# 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.”

# When Skeleton Rocking Chairs and ‘Vampire Killing Kits’ Fooled People Into Thinking They Were Rare Historical Artifacts

A vampire killing kit at Ripley's Believe It or Not! in San Francisco
Glen Bowman, Flickr // CC BY 2.0

In 2012, bizarre rocking chairs—usually dark brown, with various kinds of ornate flourishes, always in the shape of a skeleton—began popping up on sites across the internet. Gothic.org and io9 ran stories about them, and Facebook pages like Steampunk Tendencies soon followed. The chairs were sometimes described as modeled on 19th-century Russian examples—and other times described as 19th-century Russian items themselves.

The grotesque chairs were funny, but got even funnier in 2013 when someone appropriated a photo from an auction house and meme-ified it. They added a blurred effect and magnified the skeleton’s anguished, open-mouthed expression, making it seem as if it were screaming into the void—perhaps upon realizing that it must spend the rest of eternity as a rocking chair in some eccentric collector’s parlor. By early 2014, someone on 4chan had associated the meme with the words “Wake Me Up Inside (Can't Wake Up)” after lyrics from the 2003 song "Bring Me to Life" by rock band Evanescence. Then, in true internet fashion, people started adding their own text.

By then, another story had attached itself to the chairs. In 2009, the Lawrence Journal-World discussed the macabre furniture item in a column titled "Ghoulish pieces attract collectors," and suggested that the chair had something to do with a Masonic ritual.

So—aside from the joy of a good meme—what’s the deal? Was this chair used in some secret society's ceremony, or is it just a strange artifact made by some long-forgotten Russian woodworker?

## A Macabre Fantasy

According to James Jackson, the answer is neither. Jackson—the president and CEO of Jackson’s Auctions in Cedar Falls, Iowa, and a specialist in Russian art—sold the chair that was featured in several of the early news stories.

He says most of these chairs were probably made in the '90s, but were designed to look older to fool buyers into forking over more money. “These are the type of things that are created in various markets to appeal to the eclectic, exotic tastes of a wannabe fine art consumer,” Jackson tells Mental Floss. “So the person making this chair—and the guy buying it and reselling it—they understand this brain very well.”

The precise origins of the chairs Jackson's sold are murky. A couple of the chairs were sold to a third-party seller called a consignor, who then resold them to Jackson’s Auctions. Jackson suspects they were probably made somewhere in Europe—probably at a workshop where the primary goal is to “make a buck.” That would explain why no artist or craftsman's name is ever attached to the chairs.

These “fantasy chairs” were initially thought to be rare, and some sellers may have benefited from the myths and stories surrounding their origin. Over the years, people started to see more and more of these chairs at auction, which contributed to their diminishing value. Jackson said his auction house sold one of the chairs for \$2600 in 2008, but in 2012, the price dropped to \$1500. At its lowest price point, a skeleton chair sold for \$900 in Detroit, according to Jackson's database of different auction houses.

## Artifacts of the Hyperreal

Jackson says the skeleton chairs remind him of the vampire slayer kits that were popular in the '90s, and continued to be sold throughout the 2000s (they still pop up on eBay and other online auctions from time to time). Wooden trunks—purportedly full of vampire-repelling tools from the 1800s such as wooden stakes, garlic, a crucifix, and sometimes pistols—used to command high prices at auction. Sotheby’s even sold one for \$25,000 in 2011.

“It was BS,” Jackson says of the trunks, explaining that while they may have contained old tools, the pieces were assembled later for commercial purposes and given a phony backstory. “Whenever we see anything weird like that, it’s an automatic red flag. To the consumer, though, they want it to be some rare and unusual thing—and that’s not true.”

Jackson said one obvious sign that the slaying kits were inauthentic was that "they don’t show up in any literature prior to the 1990s, [and] something like that would have been written about somewhere.” In hindsight, Jackson thinks the whole scam was pretty comical. He said you had experts on TV doing careful analyses of the paper labels inside these kits, when in reality, all they had to do was use a magnifying glass to see that the letters were printed by a dot matrix.

"It’s like doing a metallurgic study on a brand new Mercedes-Benz," he said. “I didn’t have to get a microscope out and a black light and spend an hour fondling it. It’s common sense.”

