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.”

London's Trafalgar Square Gets a Poetry-Writing Red Lion

Tolga Akmen, AFP/Getty Images
Tolga Akmen, AFP/Getty Images

London’s historic Trafalgar Square just got a fifth lion, the BBC reports. The fluorescent red, AI-powered lion takes visitor-submitted words and turns them into two-line poems, which are displayed on a screen inside its mouth. The history-inspired installation is part of the ongoing festivities for the London Design Festival, which ends Sunday.

The idea comes from set designer Es Devlin, who is participating in a yearlong collaboration with Google Arts & Culture. She was inspired by another designer who remarked that Sir Edwin Landseer, who sculptured the other lions in the square in the late 19th century, "never wanted [them] to look so passive.” Landseer apparently wanted the lions to assume a more lively stance, “but Queen Victoria found it too shocking,” Devlin says.

The story of how Trafalgar Square’s lions came to be is an odd piece of history. For one, the process was painfully slow. Landseer spent four years just working up a sketch and spent hours studying the habits of lions at the London Zoo. He even waited two years for one of the zoo’s lions to die, then carted it back to his studio and kept it there until it started to decay. He was forced to throw out the animal—and his reference material—before he finished. “Which is why, if you look closely, you can see that the lions in Trafalgar Square actually have the paws of cats, rather than lions,” The Telegraph notes.

[h/t BBC]

5 Weird 1960s Covers for Classic Novels

Chaloner Woods/Getty Images
Chaloner Woods/Getty Images

There are a lot of weird and bad book covers for the classics out there, and the Internet has delighted in chronicling them.

Some are designed to mimic the look of current blockbusters, like these Twilight-style covers for novels by Jane Austen and the Brontës. Others rely on bad stock photos and inept Photoshopping for classic works that have crossed into the public domain, from The Scarlet Pimpernel to The Adventures of Huckleberry Finn.

The subset of covers for 1960s paperbacks is rich with particularly hideous findings, mostly from Penguin and Signet Classics. Shockingly, they're not made by untalented people who are bad at Photoshop. These covers were drawn by established, objectively talented, and sometimes famous illustrators like graphic design legend Milton Glaser. They were purposely executed in unorthodox, interpretive styles. But although they may be done by respected artists, their aesthetic value remains questionable. Take a look at some of the strangest below.

1. THE GREAT GATSBY BY F. SCOTT FITZGERALD // 1962

The Great Gatsby cover by John Sewell
Courtesy of Setana Books

In the baffling jacket for this Jazz Age classic, a man’s face is stretched bizarrely sideways. He appears to be wearing thick eyeliner and has some serious wrinkles around his eyes. But, let's back up for a minute: Who is this supposed to be? Surely not the title character; Gatsby doesn’t have a bald patch or a unibrow. One Twitter user who collects Gatsby editions considers this specimen to be the "oddest" one he owns.

The artist, John Sewell, was a British graphic designer working in the '60s whose print covers usually involved colored paper cut-outs. He did a cover in a similar style for F. Scott Fitzgerald's Tender is the Night, but that one is a little less weird.

2. OUR MUTUAL FRIEND BY CHARLES DICKENS // 1964

cover of Our Mutual Friend by Seymour Chwast
Courtesy of swallace99, Flickr.

The artist here is Seymour Chwast, who, along with Milton Glaser, co-founded the postmodern collective Push Pin Studios in 1954. The Push Pin style "reject[s] tradition in favor of reinvigorated interpretations of historical styles," as their website states.

And yet, the people on this cover are hideous. The eyebrows on Our Mutual Friend's Gaffer Hexam (the man in the white shirt) are at a sharp 45-degree angle, a trait rarely found in nature. Lizzie Hexam, who’s supposed to be beautiful, also looks pretty wretched.

According to the artist's biography on the Seymour Chwast Archive, "Each of his imaginary characters (even portraits of real individuals) have similar facial features—round lips, slits for eyes, bulbous noses. They never scowl, yet they are not cute." That's for sure. A quick browse through his work shows that naturalism was never his goal.

3. ADAM BEDE BY GEORGE ELIOT // 1961

Adam Bede cover by James Hill
Courtesy of swallace99, Flickr

Why is Adam Bede's hand bigger than his face? And his arm bigger than his waist? What would George Eliot think?

This one is by James Hill, the first Canadian to become a member of the American Illustrators Association. His work ranged from lurid, pulpy book covers to treatments for classics like this one to a series of paintings inspired by Anne of Green Gables.

4. CRIME AND PUNISHMENT BY FYODOR DOSTOYEVSKY // 1968

Crime and Punishment cover

Courtesy of Felt Books

The 1960s produced many psychedelic book covers, and this style spilled over into reprints of the classics. On this Dostoyevsky opus, a guy's face is replaced by a groovy rainbow with a figure in a coffin inside. While the artist is unknown, the rainbow design echoes the style of several graphic designers of the 1960s.

5. HARD TIMES BY CHARLES DICKENS // 1961

Hard Times cover
Courtesy of ElwoodAnd Eloise, Etsy

This cover for Charles Dickens's grim tale of Victorian inequality was designed by Milton Glaser, Seymour Chwast's partner in Push Pin Studios. Glaser also designed the I Love New York logo and a Bob Dylan poster that depicts the singer with a rainbow 'fro. A versatile artist, his work includes logos, posters, interior design, magazine illustrations, and, of course, book covers. But here, the heavy cross-hatching on the figures' faces, hair, and clothes nudges them into werewolf territory. The psychedelic winged horse seems like a nod to the Summer of Love, but a tavern called the Pegasus's Arms actually figures prominently in the book.

SECTIONS

arrow
LIVE SMARTER