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.”

Happy Little Mystery Solved: We Finally Know What Happened to All of Bob Ross’ Paintings

Bob Ross Inc.
Bob Ross Inc.

Bob Ross is one of the most prolific artists of the 20th century, but his works are hard to find. They aren't sold at auction houses and they rarely appear in museums. But that doesn't mean they're not out there. For each of the 403 episodes Ross filmed for The Joy of Painting, he painted three pieces: one before filming to use as a reference, one during the show, and one after for his how-to books. He painted more than 1000 works for the series and now, nearly 25 years after the painter's death, The New York Times has finally discovered where all those happy little masterpieces are hiding.

Bob Ross Inc. headquarters in northern Virginia houses stacks of boxes of Bob Ross originals. The paintings aren't kept in a climate-controlled room like you might expect to find in the back of the Louvre. Rather, they sit in a regular storage room mixed in with other Bob Ross documents and artifacts.

Knocking on the door of the building won't get you a private showing of the artwork. Bob Ross Inc. is the company that handles the Bob Ross brand, and its headquarters aren't open to the public. But the massive body of work the painter left behind is becoming slightly more accessible to fans. Earlier in 2019, Bob Ross Inc. donated some of the items in its inventory to the Smithsonian’s National Museum of American History. The donation included the paintings “On a Clear Day" and “Blue Ridge Falls,” as well as handwritten notebooks and fan letters. The museum has no plans to display the paintings as of yet, but this fall, a different set of Ross originals will be shown at the Franklin Park Arts Center in Purcellville, Virginia. The exhibit will include 24 of his paintings—the most ever displayed at one time.

Fans looking to own a happy little landscape of their own are likely out of luck. The paintings at Bob Ross Inc. aren't for sale, which means any so-called Ross paintings you see being auctioned off online are likely fakes. For tips on how to spot a counterfeit, and to see where Bob Ross's real paintings are today, watch the video from the The New York Times below.

[h/t The New York Times]

17 Artful Facts About Frida Kahlo

A visitor looks at "Self-Portrait as Tehuana or Diego on My Mind" at the Frida Kahlo Retrospective in Berlin in 2010
A visitor looks at "Self-Portrait as Tehuana or Diego on My Mind" at the Frida Kahlo Retrospective in Berlin in 2010
Sean Gallup/Getty Images

The life and work of Frida Kahlo—one of Mexico's greatest painters—were both defined by pain and perseverance. Getting to know how Kahlo lived provides greater insight into her beloved paintings, which are rich with detail and personal iconography.

1. Frida Kahlo was born and died in the same house.

Kahlo was born on July 6, 1907, in a building nicknamed “La Casa Azul” for its vivid blue exterior. There, she was raised by her mother, Matilde, and encouraged by her photographer father, Guillermo. Years later, she and her husband, Mexican muralist Diego Rivera, made it their home as well. And on July 13, 1954, Kahlo died there at age 47.

2. … and Kahlo's beloved home is now a museum.

Casa Azul is also known as The Frida Kahlo Museum. As a tribute to Kahlo, Rivera donated the house in 1958 as well as all of the artwork, created by both him and Kahlo, that it contained. Much of the interior has been preserved just the way Kahlo had it in the 1950s, making the space a popular tourist attraction that allows visitors a look at her work, life, and personal artifacts, including the urn that holds her ashes.

3. A third of Frida Kahlo's paintings were self-portraits.

Kahlo folded in symbols from her Mexican culture and allusions to her personal life in order to create a series of 55 surreal and uniquely revealing self-portraits. Of these, she famously declared, "I paint myself because I am so often alone, because I am the subject I know best."

4. A surreal accident had a big impact on Frida Kahlo's life.

On September 17, 1925, an 18-year-old Kahlo boarded a bus with her boyfriend Alex Gómez Arias, only to be forever marred when it crossed a train's path. Recalling the tragedy, Arias described the bus as "burst(ing) into a thousand pieces," with a handrail ripping through Kahlo's torso.

He later recounted, "Something strange had happened. Frida was totally nude. The collision had unfastened her clothes. Someone in the bus, probably a house painter, had been carrying a packet of powdered gold. This package broke, and the gold fell all over the bleeding body of Frida. When people saw her, they cried, ‘La bailarina, la bailarina!’ With the gold on her red, bloody body, they thought she was a dancer."

5. Kahlo’s path to painting began with that collision.

The accident broke Kahlo's spinal column, collarbone, ribs, and pelvis, fractured her right leg in 11 places, and dislocated her shoulder. Those severe injuries left her racked with pain for the rest of her life, and frequently bedbound. But during these times, Kahlo picked up her father's paintbrush. Her mother helped arrange a special easel that would allow her to work from bed. Of her life's hardships, Kahlo once proclaimed, “At the end of the day, we can endure much more than we think we can.”

6. Frida Kahlo once dreamed of being a doctor.

As a child, Kahlo contracted polio, which withered her right leg and sparked an interest in the healing power of medicine. Unfortunately, the injuries from the train accident forced the teenager to abandon her plans to study medicine.

7. Kahlo’s poor health shaped her art.

In the course of her life, Kahlo would undergo 30 surgeries, including the eventual amputation of her foot due to a case of gangrene. She explored her frustrations with her body's frailty in paintings like The Broken Column, which centers on her shattered spine, and Without Hope, which dramatically depicted a period where her doctor prescribed force-feeding. On the back of the latter, she wrote, "Not the least hope remains to me ... Everything moves in time with what the belly contains."

