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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.”

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Art
5 Things You Might Not Know About Ansel Adams

You probably know Ansel Adams—who was born on February 20, 1902—as the man who helped promote the National Park Service through his magnificent photographs. But there was a lot more to the shutterbug than his iconic, black-and-white vistas. Here are five lesser-known facts about the celebrated photographer.

1. AN EARTHQUAKE LED TO HIS DISTINCTIVE NOSE.

Adams was a four-year-old tot when the 1906 San Francisco earthquake struck his hometown. Although the boy managed to escape injury during the quake itself, an aftershock threw him face-first into a garden wall, breaking his nose. According to a 1979 interview with TIME, Adams said that doctors told his parents that it would be best to fix the nose when the boy matured. He joked, "But of course I never did mature, so I still have the nose." The nose became Adams' most striking physical feature. His buddy Cedric Wright liked to refer to Adams' honker as his "earthquake nose.

2. HE ALMOST BECAME A PIANIST.

Adams was an energetic, inattentive student, and that trait coupled with a possible case of dyslexia earned him the heave-ho from private schools. It was clear, however, that he was a sharp boy—when motivated.

When Adams was just 12 years old, he taught himself to play the piano and read music, and he quickly showed a great aptitude for it. For nearly a dozen years, Adams focused intensely on his piano training. He was still playful—he would end performances by jumping up and sitting on his piano—but he took his musical education seriously. Adams ultimately devoted over a decade to his study, but he eventually came to the realization that his hands simply weren't big enough for him to become a professional concert pianist. He decided to leave the keys for the camera after meeting photographer Paul Strand, much to his family's dismay.

3. HE HELPED CREATE A NATIONAL PARK.

If you've ever enjoyed Kings Canyon National Park in California, tip your cap to Adams. In the 1930s Adams took a series of photographs that eventually became the book Sierra Nevada: The John Muir Trail. When Adams sent a copy to Secretary of the Interior Harold Ickes, the cabinet member showed it to Franklin Roosevelt. The photographs so delighted FDR that he wouldn't give the book back to Ickes. Adams sent Ickes a replacement copy, and FDR kept his with him in the White House.

After a few years, Ickes, Adams, and the Sierra Club successfully convinced Roosevelt to make Kings Canyon a national park in 1940. Roosevelt's designation specifically provided that the park be left totally undeveloped and roadless, so the only way FDR himself would ever experience it was through Adams' lenses.

4. HE WELCOMED COMMERCIAL ASSIGNMENTS.

While many of his contemporary fine art photographers shunned commercial assignments as crass or materialistic, Adams went out of his way to find paying gigs. If a company needed a camera for hire, Adams would generally show up, and as a result, he had some unlikely clients. According to The Ansel Adams Gallery, he snapped shots for everyone from IBM to AT&T to women's colleges to a dried fruit company. All of this commercial print work dismayed Adams's mentor Alfred Stieglitz and even worried Adams when he couldn't find time to work on his own projects. It did, however, keep the lights on.

5. HE AND GEORGIA O'KEEFFE WERE FRIENDS.

Adams and legendary painter O'Keeffe were pals and occasional traveling buddies who found common ground despite their very different artistic approaches. They met through their mutual friend/mentor Stieglitz—who eventually became O'Keeffe's husband—and became friends who traveled throughout the Southwest together during the 1930s. O'Keeffe would paint while Adams took photographs.

These journeys together led to some of the artists' best-known work, like Adams' portrait of O'Keeffe and a wrangler named Orville Cox, and while both artists revered nature and the American Southwest, Adams considered O'Keeffe the master when it came to capturing the area. 

“The Southwest is O’Keeffe’s land,” he wrote. “No one else has extracted from it such a style and color, or has revealed the essential forms so beautifully as she has in her paintings.”

The two remained close throughout their lives. Adams would visit O'Keeffe's ranch, and the two wrote to each other until Adams' death in 1984.

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Dan Bell
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Design
A Cartographer Is Mapping All of the UK’s National Parks, J.R.R. Tolkien-Style
Peak District National Park
Peak District National Park
Dan Bell

Cartographer Dan Bell makes national parks into fantasy lands. Bell, who lives near Lake District National Park in England, is currently on a mission to draw every national park in the UK in the style of the maps in J.R.R. Tolkien’s The Lord of the Rings, Kottke.org reports.

The project began in September 2017, when Bell posted his own hand-drawn version of a Middle Earth map online. He received such a positive response that he decided to apply the fantasy style to real world locations. He has completed 11 out of the UK’s 15 parks so far. Once he finishes, he hopes to tackle the U.S. National Park system, too. (He already has Yellowstone National Park down.)

Bell has done various other maps in the same style, including ones for London and Game of Thrones’s Westeros, and he commissions, in case you have your own special locale that could use the Tolkien treatment. Check out a few of his park maps below.

A close-up of a map for Peak District National Park
Peak District National Park in central England
Dan Bell

A black-and-white illustration of Cairngorms National Park in the style of a 'Lord of the Rings' map.
Cairngorms National Park in Scotland
Dan Bell

A black-and-white illustration of Lake District National Park in the style of a 'Lord of the Rings' map.
Lake District National Park in England
Dan Bell

You can buy prints of the maps here.

[h/t Kottke.org]

All images by Dan Bell

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