Read it online
at the Internet Archive

Buy the book
Two hundred pages of color images
in 8.5"x8.5" paperback print
No Root 6!
  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



There are no root6 plane-filling curves within my scheme! Why? Well, I can tell you this: it has something to do with the grids. Here's that illustration I showed you earlier:


An uninspiring answer to the question (why no root6?) is that there are no grid points on either a square or triangular grid whose distance to the origin is root6. I could just leave it at that, and say "let's move on", but I seek a deeper answer. Notice also that there is no square root of 11 distance either. There is also no square root of 14 distance. The list continues in a way that is reminiscent of the erratic series of prime numbers.

In the case of the square grid, the answer is simple: each of these distances is the sum of two squares. Look no further than the Pythagorean theorem to see why this is so. But when considering the triangle grid, it is a little less obvious why we should end up with the set: root1, root3, root4, root7, root12, root13, etc. I shall leave this question open for you to explore on your own. Also, did you notice? ...there are two root13 distances - one on the square grid and one on the triangular grid. What's up with that?

Now, I must admit: earlier I claimed that all plane-filling curves have interval lengths that fall between grid points of either the square or triangular grids. But I cannot say for sure that this is true. It may be that any number of generators with arbitrary interval lengths can yield plane-filling curves (although they may evade simple mathematical analysis). I leave it up to you, dear reader/viewer/thinker, to give me an inspiring answer. You can always find me at Jeffrey@Ventrella.com. Okay, I'm afraid I am going to have to say, "let's move on now"...to the awesome root7 family.

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




Brain-filling Curves - A Fractal Bestiary
by Jeffrey Ventrella

Distributed by Lulu.com
Cover Design by Jeffrey Ventrella
Book web site: BrainFillingCurves.com

ISBN 978-0-9830546-2-7
Copyright 2012 by Jeffrey Ventrella

eyebrainbooks.com

FractalCurves.com