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 
Now we come to the root5 family. We have already met the 5dragon: 
Remember how I flipped the x values of all the segments of the TerDragon to make the Inverted TerDragon? Well, the same can be done with the 5Dragon. And, just like the inverted Ter, the inverted 5Dragon has a pinched waist. 
This next specimen is Mandelbrot's Quartet: "Each 'player', and the table between them, pertile." [16]. He claims to have "designed" it, although one could debate that such a curve is "discovered" rather than "designed". In either case, it is one of the finest selfavoiders. 
I discovered a variation of this generator, created by reversing the xflipping of each segment. I call it "Innerflip Quartet". 
Next I will show you six variations on a single generator shape. Two examples are shown on the next page. The first example has an interesting property: due to the flippings, the orientations of copies of the generator do not correspond with a continuous square grid. You can see this in the mixture of 90 and 45degree angles in the level 2 teragon. I would not have expected a curve like this to survive the fractal test. There is indeed selfcontacting in several vertices, but other than that, it is rather wellbehaved, as indicated by rendering with rounded corners. 
A close relative of this curve is shown here. 
These two specimens resolve to the same general shape, as indicated by the illustration below. 

Given the same generator, with alternate flippings, we get two gridfillers with very craggy boundaries: 

With other changes in flippings, we get the following gridfillers: 

In that last one, notice how the conifer treelike spike at the upperright corresponds to the empty gap at the bottom, rotated by 90 degrees. My brain is pertiling! 

Here are two planefilling curves of the root5 family that use a common generator shape. 


That last curve can be combined with a 180degree flipped copy of itself to make the shape of the 5Dragon...


This next curve is a selfavoider. It is followed by a similar specimen. 


Each of these last two curves can be copied four times  each copy rotated 90 degrees  and joined together to make a continuous curve. The overall shape is a replica of the Quartet (one of them is a mirrorimage of the other). This appears to be a property of many root5 curves I have shown.


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 