Brainfilling Curves: The Root 16 Square Grid Family

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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.