For those of you who follow me on twitter or instagram, you’ll be familiar with the fact that I’ve been drawing lots of these pictures, recently.

To me, mathart is all about getting surprising results. I make a lot of mathy art, and while I totally appreciate other types of art, I like math art because it helps me make sense of something. Here’s the first knotty (not technically a mathematical knot, apparently, but rather links) image that I created.

At the time, I don’t think I thought a whole lot about the math behind it, but it hung around in the back of my head for a while, and I eventually decided to do a few more drawings.

On the left, you’ll see a 5×5 grid, with 5 distinct closed links (2 pink, 2 blueish, 1 brown). The middle has 3 links, and is a 3×9 grid. The right is a 6×8 grid with 2 distinct loops. Seems likely that greatest common factor of the grid is going to be the number of loops. But I have no idea how to prove that. I just have anecdotal evidence.

A bit overwhelmed by the actual proof of the thing, I veered off to the side for a while to investigate how holes affect the number of loops, too.

The above investigation was super fascinating to me as I saw the number of loops forming and the shapes of the links changing as the holes got in their way. Except I apparently have no idea how to *count* the holes, because I’m getting weird nonsense answers.

One of the reasons I take time-lapse videos of these is that I think it helps other people join in. Looking at the final image, you miss a lot of the chances to think through what goes through my head while making them. It was surprising to me to have the blue loop jump from a rectangular shape to a jazz-hands-out-to-the-side shape in the slideshow above – it would take a lot of looking at the final image to see that. I like that it just pops out when I do the time-lapse.

Time-lapse is also just very satisfying to watch at the end of two hours of drawing.

I occasionally show these videos to my students, and by far the most common question I get is, “Wait, how long did that take?” followed by general incredulity that I would spend that long just drawing something. When I tell non-math-teacher-adults about this, the question is often, “What class is this for?” followed by a narrowing of eyes and shaking of head when I say I’m just doing it for myself. Both groups, however, are still pretty entranced by the time-lapse.

Back to the question of how many links based on the nxm grid.

There’s a lot happening here, and I don’t yet have conclusions, but I do have a LOT of notice and wonders.

- This is a study of twelves. The 12×2 has 2 loops, 12×3 has 3 loops, 12×4 has 4 loops and 12×6 has 6 links. See a pattern? These all agree about the GCF.
- Look at the dark purple color throughout. Notice how in the top image, it bounces from top to bottom and back 3 times? In the 12×3 it does so twice. In the 12×4 it does 1.5 times, and then just one full time in the 12×6. There’s gotta be something there.
- I’ve noticed that if you consider the perimeter of the grids, that each link has to consume an even number of pointy bits.
- My boyfriend, Joe, has to put up with more of these discussions that he’d perhaps like, but I’ve cajoled him into thinking through a lot of these. He asked the reasonable question of whether each link is equal in length, which is worth investigating. In the 12×6, each link has 21 distinct parts, so… yes? But each distinct “part” is not equal. If you look at the corners, those are lengthier “parts” compared to the inside rectangular pieces. I’m not totally sure how to calculate the exact length of the pieces without getting deep into calculus. I do wonder if it’s possible to make an argument about equal lengths based on structure rather than calculation, though.
- I also wonder if it’s possible to mathematically predict the shapes of the links. The bright green link is a different shape than the tealy blue link. And the lilac colored link in the 12×3 is fascinatingly symmetric. Hm…

If any of this is compelling to you, I would love it if you wanted to join me in this investigation. I don’t want someone to show up and say, “here’s the final solution, dummy, it’s already all been solved,” because I’m sure that this is already a chapter in a textbook somewhere. I don’t care about that. I care about investigating for myself and with friends what’s happening. Feel free to share ideas below, and I would love it if you did some of your own drawings and joined in on the party.

Here’s a rough attempt at explaining how to draw these: