Read it online
at the Internet Archive

Buy the book
Two hundred pages of color images
in 8.5"x8.5" paperback print
The Root 4 Square Grid 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


Now we come to the root4 square grid family. Let's return to the Koch Curve, which has four segments. If we morph the Koch generator, progressively sharpening the triangle bump in the middle, causing the first and fourth segments to come closer, the fractal dimension of the resulting curve will approach 2.


When the transformation is complete, the interval length is 2, and the second and third segments of the generator become a two-sided needle pointing upward. This is a variant of Cesaro's Sweep that I showed you earlier. It is a twice-dense gridfiller: it is self-touching among all segments (except for the segments that lie on the bottom edge).


Wolter Schraa [21] created a nice artistic image showing the transition of a curve from dimension 1 to dimension 2. A Koch Curve-like region emerges just after the middle area, and at the very end, it closes up to form the characteristic folding of Cesaro's Sweep. Here is an altered version of Schraa's image:


Now, back to a familiar question: in a square grid, how many ways can you draw four connected lines of length 1 from point A to point B, without having them self-contact? (Not counting rotations and mirror-images). The answer is 2:


Now, let's start with the first generator above (the square bump with a level segment to its right). How many plane-filling curves can we find using various flippings of this shape? Well, let's rule out a few curves that are already represented by the root2 family: the HH Dragon and the Polya Sweep. Observe that the level 2 teragons of the root2 family have the same shape (only rotated and flipped). Thus, transformed replicas of the root2 curves can be generated with this shape.

Besides these exceptions, I have discovered three unique plane-filling curves of the root4 square grid family. The first one, shown below, fills a rectangle that is tilted 45 degrees. It is a partially self-touching curve: many parts of the curve are self-touching at their vertices, but the rest of the curve is self-avoiding. It is shown below at level 6, tilted 45 degrees, with colored bars where there are gaps longer then a specific length. Can you describe the pattern of these gaps?



These next two curves are curious indeed. They appear to have variations on a similar motif.



The second specimen is shown below, pertiled four times, to make a square.


Now I'll show you the other curve that can be made using the other path from A to B that I showed. Mandelbrot included it in his book, and attributed it to Peano, calling it the "Peano Sweep". It is self-touching along some proportion of its edges. This is apparent at level 2.


For curves that are self-touching on edges, I sometimes use a different technique than rounded corners. In this case, I use a simple low-pass smoothing filter. Basically, after the points of the curve have been calculated, I adjust the position of each point to equal the average of the positions of itself plus its two neighbors. This has the effect of separating the edge-adjacencies, allowing space around the curve to breathe. Did you notice a similarity in the inner pattern to an earlier specimen of this family?







After applying this technique to make the curve breathe, I noticed a similarity to higher-level fractalizations of the Hilbert Curve, which we met earlier. And indeed, there is a direct correlation. The Peano Sweep is basically a Koch-constructed variation of the Hilbert curve, using edge-replacement instead of node-replacement. Notice that in both cases there are four tiling squares.

In the second teragon, we see that the scaled-down copies of the generators that are applied to the two bottom squares are pointing inwards. The copies of the generators applied to the top two squares remain pointing upward, in their original orientations. As I pointed out before, node-replacement requires extra connective lines, shown here in pink.

Also, notice that after several levels of fractalization, these curves start to look more similar (when using the smoothing filter technique on the Peano Sweep). Check out the main artery extending upward from the middle of the bottom, which is more noticeable on the Peano Sweep. There are also several secondary arteries - all of which correspond to the cascade of transformations used to generate the curve.




A New Slant on Fractal Dimension
Now it is time to explain a new aspect of fractal dimension, which wasn't necessary until now. Consider the illustration below. The shape at left has three segments, but the slanted one is longer than the others. Its length is root2, while the other segments have length 1. Now here's a trick: if each of these lengths are squared, and then summed, the result is 4.


Might this generator create a curve that qualifies as a member of the root4 family? Can a curve of the root4 family with only 3 segments fractalize to a plane-filling curve? Well, let's revisit the equation for fractal dimension: log N / log L. Now, instead of N representing the number of segments in the fractal generator, let's re-define N as: "the summed squares of all the segment lengths". Up until now, all the generators so far have had segment lengths of 1, and since root1 = 1, we could just refer to the number of segments. But now, we will change the definition of N to accommodate segments lengths greater than 1. And behold: the generator I have just showed you, given just the right flippings, results in a fractal curve that looks like the HH Dragon (but not quite! - look more closely). I call it V1 Dragon.


Notice that some of the blobs are bigger than others. That is because of the difference in lengths among the segments, which cascade into many different sizes. The longest segments appear to be at the left-bottom (at the start of the curve). This is related to the fact that the first segment in the generator is the longest.

But is it Plane-filling?
You might not think this a plane-filling curve, because of all these conspicuous blobs of different sizes. But remember that this curve (and in fact every curve in this book) has a limited fractal level. If we were to fractalize this curve to infinity, all of the curls would accumulate and close up to completely fill the shape. BUT...even at infinity, would the density still not be uniform throughout the shape? I shall leave this as an open question for you to ponder.

Now let's try a different flipping of this shape. Lo and behold, the resulting fractal curve looks quite different indeed.


This fractal curve is special. It is not too often that I find fractal curves that are self-avoiders. Well, this one is! Let's see it enlarged a bit, and with some cool coloring added. By the way, this curve appears to also have been discovered by artist Victor Carbajo [3].


And here's an interesting fact: if you detach the section at the lower-left, and rotate it 90 degrees, pivoting about the bottom corner, it fits snuggly into the remaining hole. Not only that, but it turns the curve into a closed loop. On the next page, I show this process, and then I show what it looks like with the interior of this closed loop filled with a solid color.


Below are two other generators that have a segment length of root2.



Look familiar? Given a specific generator shape, one set of flippings results in a right triangle while the other results in a dragon (I call this one the V2 Dragon).

Both of these fractal curves are partial gridfillers: Some of their vertices touch and some don't. This makes for some interesting internal patterning, as shown on the next page.


There are two more generators of this family that have a segment length of root2. Here's one of them:


On the next page it is shown enlarged at a higher level, and with rounded corners, so you can appreciate its meandering path.


Finally, I want to show you a member of the root4 square family, which I am especially proud of. I call it the "Dragon of Eve". It is named after Eve Peters, who was my mother's Art teacher in High School, and whose house I stayed at in my first semester in graduate school. I discovered it while living in her house.

The Dragon of Eve is a self-avoiding fractal curve. And I am very fond of it! Here it is:


Here it is enlarged at a higher level. This drawing uses a technique in which line thickness is proportional to line length. This curve reminds me a bit of the Great Wave of Kanagawa, by the Japanese artist Hokusai.


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

eyebrainbooks.com

FractalCurves.com