Tag Archives: doodling

Circling, Squaring, and Triangulating

Welcome to this week’s Math Munch!

How good are you at drawing circles? To find out, try this circle drawing challenge. There are adorable cat pictures for prizes!

What’s the best score you can get? And hey—what’s the worst score you can get? And how is your score determined? Well, no matter how long the path you draw is, using that length to make a circle would surround the most area. How close your shape gets to that maximum area determines your score.

Do you think this is a good way to measure how circular a shape is? Can you think of a different way?

Dido, Founder and Queen of Carthage.

Dido, Founder and Queen of Carthage.

This idea that a circle is the shape that has the biggest area for a fixed perimeter reminds me of the story of Dido and her famous problem. You can find a retelling of it at Mathematica Ludibunda, a charming website that’s home to all sorts of mathematical stories and puzzles. The whole site is written in the voice of Rapunzel, but there’s a team of authors behind it all. Dido’s story in particular was written by a girl named Christa.

If you have any trouble drawing circles in the applet, you might try using pencil and paper or a chalkboard. I bet if you practice your circling and get good at it, you might even be able to challenge this fellow:

The simple perfect squared square of smallest order.

The simple perfect squared square
of smallest order.

Next up is squaring and the incredible Squaring.Net. The site is run by Stuart Anderson, who works at the Reserve Bank of Australia and lives in Sydney.

The site gathers together all of the research that’s been done about breaking up squares and rectangles into squares. It’s both a gallery and an encyclopedia. I love getting to look at the timelines of discovery—to see the progress that’s been made over time and how new things have been discovered even this year! Just within the last month or so, Stuart and Lorenz Milla used computers to show that there are 20566 simple perfect squared squares of order 30. Squaring.Net also has a wonderful links page that can connect you to more information about the history of squaring, as well as some of the delightful mathematical art that the subject has inspired.

trinity-glass2-small sqBox8 wp4f6b3871_0f

Delaunay triangulationLast up this week is triangulating. There are lots of ways to chop up a shape into triangles, and so I’ll focus on one particular way known as a Delaunay triangulation. To make one, scatter some points on the plane. Then connect them up into triangles so that each triangle fits snugly into a circle that contains none of the scattered points.

Fun Fact #1: Delaunay triangulations are named for the Soviet mathematician Boris Delaunay. What else is named for him? A mountain! That’s because Boris was a world-class mountain climber.

Fun Fact #2: The idea of Delaunay triangulations has been rediscovered many times and is useful in fields as diverse as computer animation and engineering.

Here are two uses of Delaunay triangulations I’d like to share with you. The first comes from the work of Zachary Forest Johnson, a cartographer who shares his work at indiemaps.com. You can check out a Delaunay triangulation applet that he made and read some background about this Delaunay idea here. To see how Zach uses these triangulations in his map-making, you’ve gotta check out the sequence of images on this page. It’s incredible how just a scattering of local temperature measurements can be extended to one of those full-color national temperature maps. So cool!


Zachary Forest Johnson

A Delaunay triangulation used to help create a weather map.

A Delaunay triangulation used to help create a weather map.

Finally, take a look at these images that Jonathan Puckey created. Jonathan is a graphic artist who lives in Amsterdam and shares his work on his website. In 2008 he invented a graphical process that uses Delaunay triangulations and color averaging to create abstractions of images. You can see more of Jonathan’s Delaunay images here.

 armandmevis-1  fox

I hope you find something to enjoy in these circles, squares, and triangles. Bon appetit!

A Closet Full of Puzzles, Sphereland, and Math Doodles

Welcome to this week’s Math Munch!

After a few weeks off, we’re back with some exciting things to share.  First up is Futility Closet, a blog featuring “an idler’s miscellany of compendious amusements.”  The blog is full of big-worded phrases like that, but I most love the puzzles they often post – everything from chess to numbers, codes, and devilish word play.  I also love that the name of the person who wrote each puzzle accompanies it.  Take a look at the few I’ve posted below and click here for the full list of puzzles.

Here’s a puzzle called Swine Wave, by Lewis Carroll. The puzzle: Lace 24 pigs in these sties so that, no matter how many times one circles the sties, he always find that the number in each sty is closer to 10 than the number in the previous one. Want to know the solution? Click on the image above to visit Futility Closet.
This puzzle is called Project Management, by Paul Vaderlind. The question: If a blacksmith requires five minutes to put on a horseshoe, can eight blacksmiths shoe 10 horses in less than half an hour? The catch: A horse can stand on three legs, but not on two. Click on the image to visit Futility Closet for the solution!

