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.”

Amazon's Best Black Friday Deals: Tech, Video Games, Kitchen Appliances, Clothing, and More

Amazon
Amazon

This article contains affiliate links to products selected by our editors. Mental Floss may receive a commission for purchases made through these links.

Black Friday is finally here, and Amazon is offering great deals on kitchen appliances, tech, video games, and plenty more. We will keep updating this page as sales come in, but for now, here are the best Amazon Black Friday sales to check out.

Kitchen

Instant Pot/Amazon

- Instant Pot Duo Plus 9-in-115 Quart Electric Pressure Cooker; $90 (save $40)

- Keurig K-Cafe Special Edition; $190 (save $30)

- Ninja OS301 Foodi 10-in-1 Pressure Cooker and Air Fryer; $125 (save $75)

- Nespresso Vertuo Next Coffee and Espresso Machine by Breville; $120 (save $60)

- KitchenAid KSMSFTA Sifter with Scale Attachment; $95 (save $75)

- Keurig K-Mini Coffee Maker; $60 (save $20)

- Cuisinart Bread Maker; $80 (save $97)

- Anova Culinary Sous Vide Precision Cooker; $139 (save $60)

- Aicook Juicer Machine; $35 (save $15)

- JoyJolt Double Wall Insulated Espresso Mugs - Set of Two; $14 (save $10)

- Longzon Silicone Stretch Lids - Set of 14; $16 (save $11)

- HadinEEon Milk Frother; $37 (save $33)

Home Appliances

Roomba/Amazon

- iRobot Roomba 675 Robot Vacuum with Wi-Fi Connectivity; $179 (save $101)

- ASAKUKI 500ml Premium Essential Oil Diffuser; $22 (save $4)

- Facebook Portal Smart Video Calling 10 inch Touch Screen Display with Alexa; $129 (save $50)

- Bissell air320 Smart Air Purifier with HEPA and Carbon Filters; $280 (save $50)

- Oscillating Quiet Cooling Fan Tower; $59 (save $31)

- TaoTronics PTC 1500W Fast Quiet Heating Ceramic Tower; $55 (save $10)

- Vitamix 068051 FoodCycler 2 Liter Capacity; $300 (save $100)

- Ring Video Doorbell; $70 (save $30)

Video games

Sony

- Marvel's Spider-Man: Game of The Year Edition for PlayStation 4; $20 (save $20)

- The Legend of Zelda: Link's Awakening; $40 (save $20)

- Hyrule Warriors: Age of Calamity; $50 (save $10)

- Marvel's Avengers; $25 (save $33)

- The Last of Us Part II for PlayStation 4; $30 (save $30)

- LEGO Harry Potter: Collection; $15 (save $15)

- Ghost of Tsushima; $40 (save $20)

- BioShock: The Collection; $20 (save $30)

- The Sims 4; $24 (save $20)

- God of Warfor PlayStation 4; $10 (save $10)

- Days Gonefor PlayStation 4; $20 (save $6)

- Luigi's Mansion 3 for Nintendo Switch; $40 (save $20)

Computers and tablets

Microsoft/Amazon

- New Apple MacBook Pro 16 inches with 512 GB; $2149 (save $250)

- Microsoft Surface Laptop 3 with 13.5 inch Touch-Screen; $1200 (save $400)

- Lenovo ThinkPad T490 Laptop; $889 (save $111)

- Amazon Fire HD 10 Tablet (64GB); $120 (save $70)

- Amazon Fire HD 10 Kids Edition Tablet (32 GB); $130 (save $70)

- Apple iPad Mini (64 GB); $335 (save $64)

- Vankyo MatrixPad S2 Tablet; $120 (save $10)

Tech, gadgets, and TVs

Apple/Amazon

- Apple Watch Series 3 with GPS; $120 (save $79)

- Seneo Wireless Charger, 3 in 1 Wireless Charging Station; $16 (save $10)

- SAMSUNG 75-inch Class Crystal 4K Smart TV; $998 (save $200)

- Nixplay 2K Smart Digital Picture Frame 9.7 Inch Silver; $238 (save $92)

- All-New Amazon Echo Dot with Clock and Alexa (4th Gen); $39 (save $21)

- MACTREM LED Ring Light 6" with Tripod Stand; $16 (save $3)

- Amazon Fire TV Stick with Alexa Voice Remote; $28 (save $12)

- DR. J Professional HI-04 Mini Projector; $93 (save $37)

Headphones and speakers

Beats/Amazon

- Beats Solo3 Wireless On-Ear Headphones; $120 (Save $80)

- Apple AirPods Pro; $169 (save $50)

- Anker Soundcore Upgraded Bluetooth Speaker; $22 (save $8)

- Powerbeats Pro Wireless Earphones; $175 (save $75)

- JBL Boombox; $280 (save $120)

Movies and TV

HBO/Amazon

- Game of Thrones: The Complete Series; $115 (save $89)

- Jurassic World 5-Movie Set; $23 (save $37)

- Deadwood: The Complete Series; $42 (save $28)

- Back to the Future Trilogy; $15 (save $21)

Toys and Games

Amazon

- Awkward Family Photos Greatest Hits; $15 (save $10)

- Exploding Kittens Card Game; $10 (save $10)

- Cards Against Humanity: Hidden Gems Bundle; $14 (save $5)

