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.”

Looking to Downsize? You Can Buy a 5-Room DIY Cabin on Amazon for Less Than $33,000

Five rooms of one's own.
Five rooms of one's own.
Allwood/Amazon

If you’ve already mastered DIY houses for birds and dogs, maybe it’s time you built one for yourself.

As Simplemost reports, there are a number of house kits that you can order on Amazon, and the Allwood Avalon Cabin Kit is one of the quaintest—and, at $32,990, most affordable—options. The 540-square-foot structure has enough space for a kitchen, a bathroom, a bedroom, and a sitting room—and there’s an additional 218-square-foot loft with the potential to be the coziest reading nook of all time.

You can opt for three larger rooms if you're willing to skip the kitchen and bathroom.Allwood/Amazon

The construction process might not be a great idea for someone who’s never picked up a hammer, but you don’t need an architectural degree to tackle it. Step-by-step instructions and all materials are included, so it’s a little like a high-level IKEA project. According to the Amazon listing, it takes two adults about a week to complete. Since the Nordic wood walls are reinforced with steel rods, the house can withstand winds up to 120 mph, and you can pay an extra $1000 to upgrade from double-glass windows and doors to triple-glass for added fortification.

Sadly, the cool ceiling lamp is not included.Allwood/Amazon

Though everything you need for the shell of the house comes in the kit, you will need to purchase whatever goes inside it: toilet, shower, sink, stove, insulation, and all other furnishings. You can also customize the blueprint to fit your own plans for the space; maybe, for example, you’re going to use the house as a small event venue, and you’d rather have two or three large, airy rooms and no kitchen or bedroom.

Intrigued? Find out more here.

[h/t Simplemost]

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8 Times People Ruined Priceless Works of Art

Antonio Canova, Wikimedia Commons // CC BY-SA 4.0
Antonio Canova, Wikimedia Commons // CC BY-SA 4.0

“Don’t touch the art” is a simple rule, enacted by almost every gallery and museum in the world. Yet for some reason, there are a select few who choose to ignore it, either because their curiosity gets the best of them, or, in a surprising number of cases, because they're on a quest for the perfect selfie. Whatever their motives, the museum-goers below left a trail of mangled artwork in their wakes.

1. Pauline Bonaparte as Venus Victrix

If any lesson should be taken from art gallery mishaps, it’s that you should never use a valuable work of art as a piece of furniture. In July 2020, an unnamed tourist from Austria decided to luxuriate on the plaster cast of Antonio Canova’s Pauline Bonaparte as Venus Victrix (1804) at Italy’s Antonio Canova Museum to make his selfie look as casual as possible. (Bonaparte was Napoleon’s sister.) In doing so, he crumbled the toes of poor Pauline, who is depicted in the sculpture as reclining on a cushion. Surveillance footage shows the man acknowledging the loss of the extremities before walking away. Police later identified him from a museum reservation. He apologized for the accident and offered to pay for the restoration work.

2. Dom Sebastiao Statue

In 2016, a 24-year-old visiting Lisbon, Portugal, made a very bad call when he climbed onto a 126-year-old statue installed on the facade of Lisbon, Portugal's Rossio Train Station to snap a selfie. The freestanding statue, which depicted 16th century king Dom Sebastiao, toppled over and shattered on the ground. The tourist, who attempted to flee, was caught by the authorities and eventually forced to appear in front of a judge; Portugal's infrastructure department has no information about when the statue will be fixed.

3. Statua Dei Due Ercole

Hercules might have had the strength of the Gods, but unfortunately, that toughness didn't translate to sculptures of him. In 2016, two tourists visiting the Loggia dei Militi Palace in Cremona, Italy, damaged the 300-year-old Statua dei due Ercole (Statue of Two Hercules) when they climbed on it to take a selfie. The tourists were reportedly hanging off the crown of one of the marble figures—which held the town's emblem between them—when it gave way, falling to the ground. The tourists were charged with vandalism, and the government called in experts to assess the damage.

4. Ecce Homo

The most famous (read: hilarious) art "restoration" in history might be 80-year-old Cecilia Gimenez’s attempt to fix a deteriorating fresco painting at a church in Borja, Spain. Her new and improved art made international headlines and inspired endless internet memes in 2012. Saturday Night Live even worked the news into their Weekend Update segment a couple of times, with Kate McKinnon playing Gimenez.

