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.
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.
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.”
U.S. National Archives and Records Administration, Wikimedia Commons // Public domain
Amanda Crowe created remarkable works of art with just a mallet and chisel, transforming blocks of wood into bears, raccoons, deer, moose, and owls. The late Eastern Band Cherokee Indian woodcarver, who died in 2004, is being celebrated in today’s Google Doodle.
A video featuring images of some of Crowe’s works was released in recognition of Native American Heritage Month. The musical accompaniment was composed by her nephew, William “Bill” H. Crowe, Jr., and the video highlights some of Crowe’s best-known quotes.
“I carve because I love to do it,” Crowe once said. “The movement of the grains—they almost seem alive under your hands—and the beautiful tones and textures all add life to the figures you whittle.”
Crowe was born in the Qualla Boundary, North Carolina on July 16, 1928, and her uncle started teaching her how to carve wood when she was just four years old. She later commented that she was “barely old enough to handle a knife” when she first started learning, but she clearly had a knack for working with her hands.
After honing her craft throughout her childhood and teen years, she was offered a scholarship to the Art Institute of Chicago. There, she studied various media—including plaster, stone, and metal—but she always gravitated toward wood. Crowe went on to earn a master’s degree and spent some time in Mexico studying with Spanish sculptor José de Creeft.
When she finally returned home to the Qualla Boundary, she spent the next 40 years teaching art at Cherokee High School. According to Google, she is often credited with helping to restore interest in Cherokee carving—an ancestral tradition and a unique art form. Her carvings have carried her legacy around the world, having been displayed at museums in the U.S., England, Germany, and beyond.
Whether or not you’re artistically inclined, there’s a good chance that you—like millions of other people around the world—have been captivated by Bob Ross’s instructional landscape paintings and soothing voice. On what would have been Ross’s 76th birthday, we’re sharing 35 facts about the happy little legend.
1. HE KEPT AN ALLIGATOR IN THE BATHTUB AS A KID.
A lifelong animal lover, Ross was always rescuing wounded animals and nursing them back to health. As a kid growing up in Florida, this meant one rather strange addition to the family: an alligator, which he attempted to nurse back to health in the Ross family bathtub. Even in his adult life, Ross was always playing host to orphaned and injured animals, including an epileptic squirrel that lived in his empty Jacuzzi.
2. HE WAS AN AIR FORCE MASTER SERGEANT.
Ross’s quiet voice and gentle demeanor were two of his most iconic traits, which makes the fact that he spent 20 years in the United States Air Force and retired with the rank of master sergeant all the more surprising. Basically, he was the guy who told everyone else what to do.
3. HE USED TO BE QUITE THE YELLER.
Before he lent his dulcet voice to The Joy of Painting, Ross spent a lot of time yelling. "I was the guy who makes you scrub the latrine, the guy who makes you make your bed, the guy who screams at you for being late to work,” Ross once said. “The job requires you to be a mean, tough person. And I was fed up with it. I promised myself that if I ever got away from it, it wasn't going to be that way anymore."
4. BEFORE HE PAINTED HAPPY LITTLE TREES, HE PAINTED PANS.
While stationed in Alaska during his stint in the Air Force, Ross indulged his creative side by painting his now-iconic landscapes onto golden pans, which he sold for $25 apiece. Today, they can fetch as much as $7500 on eBay.
5. HE WAS INSPIRED BY BILL ALEXANDER.
From 1974 to 1982, German painter Bill Alexander hosted an art instruction show on PBS, The Magic of Oil Painting, where he shared his “wet-on-wet” oil painting technique. Ross discovered the series while working as a bartender, and became an immediate fan of the artist. He ended up studying under Alexander, who became his mentor. In fact, Ross dedicated the first episode of his own PBS show, The Joy of Painting, to Alexander. “Years ago, Bill taught me this fantastic technique,” Ross told viewers. “And I feel as though he gave me a precious gift, and I'd like to share that gift with you.”
6. WHEN ALEXANDER RETIRED, HE APPOINTED ROSS AS HIS SUCCESSOR.
In the early 1980s, as Alexander was preparing to retire, he asked Ross to take over teaching his painting classes. Ross agreed, and set out to tour the country on his own in a motor home, traveling and teaching people Alexander’s “wet-on-wet” technique. He told his wife Jane that he’d try it out for one year, and if he didn’t make enough money, he would return to Alaska.
7. HIS SIGNATURE PERM WAS AN ECONOMICAL CHOICE.
It was during Ross’s time on the road that he adopted his iconic hairstyle. Since teaching painting wasn’t an extremely lucrative profession, Ross learned to stretch every penny. One way he did this was to save money on haircuts by getting his locks permed.
Though Ross reportedly hated the permed hair, he was a businessman first, which is why he kept it. “When we got a line of paints and brushes, we put his picture on,” Bob Ross Company co-founder Annette Kowalski told Mental Floss. “The logo is a picture of Bob with that hair, so he could never get it cut. He wasn’t always happy about that.”
