Geez, These Links are Complex

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.

4 by 4

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.

link numbersI 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.

disregarded break

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. 2017-11-12 18.11.38

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?!

link 1

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. breaks connecting

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.

link net

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.


UPDATE (11/13/17)

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.

#mtbos Book Club: Weapons of Math Destruction by Cathy O’Neil


It’s today! If you’re in Minneapolis, please join us at Urban Growler Brewery at 2 pm. If you’re not, we’ll start at twitter chat at 4 pm CST using the hashtag #WMathD and #mtbos.

A HUGE thank you to Marian Dingle who graciously offered to help me formulate questions. If there’s anything good in here, that’s Marian’s doing. If there are things that are terrible and dreadful, those are mine.

In the hopes of helping people contribute best, here are the questions that will go out on Twitter starting at 4 pm.

My greatest hope is that you all take the conversation in whatever direction you like best. To that end, if you would like to ignore the questions and discuss other things, you go right ahead. These questions are simply to help stoke the conversation if we should need it.

Book Club Questions!

Q1: In reading the book, I often reflected on how WMDs have affected me in the past, and my relative awareness or lack of awareness about them. How did you react to learning about these WMDs? Which hit closest to home for you?

Q2: O’Neil had been an enthusiastic player in Wall Street until the crash happened in 2008, at which point she had a change of heart and began examining the mathematical structures that led to the recession. What are your reactions to that transformation? Is it a transformation we believe can be duplicated? Why or why not?

Q3: Many of the WMDs O’Neil outlines disproportionally affect already disadvantaged populations. If one’s goal is to do the work of anti-racism, how can we approach the proliferation of WMDs and their disproportionate impact? Would WMDs look differently if there were more quants of color? Which, if any, of these WMDs deserve wider attention? Are there WMDs we believe are more impactful than others?

Q4: The proliferation of WMDs seems to stem from our innate attraction to numeric rankings. Often, an “anti” WMD would require significant investment of time, money, and human capital to evaluate whatever metric (teacher performance, expected criminal recidivism, credit history etc.) is currently being assessed by a WMD. Is there a scale on which we believe we can replace the efficiency of WMDs with the more expensive alternative? Are algorithms our only answer to tackling big data? 

Q5: In nearly every chapter, O’Neill outlines the purported reasonableness of each WMD before raising her own objections to it.  Were you surprised by any of O’Neill’s objections? Were there any you feel were unfair? Any you feel did not go far enough?

Q6: How can we help prepare students to be proactive about how WMDs will affect them? What can we do to empower them to dismantle them? Is that even possible?




Knots, Links, & Learning

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. Rainbow Links.jpg

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.


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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.

2017-10-21 09.32.31

  • 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.

2017-10-19 21.50.20

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


Black Stats Chat summary & Thoughts

This will be quick. We had our first #mtbos book club on Black Stats a couple weeks ago and I just wanted to wrap that up with a few thoughts. I attempted a storify of our chat, and I encourage you to click on and browse that. For the tl;dr crowd though, here is my summary in several tweets, followed by two things I’ve been thinking about since the chat.

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Book Club: Black Stats by Monique W. Morris

Welcome to the first of the #mtbos Social-Justice/Racial-Equity/Let’s-All-Read-a-Darn-Book-Club posts!

Be sure to join or pop into the twitter chat THURSDAY, OCTOBER 5TH 7PM CST, using #blackstats


Seriously, we want you there. Even if you didn’t finish the book. Join!!! (Also, as I have never hosted one of these things, if you know how to do that or can give me some pointers… please do. I have no idea what I’m doing….but I’m excited nonetheless.)

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I’m mostly  housing this here to record them for myself, but if it’s of any help to any of you then all the better! There’s probably a fair amount of overlap – I’m not vetting everything as I put it here, but if I don’t get the links together, I’ll almost definitely not get around to sifting through them anyhow.  Please add stuff in the comments! 

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