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.”

The Mona Lisa Does Not Actually Cause the ‘Mona Lisa Effect’

Wikimedia Commons // Public Domain
Wikimedia Commons // Public Domain

Do you ever feel like you’re being watched? By a certain Leonardo da Vinci painting, perhaps? Scientists call it the Mona Lisa effect: the sense that the eyes of a figure in a painting or photograph are following you as you move around the room. But according to a new study in the journal i-Perception, the eyes in the Mona Lisa painting don’t actually fit the criteria.

The Mona Lisa effect is real—scholars have documented the phenomenon for nearly 2000 years. The effect doesn’t just depend on the direction of the painted figure’s gaze. The figure’s head position in the painting and the slant of the picture itself create specific geometric conditions in space, distorting the viewer’s perception of the painted person’s stare. The sensation can occur no matter where the viewer is in relation to the portrait.

Until now, according to researchers at Bielefeld University in Germany, no one had tested the effect on the Mona Lisa itself. Gernot Horstmann and Sebastian Loth, members of the university’s Cluster of Excellence Cognitive Interaction Technology, designed a study in which 24 participants viewed 15 different sections of the Mona Lisa painting on a monitor. A simple ruler was placed in front of the monitor, and each viewer marked the spot where they thought the gaze landed on the ruler, which indicated its angle.

An angle of zero meant a straight-at-the-viewer look. A slightly sideward gaze toward the viewer’s ear, corresponding to a 5-degree angle, would still prompt the sense of being watched. "But as the angle increases, you would not have the impression of being looked at,” Horstmann said in a statement.

After analyzing about 2000 assessments from participants, the researchers found that viewers felt the gaze of Leonardo da Vinci’s masterpiece to be at an angle of 15.4 degrees—looking off to their right-hand side, rather than directly at them.

“It is clear that the term Mona Lisa effect is nothing but a misnomer,” Horstmann said. But even though this particular phenomenon has been demystified, people's obsession with the painting will surely continue.

11 Secrets of Perfumers

Orlando/Three Lions/Getty Images
Orlando/Three Lions/Getty Images

Perfumers are a rare breed. These half-artist, half-scientist hybrids undergo rigorous training, memorize the smells of hundreds of ingredients, and spend decades honing their craft—which might explain why there are reportedly more astronauts than perfumers in the world, according to the BBC.

For many, the job isn't merely about peddling bottles of sweet-smelling stuff to consumers; the goal is to convey an emotion, create a beautiful moment, or jog a childhood memory. To find out what it takes to create top-notch fragrances, Mental Floss spoke with three perfumers who broke into the industry through very different paths.

1. Perfumers can identify hundreds of ingredients by smell alone.

A large perfume organ with hundreds of fragrance bottles
Mandy Aftel's perfume organ
By Joel Bernstein // Courtesy of Mandy Aftel

Master perfumers are sometimes called a nez—the French word for "nose"—for good reason. They commit hundreds of scents to memory and can distinguish between ingredients that would smell identical to the untrained nose. Many perfumers can also tell an essential oil from a synthetic material, which is no small feat. “You’re talking maybe 200 essential oils and about 1500 synthetic materials,” Jodi Wilson, a classically trained perfumer who now works as a fragrance sales manager for Orchidia Fragrances in Chicago, says of the ingredients perfumers typically employ.

The trick, she says, is to associate each smell with a distinct memory. “The more experiences you have in your life, the more memories you create, and that’s really how you remember these raw materials when you first start studying, because it reminds you of your grandmother or a flower shop or a bakery or a certain gum,” Wilson tells Mental Floss. (The link between smell and memory has actually been proven by science—one 2018 study by neurobiologists at the University of Toronto revealed that the brain not only stores information about certain scents, but also memories of when and where you first encountered them.)

2. Having a good sense of smell isn't enough to make a good perfumer.

Many perfumers have a heightened sense of smell. Jersey City-based perfumer Christopher Brosius, who founded the rebellious fragrance brand CB I Hate Perfume (a reference to his distaste for most commercial fragrances) is one of them. He realized just how strong his nose was while working briefly as a New York City cab driver—he had to roll the window down every time an “offensive” perfume wafted in his direction and made his stomach churn.