(You can see what he looked like without his trademark perm here.)
9. HE WAS “DISCOVERED” BY ONE OF HIS STUDENTS.
Though it was Alexander who got Ross started on his career path as an artist, it was Kowalski—one of Ross’s students—who put him on the pop culture map. Kowalski, who is often credited as the woman who "discovered" Ross, took a five-day instructional course with Ross in 1982, and quickly became enamored with his calming voice and positive messages.
In addition to newfound painting skills, Kowalski left the class with a new client: she became Ross’s manager, helping him broker the deal for The Joy of Painting television show with PBS, and later, a line of Bob Ross art supplies.
10. HE WORKED FOR FREE.
The Joy of Painting ran new seasons on PBS from 1983 to 1994, so even at public broadcasting rates the show must have made Ross quite a bit of loot, right? Not quite. Ross actually did the series for free; his income came from Bob Ross Inc.
Ross's company sold art supplies and how-to videotapes, taught classes, and even had a troupe of traveling art instructors who roamed the world teaching painting. It's tough to think of a better advertisement for these products than Ross's show.
11. HE COULD FILM AN ENTIRE SEASON IN ABOUT TWO DAYS.
How did Ross find the time to tape all of those shows for free? He could record a season almost as fast as he could paint. Ross could bang out an entire 13-episode season of The Joy of Painting in just over two days, which freed him up to get back to teaching lessons, which is where he made his real money.
12. THE JOY OF PAINTING WAS A WORLDWIDE HIT.
In addition to being carried by approximately 95 percent of all public television stations across America, reaching viewers in more than 93.5 million homes, The Joy of Painting was a hit outside of the U.S. as well. The show was broadcast in dozens of foreign countries, including the United Kingdom, Germany, Belgium, Switzerland, South Korea, and Turkey.
13. HE WAS PARTICULARLY BIG IN JAPAN.
The Joy of Painting was a big hit in Japan, where it aired twice a day. (His voice, however, was dubbed.) On a visit to the country, Ross was reportedly mobbed by fans.
14. ROSS LIKENED HIS POPULARITY TO A DRUG ADDICTION.
"We're like drug dealers,” Ross once said of the popularity of his painting technique. “Come into town and get everybody absolutely addicted to painting. It doesn't take much to get you addicted.”
15. VIEWERS LOVED HIM. FELLOW ARTISTS? NOT AS MUCH.
Though he was undoubtedly a pop culture phenomenon, the art world didn’t exactly embrace Ross. “People definitely know who he is," Kevin Lavin, a “struggling” painter, told The New York Times in 1991. "In his own way, he is as famous as Warhol.”
"It is formulaic and thoughtless,” sculptor Keith Frank said of Ross’s work in the same article. “Art as therapy."
“I am horrified by art instruction on television," added Abstract Expressionist Richard Pousette-Dart, who passed away the following year. "It's terrible—bad, bad, bad. They are just commercial exploiters, non-artists teaching other non-artists."
16. SOME ART SUPPLY STORES KEPT ROSS’S PRODUCTS AT A DISTANCE.
The New York Timespaid a visit to Pearl Paint Company, an art supply store in New York City, where an employee pointed to the “happy little corner" where they kept Ross’s products. "We hide them," he admitted, "so as not to offend."
17. ALEXANDER WASN’T THRILLED WITH ROSS’S SUCCESS.
Bill Alexander was one of the artists who wasn’t thrilled with Ross’s success, even though he had been his protégé. “He betrayed me," Alexander toldThe New York Times. "I invented 'wet on wet.' I trained him and he is copying me—what bothers me is not just that he betrayed me, but that he thinks he can do it better."
18. HIS HAPPY LITTLE COMMENTS WEREN’T AD LIBBED.
Though part of Ross’s appeal was his conversational tone, none of this talk of happy accidents or other happy little things was ad libbed. “He told me he would lay in bed at night and plan every word,” Kowalski once said. “He knew exactly what he was doing.”
19. HE WAS MISSING PART OF HIS LEFT INDEX FINGER.
Though you’d never know it from his painting technique, not all of Ross’s digits were intact. He lost part of his left index finger when he was a kid in a woodworking accident while working with his dad, who was a carpenter.
20. HE RARELY PAINTED PEOPLE.
While trees and wildlife often helped bring Ross’s paintings to life, he rarely painted people. In fact, he liked to keep his work as people-free as possible.
“I will tell you Bob’s biggest secret,” Kowalski told FiveThirtyEight. “If you notice, his cabins never had chimneys on them. That’s because chimneys represented people, and he didn’t want any sign of a person in his paintings.”
21. HE KEPT A TINY SQUIRREL IN HIS POCKET.
The Joy of Painting regularly featured a rotating cast of happy little animals, with a tiny squirrel named Peapod probably getting the bulk of airtime. According to Ross, Peapod liked to sit in his pocket.
22. NOT MANY PEOPLE ACTUALLY PAINTED ALONG WITH HIM.
Though The Joy of Painting was a beloved series, people didn’t seem to be watching it to learn how to be the next Picasso. It was once estimated that only 10 percent of viewers were actually painting along with Ross.
