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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.”

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Art
The Getty Center, Surrounded By Wildfires, Will Leave Its Art Where It Is
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The wildfires sweeping through California have left countless homeowners and businesses scrambling as the blazes continue to grow out of control in various locations throughout the state. While art lovers worried when they heard that Los Angeles's Getty Center would be closing its doors this week, as the fires closed part of the 405 Freeway, there was a bit of good news. According to museum officials, the priceless works housed inside the famed Getty Center are said to be perfectly secure and won't need to be evacuated from the facility.

“The safest place for the art is right here at the Getty,” Ron Hartwig, the Getty’s vice president of communications, told the Los Angeles Times. According to its website, the museum was closed on December 5 and December 6 “to protect the collections from smoke from fires in the region,” but as of now, the art inside is staying put.

Though every museum has its own way of protecting the priceless works inside it, the Los Angeles Times notes that the Getty Center was constructed in such a way as to protect its contents from the very kind of emergency it's currently facing. The air throughout the gallery is filtered by a system that forces it out, rather than a filtration method which would bring air in. This system will keep the smoke and air pollutants from getting into the facility, and by closing the museum this week, the Getty is preventing the harmful air from entering the building through any open doors.

There is also a water tank at the facility that holds 1 million gallons in reserve for just such an occasion, and any brush on the property is routinely cleared away to prevent the likelihood of a fire spreading. The Getty Villa, a separate campus located in the Pacific Palisades off the Pacific Coast Highway, was also closed out of concern for air quality this week.

The museum is currently working with the police and fire departments in the area to determine the need for future closures and the evacuation of any personnel. So far, the fires have claimed more than 83,000 acres of land, leading to the evacuation of thousands of people and the temporary closure of I-405, which runs right alongside the Getty near Los Angeles’s Bel-Air neighborhood.

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This 77-Year-Old Artist Saves Money on Art Supplies by 'Painting' in Microsoft Excel
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It takes a lot of creativity to turn a blank canvas into an inspired work of art. Japanese artist Tatsuo Horiuchi makes his pictures out of something that’s even more dull than a white page: an empty spreadsheet in Microsoft Excel.

When he retired, the 77-year-old Horiuchi, whose work was recently spotlighted by Great Big Story, decided he wanted to get into art. At the time, he was hesitant to spend money on painting supplies or even computer software, though, so he began experimenting with one of the programs that was already at his disposal.

Horiuchi's unique “painting” method shows that in the right hands, Excel’s graph-building features can be used to bring colorful landscapes to life. The tranquil ponds, dense forests, and blossoming flowers in his art are made by drawing shapes with the software's line tool, then adding shading with the bucket tool.

Since picking up the hobby in the 2000s, Horiuchi has been awarded multiple prizes for his creative work with Excel. Let that be inspiration for Microsoft loyalists who are still broken up about the death of Paint.

You can get a behind-the-scenes look at the artist's process in the video below.

[h/t Great Big Story]

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