Brainfilling Curves: The Root 8 Family

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The Root 8 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

Next we come to the root8 family. This family has some things in common with the root2 family: the generator lies on a 45 degree diagonal. Also, it can be seen as a superset of the root2 family (as well as the root4 square grid family). This is because the values 2, 4, and 8 are powers-of-two numbers (2¹ = 2; 2² = 4; 2³ = 8). We will see later that the root16 square grid family is a superset of the root2, root4 square grid, and root8 families.

We've already seen one member of the root8 family: I showed it to you at the beginning of the book as an example of how the turtle can use flippings to convert an otherwise self-crossing curve into a self-avoiding curve:

When I showed this curve to you before, it was rotated 45 degrees. Here is it shown in its native familial orientation. Notice that the generator has only 5 segments, and that three of those segments have a length of root2. These segments are responsible for the three large lobes in the 2nd teragon.

As a general rule, you can consider segments of length root2 to count as two (remember that we square the lengths when calculating fractal dimension). So in this case, two one-length segments and three segments of root2 - when squared - add up to 8: the family number.

The root8 family is quite versatile. First let's look at some curves that fill a right triangle. Two of them are shown below. Like the last curve I showed you, the generators for these curves each have 5 segments, three with a length of root2.

This last one is so interesting I decided to render it in color. It's shown on the next page.

Here's another triangle-shaped specimen. This one has even fewer segments: 4! One of the segments is length root2 and the last segment is length 2. Following the rule of squaring all lengths, as I said earlier, you can see how the sum is 8. Because of the long segment length, the result has a great variety of lengths within - with a lot of self-similar patterning.

I'm particularly fond of this one. So I made a color rendering of it, and rotated it 90 degrees. It is shown below.

Here are two more right triangles of the root8 family. These are a little less well-behaved, but interesting nonetheless.

This next specimen is a natural self-avoider. It is a relative of a root4 specimen we met earlier. On the next page I show five more curves of this family.

That last curve fractalizes into a shape that is similar to the "Twin Dragon": the result of joining two HH Dragons. But notice that it is not quite the same as the Twin Dragon; it has pinched-off babies - which each have their own pinched-off babies.

This next curve fractalizes into a pair of Twin-Dragon-like curves. Because of the similarity to the shape as the Twin Dragon, I call it the "Twin-Twin Dragon". Are the two twins holding hands? No; their babies are holding baby hands.

Speaking of dragons, the root8 family produces more dragons that are related to the HH Dragon. Here's one:

Let's see it rendered at a higher level, below. To the right is the HH Dragon.

Here's another specimen that is related to the HH Dragon:

I am impressed with how this curve is so unpredictable and irregular in its internal meandering, yet it is able to avoid any self-crossings (it does self-touch on vertices: those are separated due to the rounded-corners scheme of the drawing). Here is a rendering of two copies of this curve (one flipped 180 degrees). They are combined to make the shape of the twin dragon - which closes the loop, enabling it to be filled internally with color.

Here's another specimen that resolves to the same shape as the HH Dragon. Since the generator has shorter segments in the middle region, the mid-sections of its tergaons are rather knotted, and full of detail. This specimen's teragons are precariously self-touching (even with rounded-corners). I have rendered it below, rotated 90 degrees.

Another root8 Dragon is shown below. This one also resolves to the shape of the HH Dragon.

Here it is fractalized at a higher level, splined, colorized, and slightly rotated...for your brain-filling pleasure.

Here is a self-avoiding dragon that I was excited to discover. It appears to be a relative of the Dragon of Eve. It's like adding a smaller triangular bump onto the big triangular bump of the Dragon of Eve. It's like a curly Dragon of Eve!