The painting, a depiction of Jesus Christ by artist Elías García Martínez in the 1930s, was flaking due to moisture; Gimenez, a parishioner at the church, worked off a 10-year-old photo of the fresco while doing her restoration. When her work was revealed, Ecce Homo was redubbed "Potato Jesus." Gimenez told a Spanish TV station that she had approval to work on the fresco (which authorities deny), and had done so during the day. “The priest knew it,” she said. “I’ve never tried to do anything hidden.”

Though the church had originally planned to work with art restorers to fix the fresco, by 2014 they had changed their tune. Gimenez's artwork became a major tourist attraction, bringing 150,000 visitors from around the world and revitalizing Borja. The church charged $1.25 a head to see the artwork, which was preserved behind plexiglass, just like another very famous, memeworthy work of art: the Mona Lisa. A center dedicated to the interpretation of the new Ecce Homo opened in 2016.

5. Qing Dynasty Vases

Rule number one for entering any space with priceless art: tie your shoelaces. In February 2006, a man named Nick Flynn took the wrong staircase inside the Fitzwilliam Museum in Cambridge, England—and when he tried to change course, he accidentally stepped on his own untied shoelace and fell. With no handrails to grab, the only thing to break his fall were three Qing Dynasty vases from the 1600s and 1700s, which were sitting on a windowsill. Flynn was unhurt, but the vases, worth more than $100,000, were not so lucky: They shattered into 400 pieces.

"Although [I knew] the vase would break I didn't imagine it would be loose and crash into the other two," he said. "I'm sure I only hit the first one and that must have flown across the windowsill and hit the next one, which then hit the other, like a set of dominos." Flynn, who was reportedly banned from the museum, called the incident “just one of those unbelievably unlucky things that can sometimes happen.”

This story has something of a happy ending, though: By August 2006, Penny Bendall, a ceramic restorer, had glued one of the vases—which had broken into 113 pieces—back together for an exhibition on art restoration. "Putting the vase back together may have looked impossible to most people but actually it wasn't a difficult job—fairly straightforward," she told the Daily Mail.

6. Annunciazione

Should you be given a pass for breaking something if it was technically already broken? In 2013, a Missouri man visiting Museo dell'Opera del Duomo in Florence, Italy, wanted to see how the pinky finger of a 600-year-old statue of the Virgin Mary by Giovanni d’Ambrogio measured up next to his own. You know what happened next: The man got a little too close and damaged the statue's digit. Thankfully, the finger that he broke was made of plaster and not original to the sculpture, and art restorers grabbed it quickly before it could fall and be further damaged. The man apologized, and restorers at the museum made plans to repair the finger again. Hopefully the second fix was more permanent.

7. The Drunken Satyr

The good news is this Milan statue, which lost its left leg to an unknown selfie enthusiast in 2014, was a replica of another statue that dates back to 220 BCE. The bad news is that the replica was still very valuable and pretty old, dating back to the 1800s. Security cameras in that area of the Academy of Fine Arts of Brera weren't working when the incident occurred, but according to the Daily Mail, witnesses saw a student tourist climb onto the statue and sit on its knee to take a photo. What the student didn't realize was that the statue, made of terra cotta and plaster, had been assembled in pieces, and the leg was already partially detached; museum director Franco Marrocco told the Corriere della Sera that the museum was already planning to restore the statue before the accident.

8. The Actor

A 6-foot-tall Picasso painting is pretty hard to miss when it’s hung on a museum wall, just as the visitor who fell into one back in January 2010 discovered. A woman was attending a class at New York City's Metropolitan Museum of Art when she lost her footing and tumbled into The Actor, leaving a 6-inch tear as well as a dent in the lower right corner of the 1904 artwork. “We saw the big, coarse threads that looked sort of like a nasty jute rug,” Gary Tinterow, chairman of the museum’s department of 19th Century, Modern and Contemporary art, said in an interview. “The question was how to get Humpty Dumpty back together again.”

That process took three months. Lucy Belloli, a conservator at the Met, told The New York Times that the process involved photographing the canvas, securing flakes of paint with adhesive, and using strips of paper with rabbit-skin glue as bandages, as well as a six-week period of realigning the painting using small sand bags. ("[T]he torn portion of the canvas had to be gently coaxed back to its flat state, otherwise it would have a tendency to return to the distortion left by the accident," the Times explained.) Some retouching was also necessary. The painting was returned to the wall in April 2010 with a layer of Plexiglass to protect it; most visitors would not have been able to tell the painting was ever damaged.

This story has been updated for 2020.