Many of the drawings I do are the ones that simply seem to me as though they’ll be interesting. Sometimes they meet my expectations, sometimes they don’t, and occasionally they wildly exceed them. This drawing is of the latter quality.
I’m not actually that pleased with what it looks like. It’s not particularly pretty. I also spaced out and mixed up the purple and red in the bottom right image. But the mathematical results are really neat. Here’s what I was hoping to do with it. These are what I would call 4×4 grids, each that has exactly 1 break in it. For reference, here’s what a 4×4 looks like with no breaks.
Up until now, I’ve been thinking of this as 4 distinct links. In my head, I have labeled them 1, 2, 3 and 4 starting at the top left of the image, and moving right along the top. Thus, in this picture, dark green is 1, light green is 2, pink is 3 and purple is 4.
I now think this might be incorrect, and I should instead think of it in symmetric terms where 1 and 4 are the same link with a plurality of two, and links 2 and 3 are the same, also with a plurality of 2. Allow me to explain why.
In planning the image at the top of this post, I wanted to try putting a single break in all of the unique places a single break might exist in a 4 by 4 grid. Those are shown below. Each yellow line is a unique breaking point, with the white lines representing the duplicate breaking points that could be obtained by reflections or rotations. (Note that all the vertical breaks can become horizontal if you just rotate.)
I also disregarded the any breaks on the farther outside spaces because the resulting link would have to have a section traveling vertically rather than at the usual 45 degree angle. Maybe I should have considerd these, but they seem to be an expansion beyond the type of link I wanted to consider for now.
Okay. So, I started drawing. Part of what I like about the length of time these drawings take is that I have built in time to ponder what’s happening. I decided on the fly to color in all of the links affected by the break first. I wasn’t particularly surprised to find that each break had essentially connected two of the four links.
In fact, I thought, “That makes perfect sense!” Because if the break connected 2 links, and if I had, in fact, found all the unique breaks, I should have 4 choose 2 different images! Brilliant. So I should have a graph that connects links 1 & 2, 1 & 3, 1 & 4, 2 & 3, 2 & 4, and 3 & 4! I dutifully started to identify them aaaand… well, crap. For one thing, link 1 appears FIVE times. What the hell?!
I mean… that’s ridiculous. Link 2 appears twice, Link 3 appears three times, and Link 4 appears only twice. So are some links… more…. important (?) that others?
For that matter, there are TWO images that connect Links 1 and 4. Yet there’s only one image that connects links 2 and 3. Upon inspection, I can see why this is. There are two totally different breaks in the 1 & 4 images. One of the breaks (bottom row middle) connects to the center opening, and one of the breaks (bottom row right) doesn’t. Clearly, those are different. And the image connecting links 2 and 3 only has one unique type of link because I had disregarded the second type since it would give me a weird looking link.
Had I not abandoned the links on the furthest outside, I would have gotten two more images that would connect links 1 & 2, and links 2 & 3. The images I currently have shown are in grey below. The missing images are shown in green.
I think I can resolve some of this by no longer naming the links 1, 2, 3 & 4, but rather naming them in symmetric pairs, but it’s getting late and I should finish up some school work before tomorrow.
In the meantime, here’s a time lapse for your viewing pleasure.
If you would like to join in on the fun, please add thoughts or comments below. Please do not join in if you happen to have already solved this problem and just want to show off or give out answers.
I have created the ghastly disregarded links in the name of mathematical inquiry. I feel justified in abandoning them as horrible, but the data may be useful, so here they are.
2 thoughts on “Geez, These Links are Complex”
I am utterly delighted by how beautiful your work is. I love the breaks, the plurality, pondering and the play! Just know that we are fascinated and engaged, even if we don’t have the mental capacity at the moment to join in, except to appreciate.
Thanks for the interesting investigations into one of my favorite subjects, knots. Since I am not good, or trained, in math what you say has been particularly fascinating to me for your perspective on the subject. Brava!
Are you familiar with Paulus Gerdes? I happened to find this just yesterday: https://www.beloit.edu/computerscience/assets/Sona_2__Ch._3___Appendices_1_2.pdf