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The Root 3 Family
  A chapter from Brainfilling Curves by Jeffrey Ventrella








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 next family of plane-filling fractal curves is the root3 family. Unlike the fractal curves of the root2 family, these curves live in the triangular grid. Now I need to explain one aspect of my notation for fractal generators in the triangular grid. Look at the illustration below:


The segment values at left specify the directions that the four Koch curve segments are drawn. These values are pretty straightforward when applied to a square grid, but not so much when applied to a triangular grid. In order to use a consistent scheme for specifying segment directions in both girds, I specify all values as if they were in the square grid. So, for instance, the Koch curve is shown above with the following values: (1, 0) (0, 1) (1, -1) (1, 0). Since the Koch curve is a member of a triangular grid family, all of its segments have two transforms applied to them: a skew (notice that the top of the graph is shifted to right) and a scale (notice that the height of graph is squashed). These transformations result in grid lines that cross at 120 and 60 degree angles instead of at 90 and 45 degree angles.

Please keep this in mind whenever you are reading the segment values for generators in the triangular grid. To make it easier, I have drawn another picture at right. It shows six direction values (colored blue), which are mapped from the square grid to the triangular grid.

Here is a question regarding the triangular grid: look at the picture below. How many unique 3-segment paths can you draw from point A to point B?


As you can see, there are six ways to connect A to B using 3 segments of length 1. But in fact there are really only two, if you consider the fact that all of them are just rotations or reflections of the two examples highlighted at the top. We can ignore the rest of them because, as fractal generators, their teragons look exactly the same as the teragons of these two, just that they are rotated or reflected. Now, given these two generator shapes, consider the various ways that each of their segments can be flipped: since each segment has exactly four possible kinds of flippings, we can conclude that the number of possible flipped variations of each of these paths is:

43 = 64


Thus, there are 128 curves to test for plane-filling (64 for each of the two paths shown in the illustration). I have tested these and have found that there are ten plane-filling curves of the root3 family. One of them may already be familiar to fractal fans: the Ter-dragon:


Famed computer scientist Donald Knuth is said to have first discovered the Ter-Dragon. Unlike the HH Dragon, the Ter-Dragon has point-symmetry: its tail looks like its head...which looks like its tail. Notice also that three copies of the Ter Dragon can be combined to make a larger one. But no surprise there, right? This fractal curve just oozes with three-ness. The box at the lower-right shows how the Ter-Dragon can tile the plane.


The Ter-Dragon is our first example of a "Palindrome Curve", that is, a fractal curve which is symmetrical about its center. Palindrome Dragons have heads that look like upside-down copies of their tails. I'll be showing you more interesting properties of Palindrome fractals later on.

Based on the Ter Dragon's generator, we can create an entirely different palindrome curve simply by flipping each of the segment x values, as shown here:


This simple flip changes the resulting fractal curve from being fat in the middle to having a pinched waist. I call it the "inverted Ter-Dragon". Below is yet another variation attained from different flippings of the Ter-Dragon segments. In this case, the first and third segments have their x values flipped. It is hard to predict the outcome of these small changes...and you would probably not have guessed that the result would be a curve that completely fills a rectangle!


Just ONE flipped number. That's all it takes to transform a craggy-edged butterfly into a box.

Now let's look at the other kind of path that can connect point A to point B. This one can produce seven unique plane-filling fractal curves. They are shown on the following pages.








Check it out: the fractal curve above can be combined with another copy of itself that is rotated 180 degrees. When they are joined together they create two intertwined halves of a double-sized Ter-dragon! The more times the teragons are fractalized, the more tightly the handshake in the middle spirals inward. It reminds me a yin-yang symbol. And so, I call it the Yin Dragon. The two previous specimens are also shown at the bottom, as yin-yang pairs.


The Yin Dragon was also discovered by Tom Karzes [11], who called it "Half-TerDragon". He created the following cool pertilings:


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




Brain-filling Curves - A Fractal Bestiary
by Jeffrey Ventrella

Distributed by Lulu.com
Cover Design by Jeffrey Ventrella
Book web site: BrainFillingCurves.com

ISBN 978-0-9830546-2-7
Copyright 2012 by Jeffrey Ventrella

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FractalCurves.com