Jonathan Ferguson, a curator at the UK-based National Museum of Arms and Armour, also debunked these hunting trunks. He wrote in a blog post, “Nowhere was there evidence to support real vampire slayers carting about one of these kits.”

Still, he wrote that they were somewhat valuable as “genuine artifacts of the Gothic fiction,” and rather than being seen as fakes (since there never was a Victorian original), should be seen as "'hyperreal' or invented artifacts somewhat akin to stage, screen or magician's props."

As for the Sotheby's kit that was snatched up for \$25,000, its creation was also probably inspired by the popularity of Dracula (1897) and other late 19th century vampire lore, according to Dennis Harrington, head of Sotheby's European furniture department in New York City. Harrington notes that some of the pieces inside the kit are valuable in their own right.

"[The kit] was complete and did contain individual elements that have some intrinsic value themselves, like silver bullets and an ivory figure of Christ on the Cross (though we can no longer sell ivory items today) ..." Harrington tells Mental Floss. "The curiosity value would also have helped, and of course the golden rule of auctions is that any one lot is worth whatever someone is willing to pay for it on a particular day."

Likewise, the skeleton rocking chairs—despite not being antiques—certainly have their own unique appeal. “They’re cool, they’re neat. These are ‘man cave’ type things for the most part,” Jackson says. However, “They’re obviously not functional. You can’t sit in it comfortably.”

And what of the skeleton meme? Do the makers of these chairs know that their creation has been turned into an absurd internetism? Jackson, for his part, hadn’t heard anything about it. “I’m glad they made a joke out of [the chairs],” he said, “but I don’t know what meme means.”

# 13 Facts About the Chauvet Cave Paintings

A view taken on June 13, 2014 shows paintings of animal figures on the rock walls of the Chauvet Cave in Vallon Pont d'Arc.
Jeff Pachoud, AFP/Getty Images

Discovered by accident in 1994, the cave paintings adorning the walls of Chauvet Cave in France are among the oldest and most beautiful figurative art in human history. About 36,000 years ago, the ancient artists drew lifelike beasts that seem to gallop, crawl, and frolic through the cave’s chambers. In one stunning triptych, 50 drawings of horses, lions, and reindeer cavort across 49 feet of limestone wall. The cave paintings even impressed filmmaker Werner Herzog enough to make a documentary (available on Netflix). Here are a few more facts about the Chauvet Cave paintings.

## 1. The Chauvet Cave paintings were discovered by three local explorers.

It was December 18, 1994. French cavers Jean-Marie Chauvet, Éliette Brunel Deschamps, and Christian Hillaire had spent the day exploring the Pont d’Arc caves in the Ardèche region in southern France. They came upon an array of fallen rocks and noticed a gentle woosh of air from beneath the rock pile. Prying aside the stones, they found an aperture and dropped down into a large chamber with a high ceiling that appeared to branch off into other chambers. Their headlamps illuminated several handprints and a red ochre painting of a mammoth on the wall of one chamber. At that moment, they knew they had stumbled onto a major archaeological discovery.

## 2. Chauvet Cave was formed by an underground river.

A detail of the full-scale reproduction of frescos found at the cave of Pont-D'Arc, also known as the Chauvet Cave, on April 8, 2015 in Vallon Pont D'Arc. The frescos were reproduced by French graphic artist and researcher Gilles Tosello to replicate the Chauvet Cave, which is located in the Ardèche region of southern France.
Jeff Pachoud, AFP/Getty Images

Subterranean rivers flowing through the area's limestone hills created Chauvet Cave, along with hundreds of other gorges and caverns in the Ardèche. Chauvet Cave is about 1300 feet (roughly a quarter-mile) long with 14 chambers branching off the largest room, the Chamber of the Bear Hollows—the first one discovered by Chauvet, Brunel Deschamps, and Hillaire. This chamber, closest to the entrance, features no cave paintings; flooding is thought to have washed away any artwork. The most decorated vestibules are farthest from the entrance and include the Hillaire Chamber, Red Panels Gallery, Skull Chamber, the Megaloceros Gallery, and the End Chamber.

## 3. The Chauvet Cave painters were Aurignacians.

Aurignacians, the first anatomically modern humans in Europe, lived during the Upper Paleolithic, or Old Stone Age, between 46,000 and 26,000 years ago. (Aurignacian also refers to this time period.) Aurignacian culture is characterized by the first figurative drawings and carvings, the invention of a flaked stone tool called a burin used for engraving, bone and antler tools, jewelry, and the oldest-known musical instruments.

