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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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Google Launches World's Largest Digital Collection of Frida Kahlo Artifacts
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Fans of iconic Mexican artist Frida Kahlo have a lot of new material to sift through, thanks to Google’s launch of the largest-ever digital exhibition of artworks and artifacts related to the painter. As reported by Forbes, the “Faces of Frida” retrospective and its 800-item collection were the result of a collaboration between the Google Arts & Culture platform and 33 museums around the world.

A screenshot of Google's digital archive of Frida Kahlo artworks
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Visitors to the website can peruse rare artworks from private collections that had never been digitized until now, including View of New York, a sketch Kahlo made in 1932 while staying at the former Barbizon-Plaza Hotel. There are also personal photographs of Kahlo, as well as letters and journal entries that she penned.

Using Street View, you can even see inside the “Blue House” where she lived in Mexico City. Another feature lets visitors zoom in on high-resolution paintings, which were created using Google’s Art Camera, according to designboom.

For Google executives, the decision to celebrate the life and work of Kahlo was a no-brainer. “Frida's name kept coming up as a top contender when we started to think of what artist would be the best to feature in a retrospective,” Jesús Garcia, Google's head of Hispanic communications, told Forbes. “There's so much of her that was not known and could still be explored from an artistic perspective and life experience.”

An original artwork by multimedia artist Alexa Meade was specially commissioned for “Faces of Frida.” Photographer Cristina Kahlo, Kahlo’s great-niece, aided in the process. Check out the video below to see how she brought Kahlo's artwork to life in a living, breathing painting.

[h/t Forbes]

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15 Secrets of Caricature Artists
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iStock

The word caricature likely conjures up images of street artists on boardwalks or outside museums working up quick, humorous sketches of visitors, to the delight or dismay of their subjects. But the exaggerated illustrations of caricature include a lot more than what you see on the boardwalk—and can be more art than kitsch. We spoke to three experts in the field about the subjects caricature artists love and hate to depict, the best way to make their job harder, what they do if you don't like their drawing, and how they can tell when you really don't want to sit for a portrait.

1. THEY WANT YOU TO KNOW IT'S OLDER THAN YOU THINK.

Caricatures by Leonardo da Vinci
Wikimedia // Public Domain

Some of the greatest artists in history practiced caricature as a means to develop their skills. Eileen Owens, curator of "Biting Wit and Brazen Folly: British Satirical Prints, 1780s–1830s" at the Philadelphia Museum of Art, says Leonardo da Vinci was one of the first artists to use caricature, in the “grotesque” sketches of unusual faces and heads that populated his notebooks. (His 16th-century biographer, Giorgio Vasari, wrote that Leonardo was “so delighted when he saw curious heads, whether bearded or hairy, that he would follow anyone who had thus attracted his attention for a whole day.”) Many other well-established Renaissance artists dabbled in caricature on the side, as breaks from their rigorous training: "It was a lot more huge noses, big hair, ways to poke fun at faces. You had to be adept at drawing to know how to exaggerate," Owens says.

The form gained momentum in late-17th century Italy, when Pier Leone Ghezzi “started making funny little drawings that poked fun at well-to-do Romans and tourists,” according to Owens. From there, it spread to Britain, where it became so popular that publishing companies sprung up for the sole purpose of printing caricatures. Publishers also rented out portfolios of caricatures by the day, and hung prints in their windows, to which crowds flocked to see the latest depictions of a buffoonish Napoleon and laughable upper-crust fashions. Owens says, “This was your chance to keep up with the gossip—kind of like People magazine today.”

2. MANY OF THEM ARE SELF-TAUGHT.

A street artist paints a caricature of a girl in Prague
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Lots of caricature artists learn on the job, in part because there's not a ton of specific training available. Illustrator Tom Richmond, who spoofs movies for MAD Magazine (among other gigs), says, "Only a handful of art schools teach cartooning or caricature as a major part of curriculum, so it's hard to find instruction on how to do it. Caricature is such a specialized sort of thing, and diverse; you can’t teach it like you teach people how to draw comics, where [there's] storytelling technique and sequential art tricks and a science behind it, so to speak." Overall, what Richmond and others strive for is to “translate [your] art skill [into caricature], really lean into it—no matter how you practice.”

3. IT CAN BE GREAT TRAINING FOR OTHER ART FORMS.

Richmond says that when he teaches at workshops around the country, he always recommends—no matter what facet of the industry they are interested in—that students try their hand at live drawing, "maybe even volunteer at the local homecoming or draw for free at a daycare center." Having to work quickly with a model in front of you develops a sensitivity to gesture, to how the body leans and how weight is distributed, that's different from the skills you get "shading something for hours," Richmond explains. When you "go back to doing longer pieces, you've got an inner eye that sees things you missed before. It's great discipline for the developing eye."