- LOL Surprise OMG Remix Pop B.B. Fashion Doll; $29 (save $6)

- LEGO Ideas Ship in a Bottle 92177 Expert Building Kit; $56 (save $14)

Furniture

Casper/Amazon

- Casper Sleep Element Queen Mattress; $476 (save $119)

- ZINUS Alexis Deluxe Wood Platform Bed Frame; $135 (save $24)

- ROMOON Dresser Organizer with 5 Drawers; $59 (save $11) 

- AmazonBasics Room Darkening Blackout Window Curtains; $26 (save $5)

- Writing Desk by Caffoz; $119 (save $21)

- SPACE Seating Office Support Managers Chair; $112 (save $116)

- Rivet Globe Stick Table Lamp; $53 (save $17)

- Christopher Knight Home Merel Mid-Century Modern Club Chair; $188 (save $10)

- Walker Edison Furniture Industrial Rectangular Coffee Table; $121 (save $48)

Beauty

Haus/Amazon

- MySmile Teeth Whitening Kit with LED Light; $21 (save $12) 

- Cliganic USDA Organic Lip Balms Set of Six; $6 (save $4)

- HAUS LABORATORIES By Lady Gaga: LE RIOT LIP GLOSS; $7 (save $11)

- Native Deodorant for Men and Women Set of Three; $25 (save $11) 

- BAIMEI Rose Quartz Jade Roller & Gua Sha; $14 (save $3)

- Honest Beauty Clearing Night Serum with Pure Retinol and Salicylic Acid; $20 (save $8)

- WOW Apple Cider Vinegar Shampoo and Hair Conditioner Set; $30 (save $5) 

- La Roche-Posay Effaclar Purifying Foaming Gel Cleanser; $15 (save $5)

- wet n wild Bretman Rock Shadow Palette; $9 (save $6)

- EltaMD UV Daily Tinted Face Sunscreen Moisturizer with Hyaluronic Acid; $25 (save $6)

Clothes

Ganni/Amazon

- Ganni Women's Crispy Jacquard Dress; $200 (save $86) 

- The Drop Women's Maya Silky Slip Skirt; $36 (save $9)

- Steve Madden Women's Editor Boot; $80 (save $30)

- adidas Women's Roguera Cross Trainer; $40 (save $25)

- Line & Dot Women's Elizabeth Sweater; $74 (save $18)

- Levi's Men's Sherpa Trucker Jacket; $57 (save $41)

- Adidas Men's Essentials 3-Stripes Tapered Training Joggers Sweatpants; $28 (save $12)

- Timex Men's Weekender XL 43mm Watch; $32 (save $20)

- Ray-Ban Unisex-Adult Hexagonal Flat Lenses Sunglasses; $108 (save $46) 

- Reebok Men's Flashfilm Train Cross Trainer; $64 (save $16)

Sign Up Today: Get exclusive deals, product news, reviews, and more with the Mental Floss Smart Shopping newsletter!

New Online Art Exhibition Needs the Public’s Help to Track Down Lost Masterpieces by Van Gogh, Monet, and More

Vincent van Gogh's original Portrait of Dr. Gachet wasn't stolen, but it hasn't been seen in 30 years.
Vincent van Gogh's original Portrait of Dr. Gachet wasn't stolen, but it hasn't been seen in 30 years.
Vincent van Gogh, Wikimedia Commons // Public Domain

If you wanted to compare both versions of Vincent van Gogh’s Portrait of Dr. Gachet in person, you couldn’t. While the second one currently hangs in Paris’s Musée d'Orsay, the public hasn’t seen the original painting since 1990. In fact, nobody’s really sure where it is—after its owner Ryoei Saito died in 1996, the precious item passed from private collector to private collector, but the identity of its current owner is shrouded in mystery.

As Smithsonian Magazine reports, Portrait of Dr. Gachet (1890) is one of a dozen paintings in “Missing Masterpieces,” a digital exhibit of some of the world’s most famous lost artworks. It’s not the only Van Gogh in the collection. His 1884 painting The Parsonage Garden at Nuenen in Spring was snatched from the Netherlands’ Singer Laren museum earlier this year; and his 1888 painting The Painter on His Way to Work has been missing since World War II. Other works include View of Auvers-sur-Oise by Paul Cézanne, William Blake’s Last Judgement, and two bridge paintings by Claude Monet.

Paul Cézanne's View of Auvers-sur-Oise was stolen from the University of Oxford's art museum on New Year's Eve in 1999.Ashmolean Museum, Wikimedia Commons // Public Domain

The new online exhibit is a collaboration between Samsung and art crime expert Noah Charney, who founded The Association for Research into Crimes Against Art. It isn’t just a page where art enthusiasts can explore the stories behind the missing works—it’s also a way to encourage people to come forward with information that could lead to the recovery of the works themselves.

“From contradictory media reports to speculation in Reddit feeds—the clues are out there, but the volume of information can be overwhelming,” Charney said in a press release. “This is where technology and social media can help by bringing people together to assist the search. It’s not unheard of for an innocuous tip posted online to be the key that unlocks a case.”

The exhibition will be online through February 10, 2021, and citizen sleuths can email their tips to missingmasterpieces@artcrimeresearch.org.

[h/t Smithsonian Magazine]