In addition to the Chauvet Cave paintings, Aurignacian animal and human figurines have been found in other parts of Europe. At the Hohle Fels cave in southwestern Germany, archaeologists discovered the oldest known Venus statuette, dating from 40,000 to 35,000 years ago, and some of the oldest known bone flutes from the same time period. In Southeast Asia, a cave in Borneo bears the oldest known figurative painting, created at least 40,000 years ago.

## 4. Ancient humans visited Chauvet Cave during two separate millennia.

Picture taken on October 12, 2012 in Vallon-Pont-d'Arc of the facsimile of the Chauvet cave.
Jeff Pachoud, AFP/Getty Images

According to paleontologist Michel-Alain Garcia in Chauvet Cave: The Art of Earliest Times, radiocarbon dating of organic materials in Chauvet Cave suggest people used the cave during two different time periods. In the first, about 36,500 years ago during the Aurignacian, artists drew the majority of the Chauvet Cave paintings. They brought wood into the cave and burned it to create light and charcoal for drawing. Then, for an unknown reason, the Aurignacians abandoned the cave for about five or six thousand years, and it was taken over by cave bears. In the second instance of human use, about 31,000 to 30,000 years ago in the Gravettian period, humans left behind footprints, scorch marks from torches, and charcoal, but no artwork.

## 5. Fourteen animal species are represented in the Chauvet Cave paintings.

The most common animals in the Chauvet Cave paintings are cave lions, mammoths, and woolly rhinoceroses; all coexisted with the Aurignacians in Europe, but are now extinct. Along with depictions of cave bears, the four species make up 65 percent of the species in the paintings. The other are bison, horses, reindeer, red deer, ibex, aurochs (an extinct wild ancestor of domesticated cattle), the extinct Megaloceros deer (also called the Irish elk or giant deer), musk ox, panthers, and an owl. The paintings are notable for depicting not just figurative representations of the animals, but actual scenes that reveal the animals’ real behavior—like two woolly rhinoceroses butting horns, and a pride of lions stalking a group of bison.

## 6. Non-animal themes also pop up in Chauvet Cave paintings.

A view taken on June 13, 2014 shows paintings on the rock walls of the Chauvet cave, in Vallon Pont d'Arc.
Jeff Pachoud, AFP/Getty Images

In the middle chambers of Chauvet Cave, several walls and overhanging rocks are decorated with red dots made by human palms and stencils of human hands. In the farthest galleries of the cave, five triangular representations of a woman’s pubic area are scratched on to the walls, and one picture of a woman’s lower body similar in profile to Paleolithic Venus figurines is drawn on a stalactite-like rock pendant. Anthropologists are not sure what they’re meant to symbolize.

## 7. A prehistoric child’s footprints were discovered in Chauvet Cave.

A single track of footprints measuring 230 feet long was found in the soft clay floor of the cave’s Gallery of the Crosshatching. Researchers analyzed modern European feet that were estimated to be roughly equivalent to those of European Early Modern Humans and determined that the track was probably made by a young boy about 4.5 feet tall. Scientists were able to date the prints based on the marks left by a burning torch on the roof of the gallery. “The child regularly wiped his torch on [the vault] above his path. These charcoal marks, dated to 26,000 years ago, seem to have been placed contrary to the direction of progress on purpose, as if to mark the way back,” Garcia writes. Two bits of charcoal were retrieved from the substrate and dated to a period between 31,430 years and 25,440 years ago.

## 8. The child might have had a pet dog.

The adolescent boy’s footprints are near those of a large canid—possibly a wolf. When Garcia took a closer look, he noticed the length of the middle digit was shorter than a wolf’s, a trait more typical of a domesticated dog. But in the 1990s, when Garcia made the find, the oldest undisputed fossil evidence of a domesticated dog dated back only 14,200 years before present.

A 2017 study that built on previous research, however, compared genomes of three Neolithic dogs with those of more than 5000 canines, including modern wolves and dogs. The researchers concluded that dogs and wolves split genetically sometime between 41,500 and 36,900 years ago, and a second divergence of eastern and western dogs occurred between 23,900 and 17,500 years ago. That puts the window of domestication between 40,000 and 20,000 years ago—the same time as the Aurignacian child and his very good boy were walking through Chauvet Cave.

