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The Root 12 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



It is time to show you another overweight palindrome dragon. Like many of the others we've seen, this member of the root12 family has a typical yam-like shape: fat and lumpy in the middle; tapered at the ends.


Given the many ways you can traverse a triangular grid in the interval of root12, we should expect many more palindrome dragons. I'll show you four more. These are not at all yam-like. In fact, their boundaries are quite craggy.





Here is a root12 specimen that includes an abnormally long segment - extending the length of root7, followed by three segments of length 1. Who would have ever expected that its 5th teragon would resemble a bat's cave?



Here are two interesting specimens. The first one is based on a tiling of 12 triangles - illustrated at bottom-left. The second one is shown with two kinds of 3-way pertilings - shown at bottom-right.




Here are a couple of curves based on a common generator - it includes two segments of length root3:




Ancestry
We have met several families of fractal curves, and we have seen a number of ways in which they relate to each other. In the graph below, I show examples from all families up to root12. The prime-numbered families are placed along the top. The powers-of-two families (2, 4, 8) are shown at left. Notice that variations of the HH Dragon can be generated within each of the power-of-two families. Similarly, any root3 curve can be generated within in the root9 family, since 9 is a multiple of 3. The root10 family can generate variations of both the root2 and root5 family curves. And finally, the root12 family can generate variations of root3 and root4 (triangle grid) curves.


End of chapter.



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
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