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.”

Last Surviving Person of Interest in Isabella Stewart Gardner Museum Heist to Be Released From Prison

Federal Bureau of Investigation, Wikimedia Commons // Public domain
Federal Bureau of Investigation, Wikimedia Commons // Public domain

Almost exactly 29 years ago, two men disguised as police officers weaseled their way into Boston’s Isabella Stewart Gardner Museum and started removing prized artworks from the wall. They made off with 13 famous paintings and sculptures, representing a value of more than $500 million. It remains the largest property theft in U.S. history, but no one has ever been charged in connection with the heist.

Now, as Smithsonian reports, the last living person who may have first-hand knowledge about the heist will be released from prison this Sunday after serving 54 months for an unrelated crime. Robert (Bobby) Gentile, an 82-year-old mobster who was jailed for selling a gun to a known murderer, has been questioned by authorities in the past. In 2010, the wife of the late mobster Robert (Bobby) Guarente told investigators she had seen her husband give several of the artworks in question to Gentile—a good friend of Guarente’s—eight years prior.

A 2012 raid of Gentile’s home also revealed a list of black market prices for the stolen items. Previous testimony from other mob associates—coupled with the fact that Gentile had failed a polygraph test when he was questioned about the art heist—suggest Gentile might know more about the crime than he has let on. For his part, though, Gentile says he is innocent and knows nothing about the art or the heist.

The FBI announced in 2013 that it knew who was responsible for the museum heist, but would not reveal their names because they were dead. Still, the whereabouts of the artworks—including prized paintings by Rembrandt, Manet, Vermeer, and Degas—remain unknown. The museum is offering a $10 million reward to anyone who can provide information leading to “the recovery of all 13 works in good condition," according to the museum's website. A separate $100,000 reward will be provided for the return of an eagle finial that was used by Napoleon’s Imperial Guard.

[h/t Smithsonian]

9 Colors Named After People

Alice Roosevelt—for whom Alice Blue is named—in 1902
Alice Roosevelt—for whom Alice Blue is named—in 1902
Frances Benjamin Johnston, Library of Congress/Wikimedia // Public Domain

Throughout history, a variety of famous people have lent their names to shades of brilliant blue, shocking purple, grassy green, muddy brown, and other hues. While many of these figures are artists who were known for using or developing these hues, other color eponyms come from the scientists who invented them or those who loved to wear them. Consider this list the place where the history books meet the artist’s palette.

1. Alice Blue

A pale azure blue named for Alice Roosevelt Longworth, daughter of Theodore Roosevelt, who was known for wearing gowns of the color and thus sparking a trend for it. (She was also known for smoking in public and other forms of mischief-making, leading her father to declare: “I can either run the country or I can attend to Alice, but I cannot possibly do both.”) Her ice-blue dresses inspired the song "Alice Blue Gown" by Joseph McCarthy and Harry Tierney, which premiered in the 1919 Broadway musical Irene. ("I once had a gown that was almost new / Oh, the daintiest thing, it was sweet Alice Blue / With little forget-me-nots placed here and there / When I had it on, I walked on air.")

2. Yves Klein Blue

Visitors look at 'Monochrome Blue, without title' (1960) by French artist Yves Klein
Visitors look at Monochrome Blue, without title (1960) by French artist Yves Klein
THOMAS LOHNES/AFP/Getty Images

The artist Yves Klein was interested in art as transcendence, and he’s perhaps best known for painting monochromes in a brilliant ultramarine meant to suggest the infinity of sea and sky. (As Klein once explained, "Blue has no dimensions, it is beyond dimensions.") In 1960, he registered a formula for the color—known as IKB, or International Klein Blue—with the French government; the formula relied on ultramarine pigment mixed with a synthetic resin that wouldn't dilute the color.

During his “blue period,” Klein exhibited only blue paintings and objects, releasing a thousand and one blue balloons into the sky in Paris to celebrate one show, and serving gin, Cointreau, and blue-dye cocktails at another. Don’t copy that last idea, mixologists: everyone who drank them peed blue for days.

3. Titian Red

Visitors look at a painting by Renaissance master Titian in Rome
Visitors look at a painting by Renaissance master Titian in Rome
GABRIEL BOUYS/AFP/Getty Images

A person with red hair is sometimes said to be a Titian, after the great 16th century Venetian painter who was notably fond of painting redheads. (Examples of such paintings include Bacchus and Ariadne and Noli me Tangere, now in London's National Gallery.) In the 1960s, redheaded Barbie dolls were officially known as “Titians.” More loosely, the term has come to mean any orange-red color, although people seem to love to debate exactly what shades count.