However, many aspiring perfumers mistakenly believe that a “good nose” will get them far. “That’s like saying that if you have 20/20 vision you’re the next Picasso,” Brosius tells Mental Floss. “A keen nose is very useful, but at the end of the day I have met perfumers who were extremely talented who didn’t smell anything more sharply than anybody else. They just had the capacity to think in a different way about what they were doing with scent and combining it in unique and interesting ways.” More important than a good sense of smell is creativity, a natural talent for recognizing scents that work well together, and the “dedication to building a very particular base of knowledge and skill,” Brosius says.

3. France's Givaudan Perfumery School is the goal for many would-be top perfumers.

Jodi Wilson picks roses
Jodi Wilson picks roses for distillation while studying at the Roure Perfumery School (now called the Givaudan Perfumery School) in Grasse, France, during the 1991-92 academic year.
Courtesy of Jodi Wilson

Like many professional perfumers, Wilson was educated at what's now the Givaudan Perfumery School in France. Founded in 1946, it only accepts one or two promising students each year out of thousands of applicants—and sometimes none at all, if that year’s crop of candidates don’t live up to the school’s high standards. Former director Jean Guichard has said he hand-selected students based on their personality, talent, and motivations. “The perfumer should be a mixture between a scientist and a poet,” Guichard told the BBC. “When I meet people, I know if they have talent or not. I don’t want to have people who say, ‘I’m going to be a perfumer because they make a lot of money.’ That doesn’t interest me at all.” (And speaking of pay, Wilson says the starting salary for entry-level perfumers is about $45,000, but perfumers in New York City tend to start off a bit higher. It's not unheard of for the world's top perfumers to make six figures.)

The now-four-year Givaudan program is rigorous. First, students have to memorize about 1500 raw materials, Wilson says. Next, they learn how to build accords, which are the fragrance notes (like rose or jasmine) that form the heart of a perfume. They move on to perfume schemas (the “skeleton” of a fine fragrance, which contains 10 to 12 materials) and also learn about the culture and history of perfume. “It takes a long time to learn all of that, and that’s what you’re doing all day from 9 a.m. till 4 p.m. It’s intense,” Wilson says. If and when they graduate, they’ll have a job waiting for them at the Givaudan fragrance company, which is where they’ll learn how to make perfumes under the guidance of a seasoned professional.

4. perfume school isn’t the only way to break into the industry.

Mandy Aftel holding perfume blotters
Perfumer Mandy Aftel at work
By Foster Curry // Courtesy of Mandy Aftel

Brosius says “99.9 percent” of aspiring perfumers would benefit from attending a perfume school. However, he personally did things a little differently and learned the fundamentals of perfume-making by landing a job at Kiehl’s and completing the company’s in-house training program.

It’s even less common for a perfumer to be self-taught, but it’s not impossible. The latter camp includes Mandy Aftel, a perfumer in Berkeley, California, who dropped a fulfilling career in psychotherapy to pursue a budding passion for perfume-making. For information about natural materials, she turned to fragrance books from the early 1900s, before synthetic materials started to saturate the market. Now, her Aftelier Perfumes business uses hundreds of natural ingredients—no synthetics—to create unique fragrances, and she has a loyal clientele. Regardless of the career paths they took, all of the perfumers agreed that this career is “a continuous learning process,” as Aftel tells Mental Floss. Both Brosius and Wilson said it takes 20 to 25 years to truly master the art of perfume-making, and Aftel still calls herself a “beginner” after 30 years of working in this field.

5. Not all perfumers work with fine fragrances.

Fragrance is used in many different ways, some of which we encounter on a daily basis without realizing it. Some perfumers specialize in creating scents for “industrial application,” which could include anything from children’s toys to paint to fabric, Brosius says. In the case of toilet-bowl cleaners, cat litter, and asphalt, the goal is not necessarily to create a pleasant aroma; instead, the challenge is to mask an unpleasant one. However, many of the perfumers working on the industrial side have scientific backgrounds and tend to work for a chemical company rather than a perfume label, Wilson says.

6. Some of the materials perfumers work with are hazardous.

Some undiluted ingredients—such as cinnamon—can cause severe chemical burns if they get on one's skin. Brosius wears gloves and goggles while blending materials and says some ingredients in his studio come with a "do not open without authorization" label attached. He says, “We have a protocol here that if anything new comes in, it’s opened in specific parts of the building or even sometimes outside on the terrace so that we don’t have an accident where it’s like, ‘Oops I just spilled one single drop of aldehyde [an organic compound] and now the entire building is uninhabitable, although next week it will smell like ginger ale!”

