Monthly Archives: January 2015

Squircles, Coloring Books, and Snowfakes

Welcome to this week’s Math Munch!

Squares and circles are pretty different. Squares are boxy and have their feet firmly on the ground. Circles are round and like to roll all over the place.



Since they’re so different, people have long tried to bridge the gap between squares and circles. There’s an ancient problem called “squaring the circle” that went unsolved for thousands of years. In the 1800s, the gap between squares and circles was explored by Gabriel Lamé. Gabriel invented a family of curves that both squares and circles belong to. In the 20th century, Danish designer Piet Hein gave Lamé’s family of curves the name superellipses and used them to lay out parts of cities. One particular superellipse that’s right in the middle is called a squircle. Squircles have been used to design everything from dinner plates to touchpad buttons.

The space of superellipsoids.

The space of superellipsoids.

Piet had the following to say about the gap between squares and circles:

Things made with straight lines fit well together and save space. And we can move easily — physically or mentally — around things made with round lines. But we are in a straitjacket, having to accept one or the other, when often some intermediate form would be better. … The super-ellipse solved the problem. It is neither round nor rectangular, but in between. Yet it is fixed, it is definite — it has a unity.

"Squaring the Circle" by Troika.

“Squaring the Circle” by Troika.

These circles aren't what they seem to be.

These circles aren’t what they seem to be.

There’s another kind of squircular object that I ran across recently. It’s a sculpture called “Squaring the Circle”, and it was created by a trio of artists known as Troika. Check out the images on this page, and then watch a video of the incredible transformation. You can find more examples of room-sized perspective-changing objects in this article.

Next up: it’s been a snowy week here on the east coast, so I thought I’d share some ideas for a great indoor activity—coloring!

Marshall and Violet.

Marshall and Violet.

Marshall Hampton is a math professor at University of Minnesota, Duluth. Marshall studies n-body problems—a kind of physics problem that goes all the way back to Isaac Newton and that led to the discovery of chaos. He also uses math to study the genes that cause mammals to hibernate. Marshall made a coloring book full of all kinds of lovely mathematical images for his daughter Violet. He’s also shared it with the world, in both pdf and book form. Check it out!

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Inspired by Mashrall’s coloring book, Alex Raichev made one of his own, called Contours. It features contour plots that you can color. Contour plots are what you get when you make outlines of areas that share the same value for a given function. Versions of contour plots often appear on weather maps, where the functions are temperature, atmospheric pressure, or precipitation levels.

Contour plots are useful. Alex shows that they can be beautiful, too!

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And there are even more mathematical patterns to explore in the coloring sheets at Patterns for Colouring.

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Last up, that’s not a typo in this week’s post title. I really do want to share some snowfakes with you—some artificial snowflake models created with math by Janko Gravner and David Griffeath. You can find out more by reading this paper they authored, or just skim it for the lovely images, some of which I’ve shared below.

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I ran across these snowfakes at the Mathematical Imagery page of the American Mathematical Society. There are lots more great math images to explore there.

Bon appetit!

Reflection sheet – Squircles, Coloring Books, and Snowfakes

Harriss Spiral, Math Snacks, and SET

Happy New Year, and welcome to this week’s Math Munch!

The Harriss Spiral

Exciting news, folks! The Golden Ratio curve– that beautiful spiral that everyone adores– has evolved. And not into a freak of nature, either! Into something– dare I say it?– even more beautiful…

Meet the Harriss spiral. It was discovered/invented by mathematician and artist Edmund Harriss (featured before here and here) when he began playing around with golden rectangles. A golden rectangle is a very special rectangle, whose sides are in a particular proportion. You can read more about them here— but what’s most important to this new discovery is what you can do with them. If you make a square inside a golden rectangle you get another golden rectangle– and continuing to make squares and new golden rectangles inside of ever-shrinking golden rectangles, and drawing arcs through the squares, is one way to make the beautiful golden ratio spiral.

Edmund Harriss decided to get creative. What would happen, he wondered, if he cut the golden rectangle into two similar rectangles (same shape, just one is a scaled-down version of the other) and a square? And then what if he did the same thing to the new rectangles, again and again to make a fractal? Edmund’s new golden rectangle fractal makes this pattern, and when you draw a spiral through it, you get a lovely branching shape.

But don’t take my word for it. Math journalist Alex Bellos broke the news just this week in his article in The Guardian. His article explains much, much more than I can here– check it out to learn many more wonderful things about the Harriss spiral (and other spirals that Harriss has created…)!

(Bonus: Here’s a GeoGebra demonstration created by John Golden that builds the Harriss Spiral. It’s awesome!)

Next up is a site that sounds quite a lot like Math Munch. But it’s all games and cartoons, all the time. (Maybe that means you’ll like it better…) Check out Math Snacks, a site developed by a group of math educators at New Mexico State University. They worked hard to create games and animations that are both fun and full of interesting math.

One of my favorite games on Math Snacks is called Game Over Gopher.In this game, you have to save your carrot from an army of gophers by placing little machines that feed the gophers. Where’s the math, you may wonder? Placing the gopher-feeders and the other equipment that can help you save the carrot requires you to think carefully about geometry and coordinates.

Finally, speaking of games, here’s one of my favorites. I love to play SET, and I recently found a way to play online– either against a friend or against the computer. Click on this link to start your own game!

To play SET, you deal 12 cards. Then, you try to find a group of 3 cards that all share and all don’t share the same characteristics. For example, in the picture to the right– do you see the cards with the empty red ovals? They’re all the same shape, shading, and color (oval, empty, red), but they’re all different numbers (1, 2, and 3). Can you find any other sets in the picture? (Hint: One involves purple.)

Want to hone your SET skills without competing? Here’s a daily SET puzzle to challenge you.

Enjoy the games (and maybe invent a spiral of your own) and bon appetit!