4. Scheele's Green

Svenska Familj-Journalen, Wikimedia Commons // Public Domain

Arsenic-based green pigments were all the rage in the 19th century, coloring everything from hosiery to hats to children’s toys. The first such pigment on the scene was Scheele’s Green, discovered by Swedish chemist Carl Wilhelm Scheele in 1775. The vibrant yellow-green hue caught on, especially after it was discovered that arsenic also produced a variety of other greens, from deep emerald to pale peridot. Although Scheele and others knew how toxic these pigments were, that didn't stop the colors from being used for clothing, candles, papers, playing cards, book-bindings, and sometimes even food. In perhaps the most famous example of its use, arsenic green wallpaper graced Napoleon’s last bathroom while he suffered through his exile on St. Helena, and some think the fumes caused by his long baths may have been what killed him.

5. Isabelline

José Reynaldo da Fonseca, Wikimedia Commons // CC BY 2.5

If true, this color's origin story has to be the most off-putting in history. Once used to describe the pale champagne color of certain horse coats and bird feathers, the term Isabella-colored or isabelline is said (by no less than Isaac D'Israeli's 1791 Curiosities of Literature) to come from Isabel of Austria, the devoted daughter of Philip II of Spain.

Supposedly, when Spain laid siege to the city of Ostend in 1601, Isabella vowed not to change her undergarments until the city was taken. She expected a speedy victory, but much to her dismay (and presumably that of everyone around her), the fighting continued for three years before Spain won.

The Oxford English Dictionary dismisses this origin story, noting that Isabella as a color is first noted in 1600, a year before the siege began. But linguist Michael Quinion notes that accounts in French, German, Spanish and Italian (where isabelline has a similar color meaning) refer to the earlier Queen Isabella of Castile (1451-1504) and the siege of Granada—which means the story might just be true, even if it's about a different Isabella and a different set of 7-month-old dirty underwear.

6. Fuchsia

Heinrich Füllmaurer, Wikimedia Commons // Public Domain

Here's a more pleasant etymology: The vivid red-purple of fuchsia, the color, comes from fuchsia, the flower, which is in turn named for 16th-century German botanist Leonhart Fuchs. (His last name, by the way, comes from the German word for "fox.") And if you think fuchsia and magenta are the same color, you're closer than you might think: Magenta was originally an aniline dye named fuchsine, named after the fuchsia flower. The name was changed in 1859, the year it was patented, in honor of the French victory at the Battle of Magenta. That apparently helped the dye become a stunning success.

7. Vandyke Brown

Anthony van Dyck, Wikimedia Commons // Public Domain

This deep, warm, transparent brown was made with a high concentration of organic matter (basically: actual dirt), and was popular with the Old Masters. It was named for the innovative Flemish painter Anthony van Dyck, who often used the color in his paintings, and who also lent his name to an early photographic printing process—which also produced a brown color, but did not actually involve dirt.

8. Perkin's Mauve

Wikimedia Commons // Public Domain

Like so many scientific discoveries, the invention of synthetic dyes happened by accident. In 1856, chemistry student William Henry Perkin, then only 18, was trying to find a new way to make quinine (a popular treatment for malaria, and the ingredient that still gives tonic water its slightly bitter taste). The experiment didn't quite work as planned, but Perkin noticed some purple sludge left over in his flask after rinsing it with alcohol, and realized its potential.

His instincts were good: After Perkin patented his creation and began mass-producing it, the color swept England, becoming so popular that the magazine Punch condemned an outbreak of “the mauve measles.” The color was originally called aniline purple by Perkin, as well as Perkin's purple or Perkin’s violet. The mauve part of “Perkin’s mauve” came a few years later thanks to the French, who named it after their word for the mallow flower.

9. Hooker's Green

Thomas Herbert Maguire, Wikimedia Commons // Public Domain

The warm, grassy "Hooker's Green" is named for botanical illustrator William Hooker (1779–1832), who created a special pigment just to convey the exact green of leaves.

Bonus: Mummy Brown

A close-up of an Egyptian mummy head
A close-up of an Egyptian mummy head
iStock.com/izanbar

OK, it’s not a color named after one person, but a color named after many people—many dead people. First made in the 16th and 17th centuries, but a special favorite of the 19th century painters, this rich brown pigment was created by mixing both human and feline mummy crumbles with white pitch and myrrh. (Although we tend to think of them as protected antiquities today, people in centuries past often considered mummies just another natural resource.)

In part because of its curious components, the pigment wasn’t the most stable in the world, and it fell out of favor once its origin story became better known. According to one biography, the Pre-Raphaelite artist Edward Burne-Jones gave his tube of Mummy Brown a funeral in his garden when he discovered where it came from. The pigment was sold into the 20th century, although if you see the name “mummy brown” used today, rest assured it contains no actual corpses. Probably.

A version of this list first ran in 2016.

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