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 
Next we come to the root8 family. This family has some things in common with the root2 family:
the generator lies on a 45 degree diagonal. Also, it can be seen as a superset of the root2
family (as well as the root4 square grid family).
This is because the values 2, 4, and 8 are powersoftwo numbers (2^{1} = 2; 2^{2} = 4; 2^{3} = 8).
We will see later that the root16 square grid family is a superset of the root2, root4 square grid, and root8 families.
We've already seen one member of the root8 family: I showed it to you at the beginning of the book as an example of how the turtle can use flippings to convert an otherwise selfcrossing curve into a selfavoiding curve: 

When I showed this curve to you before, it was rotated 45 degrees. Here is it shown in its native familial orientation. Notice that the generator has only 5 segments, and that three of those segments have a length of root2. These segments are responsible for the three large lobes in the 2nd teragon.
As a general rule, you can consider segments of length root2 to count as two (remember that we square the lengths when calculating fractal dimension). So in this case, two onelength segments and three segments of root2  when squared  add up to 8: the family number. The root8 family is quite versatile. First let's look at some curves that fill a right triangle. Two of them are shown below. Like the last curve I showed you, the generators for these curves each have 5 segments, three with a length of root2. 


This last one is so interesting I decided to render it in color. It's shown on the next page. 

Here's another triangleshaped specimen. This one has even fewer segments: 4! One of the segments is length root2 and the last segment is length 2. Following the rule of squaring all lengths, as I said earlier, you can see how the sum is 8. Because of the long segment length, the result has a great variety of lengths within  with a lot of selfsimilar patterning. 

I'm particularly fond of this one. So I made a color rendering of it, and rotated it 90 degrees. It is shown below. 


Here are two more right triangles of the root8 family. These are a little less wellbehaved, but interesting nonetheless. 



This next specimen is a natural selfavoider. It is a relative of a root4 specimen we met earlier. On the next page I show five more curves of this family. 





That last curve fractalizes into a shape that is similar to the "Twin Dragon": the result of joining two HH Dragons. But notice that it is not quite the same as the Twin Dragon; it has pinchedoff babies  which each have their own pinchedoff babies. 
This next curve fractalizes into a pair of TwinDragonlike curves. Because of the similarity to the shape as the Twin Dragon, I call it the "TwinTwin Dragon". Are the two twins holding hands? No; their babies are holding baby hands. 

Speaking of dragons, the root8 family produces more dragons that are related to the HH Dragon. Here's one: 

Let's see it rendered at a higher level, below. To the right is the HH Dragon. 

Here's another specimen that is related to the HH Dragon: 

I am impressed with how this curve is so unpredictable and irregular in its internal meandering, yet it is able to avoid any selfcrossings (it does selftouch on vertices: those are separated due to the roundedcorners scheme of the drawing). Here is a rendering of two copies of this curve (one flipped 180 degrees). They are combined to make the shape of the twin dragon  which closes the loop, enabling it to be filled internally with color. 

Here's another specimen that resolves to the same shape as the HH Dragon. Since the generator has shorter segments in the middle region, the midsections of its tergaons are rather knotted, and full of detail. This specimen's teragons are precariously selftouching (even with roundedcorners). I have rendered it below, rotated 90 degrees. 


Another root8 Dragon is shown below. This one also resolves to the shape of the HH Dragon. 

Here it is fractalized at a higher level, splined, colorized, and slightly rotated...for your brainfilling pleasure. 

Here is a selfavoiding dragon that I was excited to discover. It appears to be a relative of the Dragon of Eve. It's like adding a smaller triangular bump onto the big triangular bump of the Dragon of Eve. It's like a curly Dragon of Eve! 


There is one last fractal I want to show in the root8 family: I call it "Brainfiller". Below it is colored and rotated. 


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 