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 
One way to express a square number S is to draw a square made up of S smaller squares. I've shown you fractal curves of the root4 and root9 families where square regions are filled. Now we come to the next square number: 16. Below is a curve of dimension 2 that is a partial gridfiller, and it is also partially edge selftouching. On the next page are two specimens with dimension < 2 that are "trying" to fill a square, and a 2D specimen with 4fold symmetry. 
The following two generators of the root16 square grid family have only three segments. They do not correspond to hexagonal or triangular grid lines, even though they may appear to upon first glance  they are indeed inhabitants of the square grid. In both curves, the upwardsloping segments traverse across the diagonal of two squares, thus having lengths of root5. The horizontal segments have lengths of 2. 
This curve can be copied four times and placed on a square initiator to create the closed curve shown at right. There is a vague resemblance to the structure of bronchial tubes.

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 