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The Root 16 Square Grid 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



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 self-touching. On the next page are two specimens with dimension < 2 that are "trying" to fill a square, and a 2D specimen with 4-fold 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 upward-sloping 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




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