1 Horror Vacui 2 A Very Patient Turtle Who Draws Lines 3 A Taxonomy of Fractology 4 Gallery of Specimens 
Root 2 Family  Root 3 Family  Root 4 Square Grid Family  Root 4 Triangle Grid Family 
Root 5 Family  No Root 6!  Root 7 Family  Root 8 Family 
Root 9 Square Grid Family  Root 9 Triangle Grid Family  Root 10 Family  Root 12 Family 
Root 13 Square Grid Family  Root 13 Triangle Grid Family  Root 16 Square Grid Family  Root 16 Triangle Grid Family 
Root 17 and Beyond...  5 My Brain Fillith Over  6 References  7 Acknowledgements 
The number of planefilling specimens in the root2 family is 2 (I am not counting Cesaro). The number of planefilling specimens in the root3 family (by my estimate) is 10. As we climb the family tree, the number of specimens increases nonlinearly, making exhaustive search quite an expensive proposition when many segments are involved. Consider the sheer number of possible generators if each segment has four possible flippings, and connects any two grid points, where its length is less than the interval length. I calculated the number of possible generators having up to 5 segments. Ready? 
2 seg:
3 seg: 4 seg (triangle): 4 seg (square): 5 seg: 
16
13,824 331,776 1,048,576 254,803,968 
I have done some exploration of this huge space of possibilities, using both handdrawn diagrams, and computer search algorithms, and have discovered some interesting specimens. Let's look at some of those now, as well as a few gems previously introduced by Mandelbrot and others.
First, let's start with a planefilling curve that Mandelbrot introduced in his book. It is a member of the root17 square grid family, and it is a partial gridfiller. Mandelbrot pointed out that selfavoiding curves are not only more aesthetic; they also make better models of forms found in nature (such as rivers and watershed trees). When a curve is vertex selftouching, it "knots off". In this example below, the generator itself has a pinched point. This percolates throughout the teragons, in a similar way that I showed you with the selfcrossing 7dragons. Here is Mandelbrot's Knottedoff root17 curve: 
Here is another root17 curve. It is a selfavoider, and it has a dimension of ~1.81. 
The root19 Triangle Grid Family
Let's look at a few of the fine specimens of the root19 triangle grid family. I like to start with dragons, and so here is a palindrome dragon with a pinched waist and pinched extremities. It is shown below at level 3 with rounded corners. 
Closepacking of seven circles results in a roughlyhexagonal shape: the basis for the Gosper Curve. If you completely surround these seven circles with a new layer of circles, the total number of circles is increased to 19, as shown at left. This 19cell hexagonal grid is the basis for another one of the wonderful generalized Gosper curves discovered by Fukuda, et. al [4]. It is shown below. 
I discovered a curve based on this 19cell hexagonal theme. Its dimension is less than 2, and so it has lots of open spaces, which gives it some artistic breathing room. I call it "Mandala". It is shown in color on the next page. 
Remember the AntiGosper? Well, I suspect that the following curve has some things in common: it is roughly triangular and more tightlypacked than its hexagonal cousin. 
The root20 Square Grid Family
There are five fours in twenty. And so I wondered if I could build a spacefilling curve by replacing the squares in Mandelbrot's Quartet with rotated copies of a root4 square grid family generator. Well, the Peano Sweep generator appears to do the trick: 
Here is a variation that uses a combination of the Peano Sweep generator and another foursegment shape that fractalizes to a square. It is a shape that would not normally stand on its own as a planefilling curve generator. But in the context of the whole arrangement, it works perfectly.

The following generator has a knottedoff square, like the root17 specimen we saw earlier. 
This curve is rather curious. It is a gridfiller with a bit of asymmetry. 
The root21 Triangle Grid Family
Of the root21 triangle grid family, I have found a few fat palindrome dragons. Two of them are shown below. 
Since 21 is a triangular number, we can arrange the pertiles in a roughly triangular array. Below is a triangular specimen of this family, followed by a close cousin with dimension ~1.89. On the next page is a color rendering with rounded corners.

The root25 Square Grid Family
Now we come to another square number: 25. Of the root25 square grid family, I have only one specimen to show: the Quadratic Gosper Curve. It is attributed to F.M Dekking [5], and also Doug McKenna [19]. This is a selfavoiding curve that fills a square. 
The root25 Triangle Grid Family
The root25 triangle grid family has a whale of a bumpy dragon, shown below. 
On the next page are three more specimens of this family. The third one is shown on the following page enlarged at level 3 with rounded corners. 
The root27 Triangle Grid Family
I want to show you two related specimens of the root27 triangle grid family. They were introduced in Mandelbrot's book, and they are variations of the Snowflake Sweep. Now, as we saw earlier, the snowflake sweep is a member of the root9 family, so why are these specimens root27? The reason is due to the nature of my taxonomy scheme, which requires that all generator segments lie between grid points. With a slight variation to my scheme, these curves could be represented in the root9 family, by relaxing this requirement. The smallest segments occurring within the interior of the generator would fall between the cracks, as it were. We can say that these specimens are closely related to the root9 family, except that they have a few small (and clever) genetic mutations that permit extra details to emerge in the crevices. The first specimen is one Mandelbrot called "Monkeys Tree". The second is a variant of the Snowflake Sweep, in which the whole generator is transformed and takes the place of the fifth segment. This is rendered stylistically on the next page. 
End of chapter. 
1 Horror Vacui 2 A Very Patient Turtle Who Draws Lines 3 A Taxonomy of Fractology 4 Gallery of Specimens 
Root 2 Family  Root 3 Family  Root 4 Square Grid Family  Root 4 Triangle Grid Family 
Root 5 Family  No Root 6!  Root 7 Family  Root 8 Family 
Root 9 Square Grid Family  Root 9 Triangle Grid Family  Root 10 Family  Root 12 Family 
Root 13 Square Grid Family  Root 13 Triangle Grid Family  Root 16 Square Grid Family  Root 16 Triangle Grid Family 
Root 17 and Beyond...  5 My Brain Fillith Over  6 References  7 Acknowledgements 
Brainfilling Curves  A Fractal Bestiary
by Jeffrey Ventrella Distributed by Lulu.com Cover Design by Jeffrey Ventrella 
Book web site:
BrainFillingCurves.com
ISBN 9780983054627 Copyright © 2012 by Jeffrey Ventrella 
eyebrainbooks.com 
FractalCurves.com 