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6 Math Concepts Explained by Knitting and Crochet

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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.”

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Thomas Quine, Flickr // CC BY 2.0
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Weird
Take a Peek Inside One of Berlin's Strangest Museums
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Thomas Quine, Flickr // CC BY 2.0

Vlad Korneev is a man with an obsession. He's spent years collecting technical and industrial objects from the last century—think iron lungs, World War II gas masks, 1930s fans, and vintage medical prostheses. At his Designpanoptikum in Berlin, which bills itself (accurately) as a "surreal museum of industrial objects," Korneev arranges his collection in fascinating, if disturbing, assemblages. (Atlas Obscura warns that it's "half design museum, half horror house of imagination.") Recently, the Midnight Archive caught up with Vlad for a special tour and some insight into the question visitors inevitably ask—"but what is it, really?" You can watch the full video below.

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Courtesy of Nikon
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science
Microscopic Videos Provide a Rare Close-Up Glimpse of the Natural World
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Courtesy of Nikon

Nature’s wonders aren’t always visible to the naked eye. To celebrate the miniature realm, Nikon’s Small World in Motion digital video competition awards prizes to the most stunning microscopic moving images, as filmed and submitted by photographers and scientists. The winners of the seventh annual competition were just announced on September 21—and you can check out the top submissions below.

FIRST PRIZE

Daniel von Wangenheim, a biologist at the Institute of Science and Technology Austria, took first place with a time-lapse video of thale cress root growth. For the uninitiated, thale cress—known to scientists as Arabidopsis thalianais a small flowering plant, considered by many to be a weed. Plant and genetics researchers like thale cress because of its fast growth cycle, abundant seed production, ability to pollinate itself, and wild genes, which haven’t been subjected to breeding and artificial selection.

Von Wangenheim’s footage condenses 17 hours of root tip growth into just 10 seconds. Magnified with a confocal microscope, the root appears neon green and pink—but von Wangenheim’s work shouldn’t be appreciated only for its aesthetics, he explains in a Nikon news release.

"Once we have a better understanding of the behavior of plant roots and its underlying mechanisms, we can help them grow deeper into the soil to reach water, or defy gravity in upper areas of the soil to adjust their root branching angle to areas with richer nutrients," said von Wangenheim, who studies how plants perceive and respond to gravity. "One step further, this could finally help to successfully grow plants under microgravity conditions in outer space—to provide food for astronauts in long-lasting missions."

SECOND PRIZE

Second place went to Tsutomu Tomita and Shun Miyazaki, both seasoned micro-photographers. They used a stereomicroscope to create a time-lapse video of a sweating fingertip, resulting in footage that’s both mesmerizing and gross.

To prompt the scene, "Tomita created tension amongst the subjects by showing them a video of daredevils climbing to the top of a skyscraper," according to Nikon. "Sweating is a common part of daily life, but being able to see it at a microscopic level is equal parts enlightening and cringe-worthy."

THIRD PRIZE

Third prize was awarded to Satoshi Nishimura, a professor from Japan’s Jichi Medical University who’s also a photography hobbyist. He filmed leukocyte accumulations and platelet aggregations in injured mouse cells. The rainbow-hued video "provides a rare look at how the body reacts to a puncture wound and begins the healing process by creating a blood clot," Nikon said.

To view the complete list of winners, visit Nikon’s website.

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