We’re Knot Friends – Math ∩ Programming

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its April is getting colder again.

For several summers in high school and undergrad, I was a day camp counselor. I have Previously written about how it helped me develop storytelling skillsBut recently I thought about it again because while I was cleaning out a closet full of old junk, I came across a bag of Embroidery thread. While stereotypically used to sew flowers into a pillowcase or write “home sweet home” on a hoop, in summer camps embroidery thread is used to make friendship bracelets.

For those who don’t know, a friendship bracelet is a simple form of macrame – meaning the design is constructed by tying knots, as opposed to weaving or braiding. Bracelet patterns are usually simple enough for an 8 or 9 year old to handle, albeit with some practice. They are believed to have originated among the indigenous peoples of the Americas, where knots were tied with string to keep track of time and counting, but in the United States their popularity arose among children as a symbol of gift-giving for friendship. As you say, when someone gives you a friendship bracelet, you put it on and make a wish, and you must leave it on until the bracelet falls off naturally, at which point your wish comes true.

Children took the “fall naturally” rule. is very Seriously, but in retrospect I find another aspect more fascinating. Tying friendship bracelets is a joint activity. It’s a repetitive task that you can’t do distractedly, it takes a few hours at least, and you have to stay still while you do it. But you can enjoy a common company, and in the end you made something beautiful. Children would sit in a circle, each working on their bracelet, sometimes even attaching them to each other’s backpacks in the circle of carts, chatting about whatever was on their minds. Children who were usually hyper and difficult for prison miraculously organized themselves into a peaceful and rhythmic focus. And it was nice to sit and communicate with them, when the work didn’t attract me for some reason.

Thinking about it makes me realize how little I have experienced community activities since then. It has the same feel of a family sitting together making Christmas cookies, or a group of artists sitting together painting. People complain about how hard it is to make friends in your thirties, and I wonder how much of that is simply because we don’t make time for such communal activities. We are not regularly around groups of people with the kind of free time that prompts these moments of idle connection.

Without any such thought at the time, I nevertheless developed friendship bracelet making as a specialty. I spent a lot of time teaching kids how to tie them. I’m not sure how I grew into the role. I suspect the artificial aspect of it tickled my mind, but at the time I wasn’t nearly as conscious of my love of art as I am now. I studied a dozen patterns and found a way to tie together a two-tone pattern of letters, with which I could write people’s names in a pixelated font. This impressed many teenagers.

Ten years later, this bag of yarn had managed to travel with me across the US through graduate school and many apartments, and I thought I might be able to find Math circle activity Including connections and patterns and…well, something mathematical. My attempt to make it work was a disaster, but not for the reason I thought it might be. It turns out that eight-year-olds don’t yet have enough skill to tie bracelets accurately or efficiently enough to start asking questions about possible knot patterns. Obviously I was still readjusting to the range of abilities typical of that age.

After that I figured, why don’t you try to make one again? Over the years, I have occasionally seen a pattern that was clearly not built using the techniques I was familiar with. To elaborate, I will have to briefly explain how to make a simple bracelet. Compared to other forms of fiber arts, it is quite simple, and does not require anything like a loom or knitting needles. Just the wire and something to hold the piece in place.

You begin by tying all your yarns together in a single knot at one end, glueing or pinning it to the tension, and laying out your yarns, then, using the leftmost yarn, and gradually moving it from left to right, you proceed to tie “stitches”, where a single “stitch” is made up From two manual knots of the left string on the right string. As a result of one stitch, the “leading” string (leftmost, in this case) produces the color shown above, and is “moved” to the right one position. Doing this with the same thread across all strings results in a (slightly diagonal) line of stitches of the same color. Once you complete a single row, the leading string is now at the far right, and you use the leftmost string as your new leading string.

A diagram showing how to tie a single stitch with two yarns. Considering the “red” stitch as the leading string, this “forward stitch” leaves the leading color up and moves the leading string one position to the right. Steps 3 and 5 tie the same hand knot twice, but it is the yarn placement from step 4 that shows the color in the final stitch.
A “stripe” pattern, showing the forward progress of the stitches. Image source

The stripe pattern is usually one of the first patterns to learn because it is very simple. But you can imagine that by tying strings in different orders and judiciously choosing which string is the “lead” string (ie, which color of string is shown in each stitch), you can create a variety of patterns. Some of them are pictured at the beginning of this article. However, the confusing patterns I saw could not be formed in this way, in part because, first, they were much more complex than could be constructed in the above style (obviously there is some limiting structure there). And second, they used more colors than the width of the bracelet, meaning that somehow new colored threads were switched in and out of some of the design. See, for example, these cow bracelets.

