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.


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.


© 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.”


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.


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.


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.


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.”

Paris Musées Digitized More than 100,000 Major Artworks and Made Them Downloadable

“Setting Sun on the Seine at Lavacourt” by Claude Monet
“Setting Sun on the Seine at Lavacourt” by Claude Monet
Paris Musées, CC0

The museums of Paris are home to some of the most influential artworks on Earth, and if you live outside France, you no longer need a passport to see them. As Smithsonian reports, Paris Musées—the organization behind 14 of the city's iconic museums—has digitized more than 100,000 paintings and other pieces of art and made them freely available to the public.

The institutions under Paris Musées's umbrella include the Petit Palais, Musée d’Art Moderne de la Ville de Paris, and Maison de Balzac. It started sharing the work in its inventory online in 2016, and has since uploaded more than 320,000 pictures.

Roughly a third of the images in that digital collection were published in January 2020. This recent update was part of Paris Musées's initiative toward embracing open-access art. Every one of the 100,000-plus images uploaded in this month fall under the Creative Commons Zero license, which means they are fully in the public domain. Works like "Young Ladies on the Banks of the Seine" by Gustave Courbet, “Setting Sun on the Seine at Lavacourt” by Claude Monet, and "Portrait of Ambroise Vollard” by Paul Cézanne, are now not only free to view, but free to download as well.

"Portrait of Ambroise Vollard” by Paul Cézanne
"Portrait of Ambroise Vollard” by Paul Cézanne
Paris Musées, CC0

Paris Musées eventually hopes to transition all the out-of-copyright items in its collection—which comprises roughly 1 million works—to a Creative Commons Zero license. The most recent image dump is just the first round, and other art will become available gradually as the institution carefully evaluates the copyright status of each piece. It plans to someday expand its public domain artworks to external platforms like Wikimedia Commons, but for now, you can find them on Paris Musées's website.

[h/t Smithsonian]

Apple Wants to Show Off Your Best Night Mode Photos as Part of a New Campaign

Austin Mann, Apple
Austin Mann, Apple

Calling all aspiring photographers who nabbed an iPhone 11 for the express purpose of trying out its fancy camera capabilities: It’s time for your night mode photos to see the light of day.

As Travel + Leisure reports, Apple is currently hosting a competition to find the best night mode photos taken on an iPhone 11, iPhone 11 Pro, or iPhone 11 Pro Max. You can submit your photos through January 29, after which a carefully selected team of experts will evaluate all submissions and announce the five winning images on March 4.

Judges include Arem Duplessis, the former design director of The New York Times Magazine; Darren Soh, an award-winning photographer from Singapore; Tyler Mitchell, the first black photographer to shoot the cover of American Vogue (his subject, rather memorably, was Beyoncé); and several other esteemed members of the industry.

golden gate bridge shot on iphone 11
The Golden Gate Bridge, shot on an iPhone 11 Pro.
Jude Allen, Apple

In addition to appearing on Apple’s homepage and Instagram (which has more than 21 million followers), the photos could also be featured in digital campaigns, Apple stores, third-party photo exhibitions, or even on physical billboards. In addition to all the exposure, the winners will be paid a licensing fee in exchange for granting the company complete freedom to use their work for one year.

To submit your shots, you can either share them on a public Instagram, Twitter, or Weibo account with the hashtags #ShotoniPhone and #NightmodeChallenge, or email your images to shotoniphone@apple.com—just be sure to title your files in this format: ‘firstname_lastname_nightmode_iPhonemodel.’

If you’re new to the iPhone 11 and aren’t quite sure how to snap photos in night mode, it’s easier than you might realize. The feature comes on automatically in dim or dark places and decides on a capture time for you (which you can always adjust). And if you think editing your photos afterward will increase your chances of winning the competition, that’s fine, too: Apple will accept photos edited in the app or even with non-Apple software.

You might want to avoid capturing the Eiffel Tower after dark, however—here’s why.

[h/t Travel + Leisure]