8. Kahlo didn’t see herself as a surrealist.

She rejected the label, saying, "They thought I was a Surrealist, but I wasn’t. I never painted dreams. I painted my own reality."

9. Kahlo’s tumultuous marriage sparked more pain and paintings.

Frida Kahlo with Diego Rivera and a pet dog, Mexico City, 1940s
Frida Kahlo with Diego Rivera and a pet dog, Mexico City, 1940s
Hulton Archive/Getty Images

When Kahlo met Rivera, she was a student and he was already a father of four and on his way to his second divorce. Despite a 20-year age difference, the pair quickly fell for each other, spurring Rivera to leave his second wife and wed Kahlo in 1929.

From there, they were each other's greatest fans and supporters when it came to their art. But their 10-year marriage was wrought with fits of temper and infidelities on both sides. They divorced in 1939, only to remarry a year later. Paintings like Self-Portrait with Thorn Necklace and Hummingbird, The Two Fridas, and The Love Embrace of the Universe boldly illustrated their relationship from Kahlo's perspective.

10. Kahlo grieved privately and publicly for the children she never had.

Modern doctors believe that the bus accident had irreparably damaged Kahlo's uterus, which made pregnancies impossible to carry to term. In 1932, she painted Henry Ford Hospital, a provocative self-portrait that marks one of several devastating miscarriages she suffered.

The piece would be displayed to the world in a 1938 gallery show. But Kahlo kept private personal letters to her friend, Doctor Leo Eloesser, in which she wrote, "I had so looked forward to having a little Dieguito that I cried a lot, but it's over, there is nothing else that can be done except to bear it.'" This letter, along with others from their decades-long exchange, were released in 2007, having been hidden for almost 50 years by a patron worried about their contents.

11. Frida Kahlo once arrived to an art show in an ambulance.

In 1953, toward the end of her short life, the painter was overjoyed about her first solo exhibition in Mexico. But a hospital stay threatened her attendance. Against doctors' orders, Kahlo made an incredible entrance, pulling up in an ambulance as if in a limousine.

12. Kahlo is rumored to have had several famous lovers.

When she wasn't recovering from surgery or confined to a recuperation bed, Kahlo was full of life, relishing the chance to dance, socialize, and flirt. While American sculptor Isamu Noguchi was in Mexico City for the creation of his History as Seen from Mexico in 1936, he and Kahlo began a passionate affair that evolved into a life-long friendship.

Three years later, while visiting Paris, the bisexual painter struck up a romance with the city's "Black Pearl" entertainer Josephine Baker. And many have speculated that the artist and activist also bedded Marxist revolutionary Leon Trotsky, while he and his wife Natalia stayed in Kahlo's family home after they were granted asylum in Mexico in 1936.

13. Frida Kahlo was fiercely proud of her heritage.

Though she'd lived in New York, San Francisco, and Paris, Kahlo was always drawn back to her hometown, Mexico City. She favored traditional Mexican garb, the long colorful skirts she was known for, and the Huipile blouses of Mexico’s matriarchal Tehuantepec society. Perhaps most telling, she told the press she was born in 1910, cutting three years off her age so she could claim the same birth year as the Mexican Revolution.

14. Frida Kahlo had several exotic pets.

Casa Azul boasts a lovely garden where Kahlo had her own animal kingdom. Along with a few Mexican hairless Xoloitzcuintli (a dog breed that dates back to the ancient Aztecs), Kahlo owned a pair of spider monkeys named Fulang Chang and Caimito de Guayabal, which can be spotted in Self Portrait with Monkeys. She also cared for an Amazon parrot called Bonito, who would perform tricks if promised a pat of butter as a reward, a fawn named Granizo, and an eagle nicknamed Gertrudis Caca Blanca (a.k.a. Gertrude White Shit).

15. She has emerged as a feminist icon.

Though in her time some dismissed this passionate painter as little more than "the wife of Master Mural Painter (Diego Rivera)," Kahlo's imaginative art drew acclaim from the likes of Pablo Picasso and film star Edward G. Robinson. After her death, the rise of feminism in the 1970s sparked a renewed interest in her work. Kahlo's reputation eclipsed Rivera's, and she grew to become one of the world's most famous painters.

Feminist theorists embrace Kahlo's deeply personal portraits for their insight into the female experience. Likewise, her refusal to be defined by others' definitions and the self-love shown in her proud capturing of her natural unibrow and mustache speak to modern feminist concerns over gender roles and body-positivity.

16. Kahlo’s personal style has become a vibrant part of her legacy.

Frida's art and its influence were not simply spawned from the paint she put to canvas. Her distinctive personal style has proved influential in the world of fashion, inspiring designers like Raffaella Curiel, Maya Hansen, Jean Paul Gaultier, and Dolce & Gabbana. (In 2019, Vans even launched a collection of shoes featuring her work.)

17. Frida Kahlo's work is record-breaking.

On May 11, 2016, at the first auction to put a major Frida work up for sale in six years, her 1939 painting Dos desnudos en el bosque (La tierra misma) sold for over $8 million—the highest auction price then paid for any work by a Latin American artist.

This list was first published in 2016 and updated in 2019.

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