6 Math Concepts Explained by Knitting and Crochet

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.”

Wish You Could ‘Shazam’ a Piece of Art? With Magnus, You Can

Manuel-F-O/iStock via Getty Images

While museum artworks are often accompanied by tidy little placards that tell you the basics—title, artist, year, medium, dimensions, etc.—that’s not always the standard for art galleries and fairs. For people who don’t love tracking down a staff member every time they’d like to know more about a particular work, there’s Magnus, a Shazam-like app that lets you snap a photo of an artwork and will then tell you the title, artist, last price, and more.

The New York Times reports that Magnus has a primarily crowdsourced database of more than 10 million art images. Though the idea of creating Shazam for art seems fairly straightforward, the execution has been relatively complex, partially because of the sheer quantity of art in the world. As founder Magnus Resch explained to The New York Times, “There is a lot more art in the world than there are songs.”

Structural diversity in art adds another challenge to the process: it’s difficult for image recognition technology to register 3D objects like sculptures, however famous they may be. Resch also has to dodge copyright violations; he maintains that the Digital Millennium Copyright Act applies to his app, since the photos are taken and shared by users, but he still has had to remove some content. All things considered, Magnus’s approximate match rate of 70 percent is pretty impressive.

Since the process of buying and selling art often includes negotiation and prices can fluctuate drastically, Magnus gives potential purchasers the background information they need to at least decide whether they’re interested in pursuing a particular piece. Just like browsing around a boutique where prices aren’t included on the items, a lack of transparency can be a deterrent for new customers.

Such was the case for Jelena Cohen, a Colgate-Palmolive brand manager who bought her first photograph with the help of Magnus. “I used to go to these art fairs, and I felt embarrassed or shy, because nothing’s listed,” she told The New York Times. “I loved that the app could scan a piece and give you the exact history of it, when it was last sold, and the price it was sold for. That helped me negotiate.” Through Magnus, you can also keep track of artworks you’ve scanned in your digital collection, search for artworks by artist, and share images to social media.

One thing Magnus can’t do, however, is tell you whether an artwork is authentic or not. The truth is that sometimes even art experts have trouble doing that, as evidenced by the long history of notorious art forgeries.

[h/t The New York Times]

'The Far Side' May Be Making a Comeback Online

tilo/iStock, Getty Images Plus

For the first time ever, it’s looking increasingly likely that cartoonist Gary Larson’s "The Far Side" will be available in a medium other than book collections or page-a-day calendars. A (slightly ambiguous) announcement on the official "Far Side" website promises that “a new online era” for the strip is coming soon.

From 1980 to 1995, "The Far Side" presented a wonderfully irreverent universe in which hunters had much to fear from armed and verbose deer, cows possessed a rich internal life, scientific experiments often went awry, and irony became a central conceit. In one of the more famous strips frequently pasted to refrigerator doors, a small child could be seen pushing on a door marked “pull.” Above him was a sign marking the building as a school for the gifted. In another strip, a woman is depicted looking nervously around a forest while cradling a vacuum cleaner. The caption: “The woods were dark and foreboding, and Alice sensed that sinister eyes were watching her every step. Worst of all, she knew that Nature abhorred a vacuum.”

Unlike most of his contemporaries, like Berkeley Breathed ("Bloom County") and Bill Watterson ("Calvin and Hobbes"), Larson has resisted reproduction of his work online. He famously circulated a letter to "Far Side" fan sites asking them to stop posting the single-panel strips, writing that the idea of his work being found on random websites was bothersome. “These cartoons are my ‘children,’ of sorts, and like a parent, I’m concerned about where they go at night without telling me,” he wrote.

Many obliged Larson, though the strip could still be found here and there. That he’s seemingly embracing a new method of distribution is good news for fans, but there’s no concrete evidence the now-retired cartoonist will be following in Breathed’s footsteps and producing new strips. ("Bloom County" returned as a Facebook comic in 2015.) The only indication of Larson’s active involvement is a new piece of art on the site’s landing page depicting some familiar "Far Side" characters being unthawed in a block of ice.

Larson’s comments on a return are few and far between. In 1998, he told The New York Times that going back to a strip was unlikely. “I don’t think so,” he said. “Never say never, but there’s a sense of ‘been there, done that.’” In that same profile, it was noted that 33 million "Far Side" books had been sold.

[h/t A.V. Club]

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