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6 Math Concepts Explained by Knitting and Crochet

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This crocheted Lorenz manifold gives insight "into how chaos arises." Image credit: © Hinke Osinga and Bernd Krauskopf, 2004

 
Using yarn and two pointy needles (knitting) or one narrow hook (crochet), pretty much anyone can stitch up a piece of fabric. Or, you can take the whole yarncraft thing light-years further to illustrate a slew of mathematical principles.

In the last several years, there’s been a lot of interesting discussion around the calming effects of needlecraft. But back in 1966, Richard Feynman, in a talk he gave to the National Science Teachers’ Association, remarked on the suitability of knitting for explaining math:

I listened to a conversation between two girls, and one was explaining that if you want to make a straight line…you go over a certain number to the right for each row you go up, that is, if you go over each time the same amount when you go up a row, you make a straight line. A deep principle of analytic geometry!

Both mathematicians and yarn enthusiasts have been following Feynman’s (accidental) lead ever since, using needlecraft to demonstrate everything from torus inversions to Brunnian links to binary systems. There’s even an annual conference devoted to math and art, with an accompanying needlecraft-inclusive exhibit. Below are six mathematical ideas that show knitting and crochet in their best light—and vice versa.

1. HYPERBOLIC PLANE

Courtesy of Daina Taimina

 
A hyperbolic plane is a surface that has a constant negative curvature—think lettuce leaf, or one of those gelatinous wood ear mushrooms you find floating in your cup of hot and sour soup. For years, math professors attempting to help students visualize its ruffled properties taped together paper models … which promptly fell apart. In the late ‘90s, Cornell math professor Daina Taimina came up with a better way: crochet, which provided a model that was durable enough to be handled. There’s no analytic formula for a hyperbolic plane, but Taimina and her husband, David Henderson, also a math professor at Cornell, worked out an algorithm for it: if 1^x = 1 (a plane with zero curvature, made by crocheting with no increase in stitches), then (3/2)^x means increasing every other stitch to get a tightly crenellated plane.

2. LORENZ MANIFOLD

© Hinke Osinga and Bernd Krauskopf, 2004

 
In 2004, inspired by Taimina and Henderson’s work with hyperbolic planes, Hinke Osinga and Bernd Krauskopf, both of whom were math professors at the University of Bristol in the UK at the time, used crochet to illustrate the twisted-ribbon structure of the Lorenz manifold. This is a complicated surface that arises from the equations in a paper about chaotic weather systems, published in 1963, by meteorologist Edward Lorenz and widely considered to be the start of chaos theory. Osinga and Krauskopf’s original 25,510-stitch model of a Lorenz manifold gives insight, they write, “into how chaos arises and is organised in systems as diverse as chemical reactions, biological networks and even your kitchen blender.”

3. CYCLIC GROUPS

You can knit a tube with knitting needles. Or you can knit a tube with a little handheld device called a Knitting Nancy. This doohickey looks something like a wooden spool with a hole drilled through its center, with some pegs stuck in the top of it. When Ken Levasseur, chair of the math department at the University of Massachusetts Lowell, wanted to demonstrate the patterns that could emerge in a cyclic group—that is, a system of movement that’s generated by one element, then follows a prescribed path back to the starting point and repeats—he hit on the idea of using a computer-generated Knitting Nancy, with varying numbers of pegs. “Most people seem to agree that the patterns look nice,” says Levasseur. But the patterns also illustrate applications of cyclic groups that are used, for example, in the RSA encryption system that forms the basis of much online security.

4. MULTIPLICATION

Courtesy of Pat Ashforth and Steve Plummer

 
There’s a lot of discussion about elementary students who struggle with basic math concepts. There are very few truly imaginative solutions for how to engage these kids. The afghans knit by now-retired British math teachers Pat Ashforth and Steve Plummer, and the curricula [PDF] they developed around them over several decades, are a significant exception. Even for the “simple” function of multiplication, they found that making a large, knitted chart using colors rather than numerals could help certain students instantaneously visualize ideas that had previously eluded them. “It also provokes discussion about how particular patterns arise, why some columns are more colorful than others, and how this can lead to the study of prime numbers,” they wrote. Students who considered themselves to be hopeless at math discovered that they were anything but.

5. NUMERICAL PROGRESSION

Courtesy of Alasdair Post-Quinn

 
Computer technician Alasdair Post-Quinn has been using a pattern he calls Parallax to explore what can happen to a grid of metapixels that expands beyond a pixel’s usual dimensional constraint of a 1x1. “What if a pixel could be 1x2, or 5x3?” he asks. “A 9x9 pixel grid could become a 40x40 metapixel grid, if the pixels had varying widths and heights.” The catch: metapixels have both X and Y dimensions, and when you place one of them on a grid, it forces all the metapixels in the X direction (width) to match its Y direction (height), and the other way around. To take advantage of this, Post-Quinn charts a numerical progression that’s identical on both axes—like 1,1,2,2,3,3,4,5,4,3,3,2,2,1,1—to achieve results like the ones you see here. He’s also in the process of writing a computer program that will help him plot these boggling patterns out.