Next, have you ever wondered what it would be like to visit another dimension?   In 1884, Edwin A. Abbott wrote about life in the second dimension, in a nice little book called Flatland: A Romance of Many Dimesnions.  (Fun fact: the “A” in Edwin’s name stands for Abbott.  So his name is Edwin Abbott Abbott.)  Click on that link and you can read the whole book, if you like.  The book is about a world of flat beings who have no idea that the third dimension exists.  In the book, the main character, A Square, is visited by a sphere from the unknown world “above” him.  Kind of makes me wonder whether we’re just like the characters in Flatland, three-dimensional creatures ignorant of the fourth dimension that exists “above” us…

spherelandWell, the recently released movie Flatland 2: Sphereland deals with precisely that issue.  The Math Munch team had the opportunity to preview this movie, and we loved it.  In Sphereland, the granddaughter of the Square from Flatland, Hex, and her friend Puncto try to understand some mysterious triangles that Puncto thinks will cause the disastrous end of a space exploration mission and go on an adventure to help their three-dimensional friend Spherius with a problem he brought back from the fourth dimension.

portfolio-TorusHigher dimensions can be very difficult to wrap your head around.  This movie does a great job of helping the movie-watcher to understand how higher and lower dimensions relate to each other through the plot twists and challenges that the characters face.  You can really learn a lot about dimensions and the shape of space by watching this movie.  Plus, the characters are engaging and the images are fun.  Sphereland features the voices of a number of really great actors, including Kristen Bell, Danny Pudi, Michael York, and Danica McKellar.

Want to learn more about Sphereland?  Check out the trailer:

And, here’s an interview with Danny Pudi, the voice of Puncto, and Tony Hale, who does a fantastic job as the King of Pointland:

By the way, the makers of Sphereland also made a movie of Flatland!  The Math Munch team loved that one, too.  Here’s a link to the trailer.

tumblr_mgw2ainZDX1s0payeo1_1280Finally, check out this beautiful blog of mathematical doodles by high school math student and artist Chloé Worthington!  Chloé started mathematically doodling a few years ago in… well, in class.  When she doodles in class, Chloé is better able to focus on what’s going on and makes beautiful art.   (We at Math Munch encourage you to pay attention in class while you doodle.)

Chloé does all of her doodles by hand with ink pens.  She does a lot of work with triangles, as shown here.  One of her signature doodles is this nested puzzle piece doodle:


Doodling mathematically is one of the ways that Chloé does math and shares what she loves about it with the world.  She’s a trigonometry student, too.  How do you share what you love about math – or any other subject?

Bon appetit!

Faces, Blackboards, and Dancing PhDs

Welcome to this week’s Math Munch!

What does a mathematician look like? What does a mathematician do? Here are a couple of things I ran across recently that give a window into what it’s like to be a professional research mathematician—someone who works on figuring out new math as their job.

Gary Davis, who blogs over at Republic of Mathematics, recently posted a short piece that challenges stereotypes about mathematicians. It’s called What does a mathematician look like?

Who here is a mathematician? Click through to find out!

Gary’s point is that you can’t tell who is or isn’t a mathematician just by looking at them. Mathematicians come from every background and heritage. Gary followed up on this idea in another post where he highlighted some notable mathematicians who are black women. Here’s a website called Black Women in Mathematics that shares some biographies and history. And here’s a link to the Infinite Possibilities Conference, a yearly gathering “designed to promote, educate, encourage and support minority women interested in mathematics and statistics.” Suzanne Weekes, one of the five mathematicians pictured above, was a speaker at this conference in 2010.

Richard Tapia, another of the mathematicians above, is featured in the following video. His life story both inspires and delights.

And what does this diversity of mathematicians do all day? Well, one thing they do is talk to each other about math! And though there are many new technologies that help people to do and share and collaborate on mathematics (like blogs!), it’s hard to beat a handy chalkboard as a scribble pad for sharing ideas.

At Blackboard of the Day, Mathieu Rémy and Sylvain Lumbroso share the results of these impromptu math jam sessions. Every day they post a photograph of a blackboard covered in doodles and calculations and sketches of ideas. The website is in French, but the mathematical pictures are a universal language.

Diana Davis, putting the finishing touches on a blackboard masterpiece

Sharing mathematical ideas can take many forms, and sometimes choosing the right medium can make all the difference. Mathematicians use pictures, words, symbols, sculptures, movies, songs—even dances! Let me point you to the “Dance your Ph.D.” Contest. It’s exactly what it sounds like—people sharing the ideas of their dissertations (their first big piece of original work) through dance. Entries come in from physicists, chemists, biologists, and more.  Below you’ll find an entry by Diana Davis, a mathematician who completed her dissertation at Brown University this past spring. Diana often studies regular polgyons and especially ways of “dissecting” them—breaking them up into pieces in interesting ways.

Thanks to The Aperiodical—a great math blog—for sharing Diana’s wonderful video!

Some pages from Diana’s notebooks

All kinds of mathematicians study math and share it in so many ways. It’s like a never-ending math buffet!

Bon appetit!