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.”

Tom Etherington, Penguin Press
The Covers of Jack Kerouac's Classic Titles Are Getting a Makeover
Tom Etherington, Penguin Press
Tom Etherington, Penguin Press

Readers have been enjoying classic Jack Kerouac books like The Dharma Bums and On the Road for decades, but starting this August the novels will have a new look. Several abstract covers have been unveiled as part of Penguin’s "Great Kerouac" series, according to design website It’s Nice That.

The vibrant covers, designed by Tom Etherington of Penguin Press, feature the works of abstract expressionist painter Franz Kline. The artwork is intended to capture “the experience of reading Kerouac” rather than illustrating a particular scene or character, Etherington told It’s Nice That. Indeed, abstract styles of artwork seem a fitting match for Kerouac’s “spontaneous prose”—a writing style that was influenced by improvisational jazz music.

This year marks the 60th anniversary of The Dharma Bums, which was published just one year after On the Road. The Great Kerouac series will be available for purchase on August 2.

[h/t It's Nice That]

John MacDougall, Getty Images
Stolpersteine: One Artist's International Memorial to the Holocaust
John MacDougall, Getty Images
John MacDougall, Getty Images

The most startling memorial to victims of the Holocaust may also be the easiest to miss. Embedded in the sidewalks of more than 20 countries, more than 60,000 Stolpersteine—German for “stumbling stones”—mark the spots where victims last resided before they were forced to leave their homes. The modest, nearly 4-by-4-inch brass blocks, each the size of a single cobblestone, are planted outside the doorways of row houses, bakeries, and coffee houses. Each tells a simple yet chilling story: A person lived here. This is what happened to them.

Here lived Hugo Lippers
Born 1878
Arrested 11/9/1938 — Altstrelitzer prison
Deported 1942 Auschwitz

The project is the brainchild of the German artist Gunter Demnig, who first had the idea in the early 1990s as he studied the Nazis' deportation of Sinti and Roma people. His first installations were guerrilla artwork: According to Reuters, Demnig laid his first 41 blocks in Berlin without official approval. The city, however, soon endorsed the idea and granted him permission to install more. Today, Berlin has more than 5000.

Demnig lays a Stolpersteine.
Artist Gunter Demnig lays a Stolpersteine outside a residence in Hamburg, Germany in 2012.
Patrick Lux, Getty Images

The Stolpersteine are unique in their individuality. Too often, the millions of Holocaust victims are spoken of as a nameless mass. And while the powerful memorials and museums in places such as Berlin and Washington, D.C. are an antidote to that, the Stolpersteine are special—they are decentralized, integrated into everyday life. You can walk down a sidewalk, look down, and suddenly find yourself standing where a person's life changed. History becomes unavoidably present.

That's because, unlike gravestones, the stumbling stones mark an important date between a person’s birth and death: the day that person was forced to abandon his or her home. As a result, not every stumbling stone is dedicated to a person who was murdered. Some plaques commemorate people who fled Europe and survived. Others honor people who were deported but managed to escape. The plaques aim to memorialize the moment a person’s life was irrevocably changed—no matter how it ended.

The ordinariness of the surrounding landscape—a buzzing cafe, a quaint bookstore, a tree-lined street—only heightens that effect. As David Crew writes for Not Even Past, “[Demnig] thought the stones would encourage ordinary citizens to realize that Nazi persecution and terror had begun on their very doorsteps."

A man in a shop holding a hammer making a Stolpersteine.
Artisan Michael Friedrichs-Friedlaender hammers inscriptions into the brass plaques at the Stolpersteine manufacturing studio in Berlin.
Sean Gallup, Getty Images

While Demnig installs every single Stolpersteine himself, he does not work alone. His project, which stretches from Germany to Brazil, relies on the research of hundreds of outside volunteers. Their efforts have not only helped Demnig create a striking memorial, but have also helped historians better document the lives of individuals who will never be forgotten.