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 planefilling 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 planefilling 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 planefilling 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 
Brainfilling Curves  A Fractal Bestiary
by Jeffrey Ventrella Distributed by Lulu.com Cover Design by Jeffrey Ventrella 
Book web site:
BrainFillingCurves.com
ISBN 9780983054627 Copyright © 2012 by Jeffrey Ventrella 
eyebrainbooks.com 
FractalCurves.com 