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.”

University of York
Stones, Bones, and Wrecks
UK Archaeologists Have Found One of the World’s Oldest 'Crayons'
University of York
University of York

A prehistoric chunk of pigment found near an ancient lake in England may be one of the world's oldest crayons, Colossal reports. The small object made of red ochre was discovered during an archaeological excavation near Lake Flixton, a prehistoric lake that has since become a peat wetland but was once occupied by Mesolithic hunter-gatherers. Though it’s hard to date the crayon itself, it was found in a layer of earth dating back to the 7th millennium BCE, according to a recent study by University of York archaeologists.

Measuring less than an inch long, the piece of pigment is sharpened at one end, and its shape indicates that it was modified by a person and used extensively as a tool, not shaped by nature. The piece "looks exactly like a crayon," study author Andy Needham of the University of York said in a press release.

A pebble of red ochre thought to be a prehistoric crayon
University of York

The fine grooves and striations on the crayon suggest that it was used as a drawing tool, and indicate that it might have been rubbed against a granular surface (like a rock). Other research has found that ochre was collected and used widely by prehistoric hunter-gatherers like the ones who lived near Lake Flixton, bolstering the theory that it was used as a tool.

The researchers also found another, pebble-shaped fragment of red ochre at a nearby site, which was scraped so heavily that it became concave, indicating that it might have been used to extract the pigment as a red powder.

"The pebble and crayon were located in an area already rich in art," Needham said. "It is possible there could have been an artistic use for these objects, perhaps for coloring animal skins or for use in decorative artwork."

[h/t Colossal]

Jeff J Mitchell/Getty Images
Tour the National Museum of Scotland From Home With Google Street View
Jeff J Mitchell/Getty Images
Jeff J Mitchell/Getty Images

Google's Street View technology can be used to view some amazing art, whether it's behind the walls of the Palace of Versailles in France or the Guggenheim Museum in New York. As the BBC reports, the National Museum of Scotland in Edinburgh is the latest institution to receive the virtual treatment.

The museum contains items tracing the history of the world and humanity. In the Natural World galleries, visitors will find a hulking Tyrannosaurus rex skeleton and a panorama of wildlife. In the World Cultures galleries, there are centuries' worth of art and innovation to see. The museum's permanent galleries and the 20,000 objects on display can all be viewed from home thanks to the new online experience.

Users can navigate the virtual museum as they would a regular location on Street View. Just click the area you wish to explore and drag your cursor for full 365-degree views. If there's a particular piece that catches your interest, you may be able to learn more about it from Google Arts & Culture. The site has added 1000 items from the National Museum of Scotland to its database, complete with high-resolution photos and detailed descriptions.

The Street View tour is a convenient option for art lovers outside the UK, but the museum is also worth visiting in person: Like its virtual counterpart, admission to the institution is free.

[h/t BBC]