## 9. Chauvet cave provided shelter for bears.

A view taken on June 13, 2014 shows paintings on the rock walls of the Chauvet cave, in Vallon Pont d'Arc.
Jeff Pachoud, AFP/Getty Images

Larger than modern grizzlies, cave bears spent winters in Chauvet Cave for thousands of years before humans began painting in it. They left claw scratches on the walls and dozens of tracks and footprints in the floor. In the Chamber of the Bear Hollows, researchers have found more than 300 hollows (sleeping spots that bears wore into the cave floor) and dozens of bear tracks and paw prints, made after humans stopped visiting the cave. About 2500 cave bear bones and 170 skulls were scattered throughout the cave’s main chambers. When scientists first investigated the cave in the mid-1990s, they found a cave bear skull carefully placed on a large stone in the middle of a deep chamber, in a way that only humans could have done.

## 10. The cave also provided shelter for a lot of wolves.

The floor of the Brunel Chamber, directly south of the Chamber of the Bear Hollows, showed multiple wolf prints that indicated a large number of “fissipeds” (pad-footed carnivores) had trampled the ground. Bear prints were superimposed on the wolf prints, suggesting that the bears came in after the wolves.

Not only large carnivores occupied the cave—judging from the variety of bones, it was practically a prehistoric zoo. In addition to the wolf, ibex, and bear bones, prehistorian Jean Clottes reported finding those of foxes, martens (a kind of weasel), roe deer, horses, birds, rodents, bats, and reptiles. And, yes, he also found fossilized wolf poop, indicating the wolves probably went into the cave in search of carrion.

## 11. No one knows why the Chauvet Cave paintings were created.

A detail of the full-scale reproduction of frescos found at the cave of Pont-D'Arc also known as the Chauvet cave, on April 8, 2015 in Vallon Pont D'Arc. The frescos were reproduced by French graphic artist and researcher Gilles Tosello to replicate the Chauvet Cave, located in the Ardèche region of southern France.
Jeff Pachoud, AFP/Getty Images

The purpose behind the Chauvet Cave paintings is a mystery, but some characteristics of the artwork may offer clues. Researchers have noted that the primary species depicted—cave bear, lion, mammoth, and rhinoceros—were not prey species that Aurignacians pursued for food, possibly suggesting that the paintings weren’t meant to ensure bountiful hunting.

A 2016 study hinted that the Chauvet Cave artists may have been recording contemporary events. Jean-Michel Geneste and colleagues proposed that a spray-like design in the Megaloceros Gallery was a faithful depiction of a volcanic eruption that occurred in the nearby Bas-Vivaris region between 40,000 and 30,000 years ago. If that is true, Chauvet Cave boasts the oldest known painting of volcanic activity, smoking the previous record holder—a 9000-year-old mural in central Turkey—by 28,000 years.

## 12. When Werner Herzog entered Chauvet Cave, he was overwhelmed.

Filmmaker Werner Herzog accompanied researchers into the depths of the cave system to make his 2010 documentary Cave of Forgotten Dreams (available to stream on Netflix). Herzog’s grandfather was an archaeologist, and Herzog himself once earned money as a ball boy at a tennis court to buy a book about cave art. “Even though in a way I knew what was waiting for me because I had seen photos, I was in complete and overwhelming awe,” Herzog told The A.V. Club in 2011. “The mysterious origins of it—we don’t know why they were made, and why in complete darkness and not next to the entrance.”

## 13. You can visit a scale replica of the Chauvet Cave paintings.

The world-famous Paleolithic cave paintings at Lascaux, not far from Pont d’Arc, were damaged by the exhalations of thousands of visitors after the cave was opened to the public in 1948. So, immediately after Chauvet Cave was discovered, scientists moved to protect the fragile paintings and closed it to the public; now, only scholars are allowed in during brief windows of time. But that doesn’t mean you can’t see a simulation of the artwork up close. In 2015, a scale replica of the Chauvet Cave paintings, dubbed the Caverne du Pont d’Arc, opened near the site of the actual cave. Engineers and artists faithfully recreated not just the dazzling paintings, but also the temperature, dampness, murk, and funky smell of the original.

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