23. HE REALLY DID LOVE TREES.
In 2014, FiveThirtyEight did a statistical breakdown of Ross’s work on The Joy of Painting and found that 91 percent of them included at least one tree—by far the most popular element. (And if he painted one tree, there was a 93 percent chance he’d paint a second one—though he referred to any additional trees as “friends” on the show.)
24. HIS SON, STEVE, PREFERRED LAKES.
On a few occasions, Ross’s son Steve subbed for his dad as a guest host. That same data set discovered that Steve liked happy little lakes: 91 percent of Steve’s paintings featured one (as opposed to Bob’s 34 percent).
25. HE MADE THREE COPIES OF EACH PAINTING YOU SEE IN THE JOY OF PAINTING.
Ross shot 403 episodes of The Joy of Painting and made three near-exact copies of each painting per episode. The first copy always hid off screen, and Ross referred to it while the cameras rolled (none of his on-air paintings were spontaneous). Ross painted a third copy when filming finished. This time, an assistant would stand behind him and snap photos of each brushstroke; these pictures went into his how-to books.
26. HE DIDN’T GET A WHOLE LOT OF INTERVIEW REQUESTS.
For all his worldwide popularity, there aren’t a lot of interviews with Ross. It has nothing to do with the artist being publicity-shy—it’s just that people rarely asked. “I never turn down requests for interviews,” he once said. “I’m just rarely asked.”
27. HE WAS AN MTV PITCHMAN.
For all his hokey-ness, Ross was cool enough to be asked to be a pitchman for MTV—which he deemed “The land of happy little trees.”
28. NINTENDO HAD PLANNED A SERIES OF BOB ROSS VIDEO GAMES.
Though some thought it was an April Fools’ joke, Nintendo had plans to create a series of video games based on The Joy of Painting. Unfortunately, the project ran into production problems pretty early on, so we’ll never know what might have been.
29. THE JOY OF PAINTING IS GREAT FOR INSOMNIA.
In 2001, Bob Ross Inc. media director Joan Kowalski told The New York Times how people almost seemed embarrassed to admit that Ross’s voice was the perfect solution to insomnia. “It's funny to talk to these people,'' she said. ''Because they think they're the only ones who watch to take a nap. Bob knew about this. People would come up to him and say, 'I don't want to hurt your feelings, but you've been putting me to sleep for 10 years.' He'd love it.''
Even today, Ross has become an ASMR star: On the ASMR thread on Reddit, “Bob Ross” is listed as a common trigger. A video of Ross painting a mountain has a staggering 7.65 million views, with others regularly surpassing 2 or 3 million views. Of course, not all of those are ASMR viewers, but a mounting online presence suggests they certainly deserve some of the credit.
30. HE DIDN'T SELL HIS PAINTINGS.
In a 1991 interview with The New York Times, Ross claimed he'd made over 30,000 paintings since he was an 18-year-old stationed in Alaska with the Air Force. Yet he was not one to hawk his own work. So what happened to them? When Ross died of lymphoma in 1995, most of his paintings either ended up in the hands of charity or PBS.
“One of the questions that I hear over and over and over is, ‘What do we do with all these paintings we do on television?’ Most of these paintings are donated to PBS stations across the country,” he said. “They auction them off, and they make a happy buck with ‘em. So if you’d like to have one, get in touch with your PBS station, cause … we give them to stations all over the country to help them out with their fundraisers.”
31. ROSS’S VAN WAS ONCE BURGLED OF 13 PAINTINGS.
The fact that Ross didn’t try and turn a profit from his own work doesn’t mean that you can’t find one for sale. At one point, more than a dozen of his paintings hit the black market when someone stole 13 reference paintings from Ross’s van during the show's second season.
32. HE HOPED TO DEVELOP A CHILDREN’S SHOW ABOUT WILDLIFE.
In the early 1990s, Ross was looking to branch out from art and had an idea for a kids’ show called Bob’s World, where he planned to go out into nature and teach kids about wildlife.
33. IF YOU HAPPEN TO FIND YOURSELF IN FLORIDA, YOU CAN CHECK OUT SOME OF HIS ORIGINAL WORKS.
The Bob Ross Art Workshop in New Smyrna Beach, Florida is a must-visit destination for Ross die-hards: In addition to offering art classes in Ross’s method, you’ll find a collection of the artist’s original paintings.
34. YOU CAN VIEW MORE THAN 400 OF HIS WORKS IN ONE PLACE.
Two Inch Brush—named after Ross's brush of choice for the wet-on-wet technique—is an unofficial database that organizes all 403 paintings from The Joy of Painting by season and episode.
35. HE IS A FUNKO TOY.
In August 2017, Funko released a vinyl figurine of the iconic artist/television personality. It depicts Ross dressed in his trademark jeans and button-down shirt, holding a painter’s palette. Sadly, it doesn’t come with any miniature paintings of "happy little trees."