## Houndstooth eats itself: thousoondth, the houndstooth fractal

05.23.2015 § 1 Comment

I have created a fractal I call “thousoondth”. It is based on the houndstooth pattern.

Thousoondth applies two main effects to houndstooth:

- Horizontal & Vertical Line Recursion.
- The “cusps” — my term for the protruding triangles from the top right corners of houndsteeth — recur down their respective sides, shrinking by a factor of two each copy, into infinitesimality on their neighbor corners.
- What area is annexed to one houndstooth in this way must be annexed from another, and it is the “roots” — my term for the protruding trapezoids from the left and bottom sides — of the adjacent houndsteeth which get eaten into.
- However, these roots also recur in the same way as the cusps do, eating back into cusps. This alternation while nesting continues into infinitesimality.
- I call these jagged multi-cusp/roots affectionately “jaggies”.
- The ultimate result of this effect is the asymptotic elimination of horizontal and vertical lines from the pattern. They are all replaced with diagonal lines.

- Diagonal Line Recursion.
- Once all the lines in the pattern are converted into diagonals, a second effect is applied to obscure those, so that in the end thousoondth theoretically has no lines.
- This obscuring of diagonal lines is accomplished by choosing a strategic point along them and “flipping houndsteeth” — in other words, cutting one houndstooth-shaped piece out of one side (making it the color of the opposite side) and cutting one just like it out of the opposite site (so that it’s the same color as the first side).
- Both flipped houndsteeth touch the diagonal line, and thus look like reflections of each other, or like they’ve been flipped.

- The remaining spans of the diagonal line which do not touch these flipped houndsteeth then have the same effect applied to them.
- Three such spans exist:
- a large one on one side (1/2lf the original length),
- a small one on the other (1/4th the original length),
- and the smallest one is the area between the two flipped houndsteeth’s cusps (1/8th the original length).

- These spans are the result of placing the flipped houndsteeth so that their core squares touch 5/8ths of the way along the original line.
- The flipped houndsteeth were chosen to be the largest size possible without their recurred tendrils reaching into the territory that the nearest houndsteeth from other recursive paths would reach.
- This happens to be two orders smaller.

- Depending on whether the original diagonal line arose as part of a cusp or a root, the flipped houndsteeth are placed either 5/8ths of the way up from the bottom left or 5/8ths of the way down from the top right.
- Any other way leads to overlap between jaggies.

- Three such spans exist:
- This process recurs into infinitesimality.
- At each level of depth, one may find a “supertile” — my term for the minimum area of the pattern required to tile a plane with it (as distinguished from a solid or striped sub-tile or square of it which might otherwise easily be referred to as a “tile”), which is this case is 2×2 consisting of one white square, one black square, and one of each type of striped square — along the diagonal.
- This supertile is the “as above so below” unit, that is, it is the piece which looks the same as the entire image.

Here is an animation of infinite zooming in on supertile within supertile:

Here is an animation of the effect being applied depth layer by depth layer:

And you can play around with a live version of the fractal on my homepage here: douglasblumeyer.com/thousoondth.

Here is what thousoondth looks like with only the first effect applied:

And here’s what it looks like with only the second:

**Name**

“Thousoondth” is an anagram of “houndstooth”, which is fun because it shares* this naming property with another major houndstooth variant project of mine, “othoundsto“.

In fact, I first conceived of a houndstooth fractal many years before I conceived of my Fun Musical Idea “yaw” — which othoundsto was developed as an illustration of — and at that earlier time I was referring to my fractal as othoundsto. So when I picked this fractal idea back up, I realized I’d need to rename one or the other. Having already put the yaw version of othoundsto out into the world with that name, I preferred to rename my unpublished fractal project. Besides, having given up on it years ago, I figured giving the fractal project a fresh new name could help inject some fresh new energy toward completing it. Furthermore, the name I had come up for the fractal back then turned out to suit the yaw project better: it is a rotation of the word houndstooth, and yaw deals with rotation (that of musical pitch and duration); I could not imagine rotation coming into play with my fractal, certainly at least not at fundamentally as it features in yaw.

Fortunately, it took literally less than a minute for me to come up with “thousoondth”. I chose the name because of its similarity to the word “thousandth”: an accessible number evocative of recursive division, as in a fractal.

*Yes, technically, othoundsto is not an anagram of houndstooth since it collapses the initial and terminal ‘h’s into a single ‘h’ for purposes of naturalism.

**Development**

The repo for generating thousoondth can be found on my GitHub: https://github.com/DougBlumeyer/Thousoondth. Basically, it’s a bunch of fancy stacking of black and white triangles in HTML5 Canvas.

I would like to go into more detail about my design process. I certainly did not arrive at the final version of thousoondth instantly, even after breaking away from the years-ago style of fixating on a 3×3 supertile (one solid color in the center, enough to cover a single un-wrapped houndstooth). And it is fascinating how different of a perspective I had to take on houndstooth for thousoondth than I did to understand it for othoundsto, or even Dougstooth — the former in terms of continuities and alignments, the latter in terms of weaving lattices, and this one in terms of stacked triangles. For the record, thousoondth is much more arbitrary than Dougstooth, which was designed to be an extrapolation of the principles of houndstooth, whereas thousoondth is just good fun / whatever I could come up with that worked.

However, my notes as I tried to figure this thing out are basically incomprehensible at this point, and I am a little too lazy to find the narrative in my preliminary sketches (perhaps I will throw up a Gallery of Fails, whether aborted handdrawings or coding goofs). I can say that there were moments when breakthroughs involved returning to initial, abandoned instincts — instincts that were abandoned because sub-insights had not yet been had.

And the one big theme I did finally nix was that of the thousoondth shrinking into itself in one corner, with smaller houndsteeth fundamentally overlapping larger ones; I decided that a very important aesthetic concern of mine was preserving the uninvaded square. Though the shrinking nested houndsteeth did jive pretty well with the jaggies, and fostered a fascinating ambiguous wrapping element that is lost in this version.

What’s next? Well, I suppose I could make a thousoondth Dougstooth othoundsto, combining all three of my houndstooth variant projects into another dynamically spatial composition where the 3d analog of this fractal would map onto pitch and duration (and position as the third yaw dimension) and flow over you… It would seem at this point that I must be the world’s foremost expert on houndstooth, so I will hold off on this for the time being.

[…] thought I understood Houndstooth pretty well, having created its 3D analog, as well as a fractal based on it, as well as an experimental musical work based on […]