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 
Since root4 = 2, and since square grids and triangle grids share common grid points along an axis (the horizontal "floor" of the two grids), you may suspect that there exists a family of root4 planefilling curves that live in the triangle grid. Indeed I have discovered what I believe to be all of them. 
But before I show you these planefilling curves, I would first like to show you a member of the root4 triangle grid family that is not planefilling: its fractal dimension is ~1.5849625, and it generates the famous Sierpinski Arrowhead Curve. It is identical to the Sierpinski Triangle (a solid triangle with its center triangle cut out, and then with the center triangles cut out of the remaining three, and so on)... 

When you fractalize the Sierpinski Arrowhead Curve, it converges toward the Sierpinski triangle. At each stage, it accumulates bays and peninsulas, which approach each other, getting closer and closer...but never touching. So in fact this is a selfavoiding curve. 
I was delighted when I discovered a curve that actually creates a family of Sierpinski curves! In the diagram below, I show this curve fractalized to level 6. Daddy Sierpinski sits proudly to the left, with Mommy Sierpinski to his right. To her right is daughter Sierpinski, followed by little brother Sierpinski, and then baby Sierpinski, and finally, the family pet: Turtle Sierpinski. 
Here is a higherlevel rendering of the Sierpinski family fractal curve, with some extra members of the family to the right of the pet turtle (I leave it to you to imagine who are the tiniest members of this family). 
Now it's time to look at the planefilling members of the root4 triangle grid family. I will start with one that requires no flippings in its segments. Here it is. It happens to be a palindrome: 


This fractal curve is a gridfiller, but it has a wild boundary, which has a high fractal dimension of its own. Not only that: the boundary touches itself. This fractal makes the notion of "planefilling" very fuzzy (so to speak) because the region of the plane that it fills is scattered haphazardly. All the filledin areas of fully fractalized curves that we have seen before this one had boundaries that were either straight lines or else they were fractal curves of their own...but never selftouching. Now get ready; here is that same generator with some different flippings. Its boundary is so amazingly selftouching, you might call it "selfenveloping" (but it is NOT selfcrossing, as revealed by rendering it with rounded corners). 
When highly fractalized, this curve becomes a parallelogram filled with a cacophony of triangles. 
Here is that same generator with yet another set of flippings. 
Now here is a different generator of this family. This curve is highly selfenveloping. 
On the next page is a picture showing two copies of this curve...mating. They are engaged in the most intimate embrace one can imagine. 
Here is another curve based on the same generator shape. I have rendered it below with hierarchical coloring to indicate the way nodes are formed: each node has 3way symmetry and is connected to other nodes at pinchpoints. 
Here is another curve based on the same generator shape. I have rendered it below with hierarchical coloring to indicate the way nodes are formed: each node has 3way symmetry and is connected to other nodes at pinchpoints. 

The last members of the root4 triangle grid family I will show you are curves that exactly fill an equilateral triangle. They are pseudogridfillers. 
You can think of each of the four segments of this generator as being responsible for one of
four subtriangles. The precise set of segment flippings is important, so as to avoid
edgetouching.
Zbigniew Fiedorowicz
[6] made a variation of this fractal  shown below.

Two variations are shown here, and at the bottom of the page is a portrait of all three variations at level 4. 
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 