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.”

This $49 Video Game Design Course Will Teach You Everything From Coding to Digital Art Skills

EvgeniyShkolenko/iStock via Getty Images
EvgeniyShkolenko/iStock via Getty Images

If you spend the bulk of your free time playing video games and want to elevate your hobby into a career, you can take advantage of the School of Game Design’s lifetime membership, which is currently on sale for just $49. You can jump into your education as a beginner, or at any other skill level, to learn what you need to know about game development, design, coding, and artistry skills.

Gaming is a competitive industry, and understanding just programming or just artistry isn’t enough to land a job. The School of Game Design’s lifetime membership is set up to educate you in both fields so your resume and work can stand out.

The lifetime membership that’s currently discounted is intended to allow you to learn at your own pace so you don’t burn out, which would be pretty difficult to do because the lessons have you building advanced games in just your first few hours of learning. The remote classes will train you with step-by-step, hands-on projects that more than 50,000 other students around the world can vouch for.

Once you’ve nailed the basics, the lifetime membership provides unlimited access to thousands of dollars' worth of royalty-free game art and textures to use in your 2D or 3D designs. Support from instructors and professionals with over 16 years of game industry experience will guide you from start to finish, where you’ll be equipped to land a job doing something you truly love.

Earn money doing what you love with an education from the School of Game Design’s lifetime membership, currently discounted at $49.

 

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This 10-Year-Old Is Sending Art Supplies to Hundreds of Kids in Homeless Shelters and Foster Homes

Evgeniia Siiankovskaia/iStock via Getty Images
Evgeniia Siiankovskaia/iStock via Getty Images

She may be stuck at home, but Chelsea Phaire has found a way to connect with hundreds of kids during the COVID-19 pandemic. As CNN reports, the 10-year-old from Danbury, Connecticut, has used her time in isolation to send 1500 art project packs to kids in foster homes and homeless shelters.

Phaire had been interested in starting a charity from a young age, and on her birthday in August 2019, she launched Chelsea's Charity with help from her parents. Instead of birthday gifts, Chelsea asked for art supplies, and all the items she received went to a homeless shelter in New York. The Phaires have since set up a wishlist on Amazon, so anyone can donate supplies for the art kits. One pack includes crayons, paper, markers, gel pens, coloring books, and colored pencils.

In recent months, Phaire's mission to provide resources to underserved kids has become more vital than ever. Schools around the country have closed to stop the spread of the novel coronavirus, which means kids have less access to art supplies than they did before. Young people may also be dealing with increased stress and boredom from being isolated inside. By sharing art kits, Phaire hopes to give them a healthy outlet for their struggles.

Chelsea's Charity has donated more than 1500 kits to schools, shelters, and foster homes since stay-at-home orders rolled out in March, which is more than was donated in the initiative's first five months. COVID-19 has forced Phaire to do some things differently: While she would normally get to meet many of the people she helps in person, she now sends all her donations by mail. Until it's safe to travel again, she's staying connected to kids through social media, as you can see in the video below.

[h/t CNN]