Brainfilling Curves: The Root 9 Square Grid Family

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The Root 9 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

The root9 square grid family has many interesting plane-filling curves. The square root of 9 is of course 3. And, being an integer, my scheme places the interval length horizontal, stretching across three grid cells. We already encountered one member of this family early on when I showed you the Koch curve: its friend, the Square Koch. Here is its diagram:

Now let's add a floor and a basement to this generator and see what happens. Lo and behold, we get one of the most familiar - and oldest - plane-filling curves of all time: the "Original Peano Curve" (as Mandelbrot called it: Giuseppe Peano explored many variations). Rounded corners help a lot when viewing this curve.

Here is a variation that fills the same area as the original Peano curve, but the shape it fills is a right triangle:

And here is a familiar theme once again: a generator shape can be made to create either a right triangle or a dragon, by way of alternate flippings. This variation is like a dragon (okay, maybe it's not like a dragon... let's just go with "jaggy").

Where there be dragons...there be castles. The root9 Castle below is filled with holes...holes of all sizes. It is a holey castle. No surprise: its fractal dimension is only ~1.77.

The Holey Castle is related to a large class of fractals that are riddled with holes, such as the Sierpinski Triangle and the Sierpinski Carpet (and their 3D counterparts: the Tetrix and the Menger Sponge):

Remember the variation of Cesaro's Sweep I showed you from the root4 family? It has a double-sided vertical needle. Well, I wondered if there might be something similar in the root9 square grid family ...and I came up with the generator below. Like Cesaro's Sweep, this curve is everywhere edge-self-touching except for the bottom edge. But unlike Cesaro's Sweep, it has a wonderful fractal boundary. Its 4th teragon is shown below at right. Rounded corners help only slightly to reveal the curve's trajectory.

Below is an intriguing fractal curve of dimension ~1.77.

Here are two gridfillers based on a common generator:

The sepcimen below has a generator with a 2-length segment corresponding to a 2x2 square (shown in purple at right). It has a lot of self-touching edges, and so I used the low-pass smoothing filter to render level 5 with filled-in areas to show the interesting self-similarity.