Tag Archives: physics

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

Visualizations, Inspirations, and the Super Ultimate Graphing Challenge

Welcome to this week’s Math Munch!

Jason Davies

Meet Jason Davies, a freelance mathematician living in the UK. Growing up in Wales (one of the 4 countries of the United Kingdom) his classes were taught in Welsh. This makes Jason one of only about 611,000 people that speak the language, only 21.7% of the population of Wales! Imagine if only 1/5 of France spoke French!! These statistics are from a 2004 study, so the numbers may have changed a bit, but they still say something interesting don’t they?

Prime Seive

Jason is all about what numbers and pictures can tell us.  Since graduating from Cambridge, he’s been doing all sorts of data visualization and computer science on his own for various companies and IT firms. I originally found Jason through a link to his Prime Seive visualization, but take a look at his gallery and you’re bound to find something beautiful, interesting, interactive, and cool. I’ve linked to some of my favorites below.

Interactive Apollonian Gasket

Rhodonea Curves

Set Partitions

I asked Jason a few questions about his interest in data visualization and math in general. Here’s a tasty little excerpt:

MM: What’s the most important trait for a mathematician to have? Is there one?

JD: Persistance is always useful in maths! I think the stereotype is to be analytical and logical, but in fact there are many other traits that are highly important, for instance communication skills. Mathematics is passed on from person to person, after all, so being able to communicate ideas effectively is dynamite.

MM: Do you have a message you’d like to give to young mathematicians?

JD: The world needs you!

Read the rest in our Q&A with Jason Davies, and you can see all of our interviews on the Q&A page we’ve just created.

Up next, a beautiful and inspiring video from Spain. The video is actually called Insprations, and it comes to us from Etérea Studios, the online home of animator Cristóbal Vila. In the intro he says, “I looked into that enormous and inexhaustible source of inspiration that is Escher and tried to imagine how it could be his workplace, what things would surround an artist like him, so deeply interested in science in general and mathematics in particular.”

I’d die to have an office like this!

It gets better.  Cristóbal added a page explaining all of the wonderful maths in the video. Click to read about Platonic solids, tilings, tangrams, and various works of art by M.C. Escher.

Finally, a nifty new game that explores the relationship between graphs and different kinds of motion. Super Ultimate Graphing Challenge is a game developed by Physics teacher Matthew Blackman to help his students understand the physics and mathematics of motion. You might not understand it all when you start, but keep playing and see what you can make of it. If you need a bit of help or have something to say, post it in our comments, and we’ll happily reply.

Bon appetit!