Brainfilling Curves: The Root 13 Square Grid Family

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The Root 13 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 it is time to look at the root13 square grid family. I will start with a curve I discovered that is a self-avoider analogous to the Gosper curve: it requires two kinds of flippings (normal and double-flipped), and it roughly corresponds to a regular polygonal tiling - in this case, the square. Take a square of 9 squares, and attach one square on each edge. That is the tiling for this fractal curve. Since Mandelbrot referred to space-filling curves as "Peano Curves", and also referred to the Gosper Curve as "Peano-Gosper", I call this one the "13-Peano-Gosper".

There are several variations of this curve with dimension less than 2. I'll show you two on the next page, followed by a filled-in version of the second one.