4. THEY’RE NOT (NECESSARILY) OUT TO MOCK YOU.

Caricatures have been defined as "portrait[s] with the volume turned up." But that doesn't mean they have to be mean-spirited. Richmond says, “Caricature is a depiction of someone in a humorous way, but at its best it has a narrative behind it—you’re pointing out something about their presence, not just making fun of their features.” He explains that he’s not examining someone’s face to find a nose or a chin or dimples to blow out of proportion, but "trying to understand who you are as a person and exaggerate that.”

"I want to make [clients] smile or laugh," says CeCe Holt, who sketches at events and amusement parks, and is also business manager for the non-profit International Society of Caricature Artists (ISCA). "I never want to make anybody cry."

5. THEY DON’T SWEAT IT WHEN SOMEONE DOESN’T LIKE THEIR LIKENESS …

Just because caricaturists strive to capture your essence doesn't mean you're going to like it. People can be in denial about their appearance, with a radically different idea of their weight, for instance, or even whether they have freckles. In Holt’s experience, party guests usually don’t make a fuss about their caricatures, since they haven’t directly paid for them. But when the occasional amusement park patron kicks up a fuss, “I just say I’m sorry and move on to the next person.”

Richmond is similarly blasé, pointing out that when a caricaturist is drawing a quick sketch for $15, the occasional bad portrait is bound to sneak in. "Sometimes they refuse to pay, or come back later and want their money back. Live caricature can be hair-raising, which is why I prefer working with art directors."

6. … BUT SOMETIMES CUSTOMERS RETALIATE.

Christopher Walken's caricature in the foreground at Sardi's following its unveiling in 2010.
Christopher Walken's caricature in the foreground at Sardi's following its unveiling in 2010.
Jemal Countess/Getty Images

Occasionally, customers do try to turn the tables. Ipecacxink, a caricature artist at a Midwest theme park, writes in a Reddit AMA about a boy she accidentally made very upset with her drawing. "I went to lunch right after I did it. Apparently while I was gone, he came back and drew a circle with spikey hair, glasses, and frowny eyebrows and a note that said, 'How do you like someone making fun of you?!' under it. He then placed it on my chair. It was hilarious. I saved it."

At Sardi's—the Times Square tourist destination known for its wall of caricatures—some of the celebrities depicted have gotten mad enough to take down their pictures, the restaurant's owner told AMNew York. It used to be that the in-house caricaturist (who's paid in meals instead of money) would hand over unfinished versions to the subjects first, to get the seal of approval, before going on to later exaggerate their features. That's stopped, but these days the caricatures have become less humorous, and more like regular portraits—which helps keep the peace between the restaurant and its famous clientele.

7. THEY CAN DO PORTRAITS IN AS LITTLE AS THREE MINUTES.

When she’s sketching guests at amusement parks like Worlds of Fun in Kansas City, Missouri, Holt aims to churn out a black-and-white portrait in three minutes. Working at a wedding reception, where she might add color, six minutes is the max. Much of this has to do with fitting in as many guests as possible—“You have to be fast to get through the crowd or they’ll leave,” she says.

For Holt, the need for speed means she has to “go with her instincts; there isn’t time to second-guess” a depiction. For Richmond, working quickly means caricaturists develop a "sixth sense" for how to capture expressions: “You develop an instinct for people, whether they’re energetic and outgoing, or more quiet." Some of that means honing in on their signature details: "Friends behind will be going, 'It’s the smile! That's exactly how he looks!'" Richmond says.

8. BORING-LOOKING CUSTOMERS ARE THE HARDEST.

Man's hands with pencils drawing a woman's portrait
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The caricaturist's worst fear is the customer who comes in looking exactly like the girl (or guy) next door. "Most people are surprised to hear that what I consider to be the most difficult sort of person to draw is one that is completely average looking," caricaturist GertrudisSlugworth, who works at a theme park, wrote on Reddit."I will get a bland looking individual every once in a while, and when it happens I usually try to focus more on things like clothes, hair, or jewelry to get a decent likeness."

On the other hand, people who are naturally distinctive-looking are often artist favorites. Richmond says he particularly loves drawing Slash, the guitarist from Guns N’ Roses. “He’s already funny looking, with no features, just glasses, hair, and a big top hat, so you don’t have to work that hard,” he says. “You can just do him standing there with his guitar by his ankles, like he plays it, or exaggerate how he puts his head back, which shows a lot about him as a player.”

