalCurves.com  Bubble Tree Algorithm 