Tag Archives: spirals

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:

Slides and Twists, Life in Life, and Star Art

Welcome to this week’s Math Munch!

I ran across the most wonderful compendium of slidey and twisty puzzles this past week when sharing the famous 15-puzzle with one of my classes.  It’s called Jaap’s Puzzle Page and it’s run by a software engineer from the Netherlands named Jaap Scherphuis. Jaap has been running his Puzzle Page since 1999.

Jaap Scherphuis
and some of his many puzzles

Jaap first encountered hands-on mathematical puzzles when he was given a Rubik’s Cube as a present when he was 8 or 9. He now owns over 700 different puzzles!

Jaap’s catalogue of slidey and twisty puzzles is immense and diverse. Each puzzle is accompanied by a picture, a description, a mathematical analysis, and–SPOILER ALERT–an algorithm that you can use to solve it!

On top of this, all of the puzzles in Jaap’s list with asterisks (*) next to them have playable Java applets on their pages–for instance, you can play Rotascope or Diamond 8-Ball. Something that’s especially neat about Jaap’s applets is that you can sometimes customize their size/difficulty. If you find the 15-puzzle daunting, you can start with the 8-puzzle or even the 3-puzzle instead. The applets also have a built in solver. I really enjoy watching the solver crank through solving a puzzle–it’s so relentless, and sometimes you can see patterns emerge.

Over ten solves, I found that the autosolve for the 15-puzzle averaged 7.1 seconds. How long do you think on average the 63-puzzle would take to solve?

You can read more about Jaap in this interview on speedcubing.com or on his about page.


The 15-puzzle


Diamond 8-ball

Next, I recently read about an amazing feat: Brice Due created a copy of Conway’s Game of Life inside of a Game of Life! This video shows you what it’s all about. It starts zoomed in on some activity, following the rules of Life. The it zooms out to show that this activity conspires to make a large unit cell that is “turned on.” This large cell was dubbed a “OTCA metapixel” by its creator, where OTCA stands for Outer Totalistic Cellular Automata.

Finally, the video zooms out even more to show that this cell and others around it interact according to the rules of Life! The activity at the meta-level that is shown at the end exactly corresponds to the activity on the micro-level that we began with.  Check it out!

This metapixel idea has been around since 2006, but the video was created just recently by Philip Bradbury. It was made using Golly, a cellular automata explorer that is one of my favorite mathematical tools.

Last up, some star art! (STart? STARt? st-art?)  It turns out that the Math Munch team members all converged toward doing some StArT this semester as a part of our mathematical art (MArTH) seminar. Here is some of our work, for your viewing pleasure. Bon appetit!

by Anna Weltman

by Anna Weltman

Stars of the Mind’s Sky
by Paul Salomon

Star Ring 24
by Paul Salomon

300 Stars in Orbit
by Paul Salomon

by Justin Lanier

Noodles, Flowsnake, and Symmetry

Welcome to this week’s Math Munch!

Gemelli, by Sander Huisman

Gemelli, by Sander Huisman

How much do you like pasta?  Well, these mathematicians and scientists from around the world like pasta so much that they’ve been studying its shape mathematically!  Check out this New York Times article about Sander Huisman, a graduate student in physics from the Netherlands, and Marco Guarnieri and George L. Legendre, two architects from London, who have all taken up making graphs of and equations for pasta shapes.  Sander posts his pasta-graphs on his blog.  Legendre wrote this book about math and pasta, called Pasta By Design.  Legendre has even invented a new type of pasta, shaped like a Mobius strip (see last week’s Math Munch for lots of cool things with Mobius strips), which he named after his baby daughter, Ioli!

Some of Legendre’s pasta plots

Next, here comes the flowsnake.  Wait – don’t run away!  The flowsnake is not a terrifying monster, despite it’s ominous name.  It is a space-filing curve, meaning that the complete curve covers every single point in a part of two-dimensional space.  So if you were to try to draw a flowsnake on a piece of paper, you wouldn’t be able to see any white when you were done.  It’s named flowsnake because it resembles a snowflake.

The flowsnake curve

A single piece of the flowsnake curve.

Units of flowsnake fit together like puzzle pieces to fill the plane

Finally, check out this awesome online symmetry-sketcher, called Symmetry Artist!  Here, you can make doodles of all kinds and then choose how you want to reflect and rotate them.  Fun!

Bon appetit!

Mazes, Spirals, and Paper Folding

Welcome to Math Munch!   Here you will find links to lots of cool mathy things on the internet.  We’ll post some new items each week for you to enjoy.  We hope you are as inspired and excited by these creations as we are!

Maze A Day is a blog where Warren Stokes publishes new mazes he has created.  Every day!  What a cool project.

A number spiral

Here’s another maze that was submitted by truff.

Number Spiral is a website that shares some cool number spirals and some deep patterns that have been found in prime numbers.  I like how the author Robert Stacks both gives a very simple introduction to his work and carries it through to very complex mathematics.

Finally, here is a short video about the work of paper engineer Matt Shlian at the University of Michigan.  A favorite quote: ” I think there’s this great crossover right now between science and art that the art students don’t know anything about and the scientists don’t know that artists are out there that exist that can help them figure out some of these things.”

Bon Appetit!

Paper Cuts from Michigan Daily on Vimeo.