Monthly Archives: June 2012

Turing, Nets, and More Yoshimoto

Welcome to this week’s Math Munch!

The Turing Tenner

What you see there is a 10 pound note. You know, British money. So who’s that guy on there? Must be a president or king or prime minister or something, right? NO! That’s Alan Turing, one of the most important mathematicians of the 20th century. During WWII, he was a codebreaker for the Allies, intercepting German submarine codes. His analysis of the Enigma Machine was a huge turning point in the war. (video explanation)

In England they put the queen on one side of the money, but the other’s used for significant Brits. Charles Darwin is currently on the 10 pound note, but these things change, and there’s a petition to get Turing on the ten. A Turing Tenner, as they call it. It’s all part of Turing’s 100th birthday celebration.

Google’s homage to Alan Turing

Since Turing did some of the earliest work on computing theory and artificial intelligence, Google paid tribute to the computer legend with a recent doodle. It’s a fantastic little puzzle game based on his work. I’ll let you figure it out, but definitely try this one. Click here to play!

In last week’s munch, Justin introduced us to the Yoshimoto Cube, and we’ve kept on thinking about it.  Here’s a couple simple templates for making one cubelet.  (template 1, template 2)  Make 8 of those and hinge them together with some tape.  I made a short video to show you how to connect them.  But it didn’t end there!

A flat template for a 3D model like that is called a net or a mesh.  Do you know any nets for a cube?  There’s actually lots.  Check out this site, where it’s your job to figure out which nets fold up into a cube and which ones don’t.  It’s a lot of fun.  Here’s another net site showing lots of nets for a pyramid, dodecahedron, and a whole bunch of other solid shapes.  How many do you think there are for a tetrahedron?  Can you design one for an octahedron?

The Monster Mesh

I spent some time this week trying to design a better net for the

The Mega-Monster Mesh
A one-sheet model for the Yoshimoto cube.

Yoshimoto cube, and I think I succeeded!  The tape on my hinges kept breaking, so I wanted to try to make paper hinges.  With my first attempt, which I called The Monster Mesh, I was able to design a net for half of the star.  Down from 8 tape hinges to 2 was a big improvement, but last night I got it perfect!  Using my new version, The Mega-Monster Mesh,  you can make the entire cube without any taped hinges!  The model is pretty complicated, so if you want to give it a shot, feel free to email us at with any questions.

Finally, something I’m really really proud of.  Justin and I spent most of Sunday afternoon on the floor of my apartment making a stop motion animation of with Yoshimoto Cube models.  It’s called “Yoshimoto Friends,” and we hope you love it as much as we do.  (We used the free iMotionHD app for iPad and iPhone, in case you want to make your own stop motion animation.)

Bon appetit!


I made another video showing how the mega-monster mesh folds up.  Here it is, acting like a transforming bug!

Music Box, FatFonts, and the Yoshimoto Cube

Welcome to this week’s Math Munch!

The Whitney Music Box

Jim Bumgardner

Solar Beat

With the transit of Venus just behind us and the summer solstice just ahead, I’ve got the planets and orbits on my mind. I can’t believe I haven’t yet shared with you all the Whitney Music Box. It’s the brainchild of Jim Bumgardner, a man of many talents and a “senior nerd” at Disney Interactive Labs. His music box is one of my favorite things ever–so simple, yet so mesmerizing.

It’s actually a bunch of different music boxes–variations on a theme. Colored dots orbit in circles, each with a different frequency, and play a tone when they come back to their starting points. In Variation 0, for instance, within the time it takes for the largest dot to orbit the center once, the smallest dot orbits 48 times. There are so many patterns to see–and hear! There are 21 variations in all. Go nuts! In this one, only prime dots are shown. What do you notice?

You can find a more astronomical version of this idea at SolarBeat.

Above you’ll find a list of the numerals from 1 to 9. Or is it 0 to 9?

Where’s the 0 you ask? Well, the idea behind FatFonts is that the visual weight of a number is proportional to its numerical size. That would mean that 0 should be completely white!

FatFonts can also be nested. The first number below is 64. Can you figure out the second?

This is 64 in FatFonts.

What number is this?
Click to zoom!

FatFonts was developed by the team of Miguel NacentaUta Hinrichs, and Sheelagh Carpendale. You can see some uses that FatFonts has been put to on their Gallery page, and even download FatFonts to use in your word processor. Move over, Times New Roman!

