Genuary 2021 Revisiting the Penrose Tiles (Day 29)

 

I wasn't super satisfied with my Penrose Tile drawing from Day 29, so I modified it (perhaps imperceptibly), and had another go at it. It's better, but I think there's even more to be done.

Also, I wanted to describe my approach, which comes, in part from the discussion here: http://www.math.ubc.ca/~cass/courses/m308-02b/projects/schweber/penrose.html talking about beginning with a known good tiling and then subdividing from there.

I won't explain them here, (read the above linked page for more detail) but I'm using two different shapes of rhombuses, "tall" and "short". In the earlier version, my seed pattern was a "wheel" or flower shape made up of 10 "tall" rhombuses arrayed around a point at the center of the page.

I was dissatisfied with this for a couple of reasons:

1. It's not "generative" enough - it's some natural origin of some pattern, lining up with the center of the page. Bleh.
2. There's too much symmetry. Penrose tiles are supposed to be this magnificent non-periodic pattern, but sitting right at the center of the screen is a 5-fold rotational axis of symmetry. Bleh. Not just a rotational axis, but 5 different planes of mirror symmetry.

So, I went in to modify the generator in two ways to fight this to some degree:

1. My new generator wouldn't use a symmetric "wheel", it would generate all possible patterns of rhombuses coming together at a vertex, and choose one. Also, rotate it, which is easy.
2. The new generator should also offset the center of the page, relative to the center of the "wheel". This is also easy. Indeed, my earlier Genuary code was repurposed from some earlier code where I was making laser cut jigsaw puzzles, and that code had an offset. I deleted the offset to make porting my old code to my Genuary codebase easier. Putting it back is not a big deal.

This is what a wheel looks like. The blue triangles are from "Tall" rhombuses, and red are from "Short". I didn't make up the names. Note that there are edge labelings that force the triangles to alternate, to keep from falling into a degenerate, boring, tiling.

I am already happy at this point that my wheel is asymmetric, look at all that asymmetry. Mission accomplished.

Subdividing all of those triangles once.

I won't describe the subdivision here (read the linked page), but you can see that the tall triangles turn into a tall and a short triangle and the short triangles turn into a tall and two short triangles. And, whaaaat? We now have a symmetric red star inside a symmetric blue ring. There's asymmetry outside the blue ring, but we have to move a ways to get there.

Subdividing a second time, the symmetry hasn't gone away, nor will it. 

It's harder to be certain about the symmetry as the subdivisions increase, but you can feel its presence.

Ok, but if I shift it off the side of the page, it's not so bad, right?

Zooming in (and subdividing to maintain approximately the same level of detail). The point of symmetry is still staring at me from the top-middle of the page, like the eye of Sauron, in the middle of the red spot on Jupiter. Checkmate.

Let's see how I did, relative to my objectives:

1. More "generative"? Check, I guess? I like the fact that my seed "wheel" has more variety to it. And I'm rotating and centering in ways that mean that the next one I generate will be somewhat different from this drawing.
2. Less symmetry? Meh, let's give that a C. The rotational center is not at the center of the page, so that's good. I could actually modify the code to ensure that my recenter-and-zoom code moved the rotational center off the page, that wouldn't be too hard.

Another way to carry this forward would be to throw out the "wheel" seed altogether and just start with a rhombus and zoom in to that. Or even a triangle.

I really should give it another go - the zooming wouldn't be too hard, and I'm happy with the output in other respects.

One other thing that I should revisit for the next pass is my "edgepool" system - it's doing a pretty good job of accumulating paths and giving them back to me in a left-to-right draw order, which makes the page fill up in a sane order, mostly. This also does a good job of lessening (perhaps not optimizing) the pen up movements, which makes the drawing more efficient. So that's all good. But a few things I could do:

1. Right now, the paths are sorted based on the x-coordinate (or "abscissa") of vertex 0 on the path. I had considered checking to see if the last vertex is lower in x, and if so, flipping, which for this drawing wouldn't do anything, as all the drawn shapes are closed (unless you're clipping off the edge of the page, but that's not important right now). What I could do, instead, is to find the minimum x for all the points on the path, and use that to sort. This would avoid a little bit of "swiss cheese" effect where I'm filling the page with things and some things almost entirely to the right of some other things draw first. This would fill more evenly
2. Don't just sort on x, also sort on y. I don't think this would have a lot of effect, as floating point, it'll getcha 99.99% of the time. But if there were three paths with "the same" x value, maybe draw them top to bottom, rather than middle, top, bottom, or some other weird order.
3. A variant on the above would be to bucket the paths in x, and then within a bucket, sort in y. This would work around the floating point issue where things aren't ever going to match their x coordinates exactly. I could even just round the (x, y) pairs to integers, and it'd be pretty close to this.

I'm stepping away from this for a little bit to work on revisiting a different piece. Maybe I'll post about that. Maybe I'll post about coming back to this. Stay tuned.