Tag Archives: snowflakes

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

Math Meets Art, Quarto, and Snow!

Welcome to this week’s Math Munch!

article-0-19F9E81700000578-263_634x286… And, if you happen to write the date in the European way (day/month/year), happy Noughts and Crosses Day! (That’s British English for Tic-Tac-Toe Day.) In Europe, today’s date is 11/12/13– and it’s the last time that the date will be three consecutive numbers in this century! We in America are lucky. Our last Noughts and Crosses Day was November 12, 2013 (11/12/13), and we get another one next year on December 13 (12/13/14). To learn more about Noughts and Crosses Day and find out about an interesting contest, check out this site. And, to our European readers, happy Noughts and Crosses Day!

p3p13Speaking of Noughts and Crosses (or Tic-Tac-Toe), I have a new favorite game– Quarto! It’s a mix of Tic-Tac-Toe and another favorite game of mine, SET, and it was introduced to me by a friend of mine. It’s quite tricky– you’ll need the full power of your brain to tackle it. Luckily, there are levels, since it can take a while to develop a strategy. Give it a try, and let us know if you like it!


Looking to learn about some new mathematical artists? Check out this article, “When Math Meets Art,” from the online magazine Dark Rye. It profiles seven mathematical artists– some of whom we’ve written about (such as Erik and Martin Demaine, of origami fame, and Henry Segerman), and some of whom I’ve never heard of. The work of string art shown above is by artist Adam Brucker, who specializes in making “unexpected” curves from straight line segments.

gauss17_smallAnother of my favorites from this article is the work of Robert Bosch. One of his specialities is making mosaics of faces out of tiles, such as dominoes. The article features his portrait of the mathematician Father Sebastien Truchet made out of the tiles he invented, the Truchet tiles. Clever, right? The mosaic to the left is of the great mathematician Gauss, made out of dominoes. Check out Robert’s website to see more of his awesome art.

Finally, it snowed in New York City yesterday. I love when it snows for the first time in winter… and that got me wanting to make some paper snowflakes to celebrate! Here’s a video by Vi Hart that will teach you to make some of the most beautiful paper snowflakes.

Hang them on your windows, on the walls, or from the ceiling, and have a very happy wintery day! Bon appetit!

Mathpuzzle, Video Contests, and Snowflakes

Welcome to this week’s Math Munch!


One of my favorite math sites on the internet is mathpuzzle. It’s written and curated by recreational mathematician Ed Pegg Jr. About once a month, Ed makes a post that shares a ton of awesome math—interesting tilings, tricky puzzles, results about polyhedra and polyominos, and so much more. Below are some of my favorite finds at mathpuzzles. Go to the site to discover much more to explore!


Shapes that three kinds of polyominoes can tile.


Erich Friedman’s 2012 holiday puzzles


A slideable, flexible hypercube you can hold in your hands! Video below.


Next, have you ever wanted to be a movie star? How about a math movie star? Then there are two math video contests that you should know about. The first is for middle schoolers— the Reel Math Challenge. It’s run by MATHCOUNTS, which has for many years run a middle school problem solving contest. (I competed in it when I was in middle school.) This is only the second year for the Reel Math Challenge, but lots of videos have already been created. You can check them out here.

MathovisionThe second contest is for high schoolers and is called Math-O-Vision. The challenge is to make a video that shows “the way Math fills our world.” Math-O-Vision is sponsored by the Dartmouth College Math Department and the Neukom Institute.

makeaflakeFinally, here’s a fun little applet I found called Make-a-Flake. You can use it to make intricate digital snowflake designs.


Two snowflakes from the Make-a-Flake gallery.

Of course, it’s a lot of fun to make non-virtual snowflakes as well—find a pair of scissor and some paper and go for it! For basic instructions, head over to snowflakes.info. And for some inspiration, check out this Flickr group!

Bon appetit!