This past week, Paul pointed me to this cool video by George Hart about interlocking complementary polyhedra that together form a cube. It reminded me of something I saw for the first time a few years ago that just blew me away. You have to see the Yoshimoto Cube to believe it:

In addition to its more obvious charms, something that delights me about the Yoshimoto Cube is how it was found so recently–only in 1971, by Naoki Yoshimoto.  (That other famous cube was invented in 1974 by Ernő Rubik.) How can it be that simple shapes can be so inexhaustible? If you’re feeling inspired, Make Magazine did a short post on the Yoshimoto Cube a couple of years that includes a template for making a Yoshimoto Cube out of paper. Edit: These template and instructions aren’t great. See below for better ones!

Since it’s always helpful to share your goals to help you stick to them, I’ll say that this week I’m going to make a Yoshimoto Cube of my own. Begone, back burner! Later in the week I’ll post some pictures below. If you decide to make one, share it in the comments or email us at

We’d love to hear from you.

Bon appetit!


Here are the two stellated rhombic dodecahedra that make the Yoshimoto Cube that Paul and I made! Templates, instructions, and video to follow!

Here are two different templates for the Yoshimoto cubelet. You’ll need eight cubelets to make one star.

And here’s how you tape them together:

Knots, Torus Games, and Bagels

Welcome to this week’s Math Munch!

The things we have lined up for you this week have to do with a part of math called topology.  Topology is like geometry in many ways, except the shapes you study aren’t rigid.  Instead, you can twist, stretch, squish, and generally deform them in any way you like, so long as you don’t rip any holes or attach things that weren’t already attached.  One of the reasons why topology is interesting is that you get to play with new and fascinating shapes, like…

… knots!  This nifty site, Knot Theory Online, is full of interesting information about the study of mathematical knots and its history and applications.  For some basic information, check out the introduction to knots page.  It talks about what a knot is, mathematically speaking, and some ways that mathematicians answer the most important question in knot theory: is this knot the unknot?  The site also has some fun games in which you can play with transforming one knot into another.  Here’s my favorite: The Hunt for the Elusive Trefoil Knot.

Knots can also be works of art – and this site, Knot Plot, showcases artistic knots at their best.  Here are some images of beautiful decorative knots.

A really cool thing about knot theory is that it is a relatively new area of mathematical research – which means that there are many unsolved knot theory problems that a person without a lot of math training could attempt!  Here’s a page of “approachable open problems in knot theory,” compiled by knot theorist and Williams College professor Colin Adams.

One of a topologist’s favorite objects to study is one that you might encounter at breakfast – the torus, or donut (or bagel).  To get a sense for what makes a torus topologically interesting and for what life might be like if you lived on a torus (instead of a sphere, a topologically different surface), check out Torus Games.  Torus Games was created by mathematician Jeff Weeks.  You can play games that you’d typically play on a plane, in flat space – such as Tic-Tac-Toe, chess, and pool – but on a torus (or a Klein bottle) instead!

A maze – on a torus!

By the way, you can find Torus Games and other cool, free, downloadable math software on our new page – Free Math Software.  You’ll find links to other software that we love to use – such as Scratch and GeoGebra, and another program by Jeff Weeks called Curved Spaces.

All this talk of tori making you hungry?  Go get your own tasty torus (bagel), and try this fun trick to slice your bagel into two linked halves.  This topologically delicious breakfast problem was created by mathematical artist George Hart.

Bon appetit!  (Literally, this time.)

The Fractal Foundation, Schoolhouse Rock, and More

Welcome to this week’s Math Munch!

Triangle Cutout Fractal

Up first, check out the Fractal Foundation.  They’re mission is simple: “We use the beauty of fractals to inspire interest in Science, Math and Art.”  If you played around with recursive drawing a few weeks ago, then perhaps you were as inspired by fractals as they hope you’ll be.  If you’re not really sure what fractals actually are, here’s a great one-page explanation from the Fractal Foundation website.  They also have an excellent page of “fractivities,” including instructions for the beautiful paper cutout fractal pictured on the right.  If you want to have your mind blow, check out their fantastic page of fractal videos.  Just amazing.

Next up, have you ever heard of Schoolhouse Rock?  It’s a series of rocking animated music videos that originally ran on TV from 1973 to 1985.  Vintage math goodness!  They cover all kinds of educational stuff like grammar and history, but I totally love the math videos, and a few of them are on YouTube!  Down below you can watch two of my favorites, and you can find the others here.  if you poke around YouTube, you could probably find a few more as well.

Finally, a few additions to our resource pages.  For the Math Games page, we’re adding Linebounder.  You and the computer battle to draw a line towards your goal.  I had a really hard time with this at first, but there are certain strategies that the computer simply cannot beat.  You just have to find them.  Also new is Shift, another fun game that plays with the relationship between figure and ground.  For the new Math Art Tools page, we’re adding Tessellate!  It’s an interactive applet that lets you make custom tiles to cover the plane.  Here’s a few examples I just made.

Bon appetit!