Brainfilling Curves: Root 17 And Beyond

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Root 17...and Beyond
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 number of plane-filling specimens in the root2 family is 2 (I am not counting Cesaro). The number of plane-filling specimens in the root3 family (by my estimate) is 10. As we climb the family tree, the number of specimens increases non-linearly, making exhaustive search quite an expensive proposition when many segments are involved. Consider the sheer number of possible generators if each segment has four possible flippings, and connects any two grid points, where its length is less than the interval length. I calculated the number of possible generators having up to 5 segments. Ready?

2 seg:
3 seg:
4 seg (triangle):
4 seg (square):
5 seg:

16
13,824
331,776
1,048,576
254,803,968

I have done some exploration of this huge space of possibilities, using both hand-drawn diagrams, and computer search algorithms, and have discovered some interesting specimens. Let's look at some of those now, as well as a few gems previously introduced by Mandelbrot and others.

First, let's start with a plane-filling curve that Mandelbrot introduced in his book. It is a member of the root17 square grid family, and it is a partial gridfiller. Mandelbrot pointed out that self-avoiding curves are not only more aesthetic; they also make better models of forms found in nature (such as rivers and watershed trees). When a curve is vertex self-touching, it "knots off". In this example below, the generator itself has a pinched point. This percolates throughout the teragons, in a similar way that I showed you with the self-crossing 7-dragons. Here is Mandelbrot's Knotted-off root17 curve:

Here is another root17 curve. It is a self-avoider, and it has a dimension of ~1.81.

The root18 Square Grid Family
I have one specimen to show of this family. It is a gridfiller. It is shown at right at level 3, rotated by 45 degrees. Personally, I find nothing attractive about this specimen. It looks like a poorly-made gingerbread man. But I am including it in the book...even boring specimens deserve to be seen.

The root19 Triangle Grid Family
Let's look at a few of the fine specimens of the root19 triangle grid family. I like to start with dragons, and so here is a palindrome dragon with a pinched waist and pinched extremities. It is shown below at level 3 with rounded corners.

Close-packing of seven circles results in a roughly-hexagonal shape: the basis for the Gosper Curve. If you completely surround these seven circles with a new layer of circles, the total number of circles is increased to 19, as shown at left. This 19-cell hexagonal grid is the basis for another one of the wonderful generalized Gosper curves discovered by Fukuda, et. al [4]. It is shown below.

I discovered a curve based on this 19-cell hexagonal theme. Its dimension is less than 2, and so it has lots of open spaces, which gives it some artistic breathing room. I call it "Mandala". It is shown in color on the next page.

Remember the Anti-Gosper? Well, I suspect that the following curve has some things in common: it is roughly triangular and more tightly-packed than its hexagonal cousin.

The root20 Square Grid Family
There are five fours in twenty. And so I wondered if I could build a space-filling curve by replacing the squares in Mandelbrot's Quartet with rotated copies of a root4 square grid family generator. Well, the Peano Sweep generator appears to do the trick:

Here is a variation that uses a combination of the Peano Sweep generator and another four-segment shape that fractalizes to a square. It is a shape that would not normally stand on its own as a plane-filling curve generator. But in the context of the whole arrangement, it works perfectly.

The following generator has a knotted-off square, like the root17 specimen we saw earlier.

This curve is rather curious. It is a gridfiller with a bit of asymmetry.

The root21 Triangle Grid Family
Of the root21 triangle grid family, I have found a few fat palindrome dragons. Two of them are shown below.

Since 21 is a triangular number, we can arrange the pertiles in a roughly triangular array. Below is a triangular specimen of this family, followed by a close cousin with dimension ~1.89. On the next page is a color rendering with rounded corners.

The root25 Square Grid Family
Now we come to another square number: 25. Of the root25 square grid family, I have only one specimen to show: the Quadratic Gosper Curve. It is attributed to F.M Dekking [5], and also Doug McKenna [19]. This is a self-avoiding curve that fills a square.

The root25 Triangle Grid Family
The root25 triangle grid family has a whale of a bumpy dragon, shown below.

On the next page are three more specimens of this family. The third one is shown on the following page enlarged at level 3 with rounded corners.

The root27 Triangle Grid Family
I want to show you two related specimens of the root27 triangle grid family. They were introduced in Mandelbrot's book, and they are variations of the Snowflake Sweep. Now, as we saw earlier, the snowflake sweep is a member of the root9 family, so why are these specimens root27? The reason is due to the nature of my taxonomy scheme, which requires that all generator segments lie between grid points. With a slight variation to my scheme, these curves could be represented in the root9 family, by relaxing this requirement. The smallest segments occurring within the interior of the generator would fall between the cracks, as it were. We can say that these specimens are closely related to the root9 family, except that they have a few small (and clever) genetic mutations that permit extra details to emerge in the crevices. The first specimen is one Mandelbrot called "Monkeys Tree". The second is a variant of the Snowflake Sweep, in which the whole generator is transformed and takes the place of the fifth segment. This is rendered stylistically on the next page.

This specimen was rendered with loving curves and printed on the hardback cover of one of the editions of Mandelbrot's book. It is reproduced at left.

The two final specimens I want to show you are of the root29 triangle grid and root36 square grid families. These both have dimensions less than 2, but they are self-avoiding, and so they do not require rounded corners. Their diagrams are shown below, and on the next two pages are color renderings. The second one is my attempt at filling the Koch Snowflake. Enjoy the last two specimens of this book!