7. They want you to know your aromatherapy lotion might not be made of rose, jasmine, or whatever the bottle claims it contains.

Labels can be deceptive. If you’re buying an “aromatherapy” lotion or shower gel that claims to have rose, sandalwood, or jasmine in it but costs $15, that’s a red flag. According to Wilson, these ingredients can cost many thousands of dollars per pound. Wilson says it’s far more likely that cheaper products contain just a drop or two of the natural oils advertised—for the sake of being able to list them on the label—plus a host of synthetic ingredients that mimic the smell.

8. They're not always working on fragrances they like.

Marketing is a huge part of the cost of the perfume, especially on the higher end; the perfume industry spent around $800 million on marketing in 2016, according to Bloomberg. “Ninety percent of the time, the cost of the juice in that bottle is fractional,” Brosius says.

Marketing demands are also one reason why perfumers don't always get to follow their nose—and their creativity. “Most perfumers who work at large houses are not so happy with their job all the time,” Brosius says. “For every fine fragrance they get to work on, they’re compelled to work on a ton of crap fragrances as well. Much of it is entirely dependent on the whim of the marketing company.”

Companies are also more risk-averse, Wilson says—and the perfumes themselves now aren’t always built to last. “It used to be that a ‘classic’ was considered to last for 20 years—so your Chanel 5 and things of that nature,” Wilson says. “Now, it’s very rare to have a perfume that stays around for even 10 years.”

9. The smell of puppies is impossible to replicate—but perfumers are trying.

A bottle of Soaked Earth accord from CB I Hate Perfume
Kevin O'Mara, Flickr // CC BY-NC-ND 2.0

Brosius has taken on some ambitious projects over the years, including fragrances imitating the smells of snow and wet earth, but some scents are harder to capture than others. That’s because the aroma chemicals needed to replicate certain smells haven't been created yet. This can be said of gasoline, champagne and certain wines, and some animal smells. “Particularly with puppies and kittens, the molecules needed to accurately reproduce those smells don’t exist in the perfumer’s palette. You can’t solvent extract puppies and kittens for their smell," Brosius says, describing a method that involves applying a chemical solvent to a raw material—such as a flower—to extract its aroma.

However, he’s had success creating "a context that’s so strong that people are convinced that they’re smelling something that isn’t there," he says. For instance, his roast beef fragrance doesn’t contain roast beef or anything like it, but it does contain notes of parsley and black pepper. That scent in particular, and a few others like it, aren't meant to be worn on the body. Brosius says some of his fragrances are more like modern-day "smelling salts," where the goal is to revive you, in a sense, by relieving stress. "All you have to do is open the bottle, breathe in, and your system will automatically reset to calm," he says.

10. Perfumers sometimes work with whale poop.

A small bowl with ambergris in it
Peter Kaminski, Flickr // CC BY 2.0

Perfume-makers work with some unusual ingredients, and ambergris is certainly at the top of the list. This rock-like material comes from the excrement of sperm whales and occasionally washes up on shore. Aftel is fortunate enough to have some on display at the olfactory history museum she operates, called the Aftel Archive of Curious Scents. To convert the solid mass of crushed up squid and cuttlefish bits into an aromatic oil, she had to mash it up with a mortar and pestle, then add alcohol, heat it, and let it age. So what does it smell like in liquid form? “Heaven,” Aftel says. “It’s just ambery and shimmery. It’s a miracle of transformation.” Besides, Herman Melville mentioned it in Moby Dick and it used to be a 17th-century ice cream flavor, so you know it has to be good.

11. They keep wool nearby to combat nose fatigue.

Wool is the olfactory equivalent of eating sorbet in between courses. If you’re smelling the same scent for a prolonged period of time, or sniffing too many perfumes in a row, your odor receptors will habituate and stop sending those signals to your brain. This is officially called olfactory fatigue, and it explains why you might stop noticing a smell after a couple of minutes.

“If you smell a lot of scented materials, a lot of times your nose will just kind of conk out,” Aftel says. She keeps some wool nearby to help reset her sense of smell, and three big whiffs does the trick. So why does this work? Aftel says one theory is that the lanolin in wool absorbs and neutralizes odors, giving the brain a rest from sensory overload. As for those coffee beans you might see in some perfume shops? Those "definitely don't work," Aftel says.

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