Brainfilling Curves: The Root 5 Family

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The Root 5 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 root5 family. We have already met the 5-dragon:

Remember how I flipped the x values of all the segments of the Ter-Dragon to make the Inverted Ter-Dragon? Well, the same can be done with the 5-Dragon. And, just like the inverted Ter, the inverted 5-Dragon has a pinched waist.

This next specimen is Mandelbrot's Quartet: "Each 'player', and the table between them, pertile." [16]. He claims to have "designed" it, although one could debate that such a curve is "discovered" rather than "designed". In either case, it is one of the finest self-avoiders.

I discovered a variation of this generator, created by reversing the x-flipping of each segment. I call it "Inner-flip Quartet".

Next I will show you six variations on a single generator shape. Two examples are shown on the next page. The first example has an interesting property: due to the flippings, the orientations of copies of the generator do not correspond with a continuous square grid. You can see this in the mixture of 90 and 45-degree angles in the level 2 teragon. I would not have expected a curve like this to survive the fractal test. There is indeed self-contacting in several vertices, but other than that, it is rather well-behaved, as indicated by rendering with rounded corners.

A close relative of this curve is shown here.

These two specimens resolve to the same general shape, as indicated by the illustration below.

Given the same generator, with alternate flippings, we get two gridfillers with very craggy boundaries:

With other changes in flippings, we get the following gridfillers:

In that last one, notice how the conifer tree-like spike at the upper-right corresponds to the empty gap at the bottom, rotated by 90 degrees. My brain is pertiling!

Here are two plane-filling curves of the root5 family that use a common generator shape.

That last curve can be combined with a 180-degree flipped copy of itself to make the shape of the 5-Dragon...

This next curve is a self-avoider. It is followed by a similar specimen.

Each of these last two curves can be copied four times - each copy rotated 90 degrees - and joined together to make a continuous curve. The overall shape is a replica of the Quartet (one of them is a mirror-image of the other). This appears to be a property of many root5 curves I have shown.