6. MÖBIUS BAND

Courtesy of Cat Bordhi

 
A Möbius band or strip, also known as a twisted cylinder, is a one-sided surface invented by German mathematician August Ferdinand Möbius in 1858. If you wanted to make one of these bands out of a strip of paper, you’d give an end a half-twist before attaching the two ends to each other. Or, you could knit one, like Cat Bordhi has been doing for over a decade. It ain’t so simple to work out the trick of it, though, and accomplishing it requires understanding some underlying functions of knitting and knitting tools—starting with how, and with what kind of needles, you cast on your stitches, a trick that Bordhi invented. She keeps coming back to it because, she says, it can be “distorted into endlessly compelling shapes,” like the basket pictured here, and two Möbii intersecting at their equators—an event that turns Möbius on its ear by giving it a continuous “right side.”

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iStock // Ekaterina Minaeva
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Man Buys Two Metric Tons of LEGO Bricks; Sorts Them Via Machine Learning
May 21, 2017
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iStock // Ekaterina Minaeva

Jacques Mattheij made a small, but awesome, mistake. He went on eBay one evening and bid on a bunch of bulk LEGO brick auctions, then went to sleep. Upon waking, he discovered that he was the high bidder on many, and was now the proud owner of two tons of LEGO bricks. (This is about 4400 pounds.) He wrote, "[L]esson 1: if you win almost all bids you are bidding too high."

Mattheij had noticed that bulk, unsorted bricks sell for something like €10/kilogram, whereas sets are roughly €40/kg and rare parts go for up to €100/kg. Much of the value of the bricks is in their sorting. If he could reduce the entropy of these bins of unsorted bricks, he could make a tidy profit. While many people do this work by hand, the problem is enormous—just the kind of challenge for a computer. Mattheij writes:

There are 38000+ shapes and there are 100+ possible shades of color (you can roughly tell how old someone is by asking them what lego colors they remember from their youth).

In the following months, Mattheij built a proof-of-concept sorting system using, of course, LEGO. He broke the problem down into a series of sub-problems (including "feeding LEGO reliably from a hopper is surprisingly hard," one of those facts of nature that will stymie even the best system design). After tinkering with the prototype at length, he expanded the system to a surprisingly complex system of conveyer belts (powered by a home treadmill), various pieces of cabinetry, and "copious quantities of crazy glue."

Here's a video showing the current system running at low speed:

The key part of the system was running the bricks past a camera paired with a computer running a neural net-based image classifier. That allows the computer (when sufficiently trained on brick images) to recognize bricks and thus categorize them by color, shape, or other parameters. Remember that as bricks pass by, they can be in any orientation, can be dirty, can even be stuck to other pieces. So having a flexible software system is key to recognizing—in a fraction of a second—what a given brick is, in order to sort it out. When a match is found, a jet of compressed air pops the piece off the conveyer belt and into a waiting bin.

After much experimentation, Mattheij rewrote the software (several times in fact) to accomplish a variety of basic tasks. At its core, the system takes images from a webcam and feeds them to a neural network to do the classification. Of course, the neural net needs to be "trained" by showing it lots of images, and telling it what those images represent. Mattheij's breakthrough was allowing the machine to effectively train itself, with guidance: Running pieces through allows the system to take its own photos, make a guess, and build on that guess. As long as Mattheij corrects the incorrect guesses, he ends up with a decent (and self-reinforcing) corpus of training data. As the machine continues running, it can rack up more training, allowing it to recognize a broad variety of pieces on the fly.

Here's another video, focusing on how the pieces move on conveyer belts (running at slow speed so puny humans can follow). You can also see the air jets in action:

In an email interview, Mattheij told Mental Floss that the system currently sorts LEGO bricks into more than 50 categories. It can also be run in a color-sorting mode to bin the parts across 12 color groups. (Thus at present you'd likely do a two-pass sort on the bricks: once for shape, then a separate pass for color.) He continues to refine the system, with a focus on making its recognition abilities faster. At some point down the line, he plans to make the software portion open source. You're on your own as far as building conveyer belts, bins, and so forth.

Check out Mattheij's writeup in two parts for more information. It starts with an overview of the story, followed up with a deep dive on the software. He's also tweeting about the project (among other things). And if you look around a bit, you'll find bulk LEGO brick auctions online—it's definitely a thing!

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Nick Briggs/Comic Relief
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What Happened to Jamie and Aurelia From Love Actually?
May 26, 2017
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Nick Briggs/Comic Relief

Fans of the romantic-comedy Love Actually recently got a bonus reunion in the form of Red Nose Day Actually, a short charity special that gave audiences a peek at where their favorite characters ended up almost 15 years later.

One of the most improbable pairings from the original film was between Jamie (Colin Firth) and Aurelia (Lúcia Moniz), who fell in love despite almost no shared vocabulary. Jamie is English, and Aurelia is Portuguese, and they know just enough of each other’s native tongues for Jamie to propose and Aurelia to accept.

A decade and a half on, they have both improved their knowledge of each other’s languages—if not perfectly, in Jamie’s case. But apparently, their love is much stronger than his grasp on Portuguese grammar, because they’ve got three bilingual kids and another on the way. (And still enjoy having important romantic moments in the car.)

In 2015, Love Actually script editor Emma Freud revealed via Twitter what happened between Karen and Harry (Emma Thompson and Alan Rickman, who passed away last year). Most of the other couples get happy endings in the short—even if Hugh Grant's character hasn't gotten any better at dancing.

[h/t TV Guide]

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