This cute cow couldn’t be made with a simple stitch pattern. Image source

Otherwise, having no experience with fiber arts, I was clueless and curious as to how this could be done. After some searching I found the so-called Alpha Bracelets, which opened the case wide.

Instead of using threads both as a structure to hold knots and as things that tie the knots, the alpha bracelet has threads that run along the length of the bracelet, and have no purpose other than to tie knots on them. By analogy with weaving (which I knew nothing about a few months ago), they notice warp and weft strings, whereas “classic” bracelets do not. And because we tie knots, the color of the “twisted” threads is never shown, except at the ends when they are tied.

To get more colors, there is a slightly complex process of “tying” a new thread, where the old leading thread is threaded between the two top knots of a new stitch and passes under the whole composition. Masha ties, a bracelet YouTuber, has perhaps the most popular tutorial on the internet on how to make alpha bracelets. But through this search I also discovered the site, which has a compendium of different templates. The diagrams on this site made one clear difference to me between “classic” and alpha bracelets: the stitches of classic bracelets are laid on a sheared lattice, while alpha bracelets are laid on a standard Euclidean grid. And you can easily create a markup outlining how to tie a pattern.

Classic bracelet diagram from braceletbook.
Part of an alpha pattern diagram from the bracelet book.

The alpha technique allows you to draw pixel art into your bracelet. And elaborate alpha patterns tend to be much larger than is practical to wear on your wrist. It actually becomes a kind of miniaturized macrame carpet.

So I wanted to try my hand at it. Since I’m now in my thirties and friendship isn’t what it used to be, I wasn’t sure what kind of bracelet to make. Luckily my toddler loves Miyazaki movies, so I made him this without a face bracelet.

It’s a little rough around the edges, but not bad for my first. And the toddler doesn’t care. He’s just happy to have a faceless friend. I then started a new pattern, which is currently about 80% done. Continuing the Japanese theme, this is a Hokusai look big wave.

If you look closely you can see a few places where I messed up, the worst being the bottom right side where I overtightened a few stitches at the edge causing the edge to skew. Because it was so big I attached the end to a small dowel, making it look like a scroll.

Again, since Alpha Bracelets are knotty pixel tapestries, I figured why not put them on my wall and make a tiny gallery. And there are always a handful of contemporary artists whose art I admire, but their prices are too high, or their best works have been sold, and they don’t make prints. So I will never get to put on my wall. take for example, Kelly Rimtsen, is known for dramatically posed women in colorful dresses from the 1950s with power tools. I emailed her years ago asking about prints and she replied, “I don’t do prints.” Today she probably does, but it’s still very hard to find prints of her good works.

The first time I saw one of her pieces (at a restaurant on Newberry Street in Boston), it really shocked me. But as I saved enough money to afford what her art once cost, so she gained enough fame to keep her prices impractical all the time. I even tried to draw my own imitation of one of her paintings, though it’s not that good.

So instead I decided to convert one of her creations into pixel art, and tie a friendship bracelet tapestry myself. Here is my pixel art in progress. It still needs some cleaning, and I’m not sure how to get exactly the right colors of thread, but I’m working on it.

My advanced pixel version of one of Rimtzen’s creations, which I mean as a friendship bracelet

In my life, this craft has moved quite far from community bonding and gifts. But it still scratches a certain itch to work with my hands, and the slow, steady progress toward building something that isn’t hindered by anything outside of your own effort. Plus, each stitch only takes a few seconds to tie, and unlike woodworking or knitting, there’s no set up/hang up/take down time. You just place the wires. Having an ongoing project at my desk gives me something quick to do when my plans are being made, or when I’m in a listen-only meeting. Instead of opening a social media site for an empty dopamine hit, or getting upset about someone else’s bad takes, or playing a game of ball chess, I can do 1/500th of something that beautifies my life.


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