9. THEY MAY CHANGE THEIR TECHNIQUE TO SUIT THE WAY YOU LOOK.

When she first started in the business, Holt says she dreaded drawing people who weren’t thin; she was afraid they might take offense at her portraits, although she didn’t intend any. Over the years she’s honed a technique in which she draws faces using a soft line that thickens toward the bottom. The result is “Cute, but they still feel like it looks like them,” Holt says.

GertrudisSlugworth writes that for people with obvious deformities, she may forego exaggerations, even though those are normally the hallmarks of caricature: "I find the best way to handle it is to go more realistic than exaggerated, depending on their attitude. Sometimes if it's an easy fix (e.g missing an eye), the customer will just ask to be drawn as 'normal.' For the most part though, people recognize any obvious deformities they have, and accept your portrayal of them."

10. STREET ARTISTS HAVE AN ADVANTAGE.

Tourists look at caricaturists in Rome
PATRICK HERTZOG/AFP/Getty Images

Richmond says that artists "sitting in front of a museum while the subject is in front of them have more of an advantage" than he does when it comes to creating an expressive caricature, since he often has to work from photos, which don't show gesture and personality in the same way. "When I'm working from 2D photos, all you’ve got is what the photo shows you, and it's basically superficial. It doesn’t really do it."

Holt agrees: "Working from a picture is different from getting your first instincts from a person." When a freelance client wants her to draw someone from photos, she says she'll at least ask for multiple photos to work from, especially body shots, which help to show posture—yet another indicator of the subject's personality.

11. THEY'RE INCREASINGLY IN DEMAND.

Richmond says that although staff cartoonists may be disappearing at newspapers as that industry shrinks, editorial cartooning—which often relies on caricature—“is experiencing a boom right now." Some of this is thanks to the heated political climate, he notes. But there's a deeper reason, too: "Most media stories, TV shows, or articles are, at bottom, about people and need images of people to illustrate [them]," Richmond says. "Caricature is one thing you can’t do with a camera, so when you need a humorous touch, caricature is a great solution."

12. THERE'S A CARICATURIST CONVENTION.

The ISCA hosts an annual convention each November that draws hundreds of caricaturists from around the world. Aside from a week of guest speakers, seminars, and demonstrations, the main attraction is a days-long competition in which the artists draw each other for prizes in categories like best color technique and most humorous. (The big award there is called the Golden Nosey.) Richmond says, “The variety of styles [there] is crazy: acrylic painting, pastels, airbrush, sculpture, and everything in between.” Holt says there's even an artist who spits ink out of his mouth.

13. THEY MIGHT HIDE THINGS IN THEIR PORTRAITS.

Artwork by Al Hirschfeld on display at The New York Botanical Garden in 2011
Astrid Stawiarz/Getty Images

Richmond says that a favorite stylist of his is the late Al Hirschfeld, who for decades hid his daughter’s name, Nina, in his cartoons of cultural icons for The New York Times. (Hirschfeld would append the number of Ninas to his signature, creating a kind of game for readers). Ipecacxink says she "used to draw a picture of my face in [subject's] pupils sometimes. Really tiny. Or, I used to draw a little radioactive symbol somewhere in the drawing. We had to wear these god-awful neon yellow shirts to work, and I always felt we were radioactive."

14. THEY CAN TELL WHEN YOU DON'T WANT A DRAWING.

Occasionally, parents, friends, or partners will purchase a drawing for someone who just isn't interested. In that case, the caricaturist can probably pick up on it: "They either wouldn't look at you, wouldn't smile, or just sit down funny," ipecacxink writes. "I tried to handle it professionally. I would talk, if they wouldn't talk, I'd be quiet, but smile like an idiot when it was all said and done ... I always tried to be friendly to lessen the likelihood of them leaving without paying."

15. THEY MIGHT BE SWAPPING THEIR PENCILS FOR A TABLET.

Some contemporary caricaturists paint portraits, like Owens’s traditional satirical masters once did. They may also be adept with other analog media, like bullet-tip markers, color sticks (basically colored pencils with no wood casings), pen and brush, and paper. But thanks to the changing needs of publications in an online age, which want all files submitted electronically, caricature artists working in their studios have also gone digital. Holt sometimes works on an iPad Pro with an Apple Pencil in Procreate. Richmond now does all his coloring on a computer or a tablet. “[A tablet’s] so convenient, because it’s like having unlimited amounts of paper, and your pencil never needs to be sharpened, and all your tools fit in a tiny bag,” he says. “But it’s still about the creativity behind it. Computers can’t do it all on their own.”

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