Author Archives: Anna Weltman

Stomachion, Toilet Math, and Domino Computer Returns!

Welcome to this week’s Math Munch!

I recently ran across a very ancient puzzle with a very modern solution– and a very funny name. It’s called the Stomachion, and it looks like this:

Stomachion_850So, what do you do? The puzzle is made up of these fourteen pieces carved out of a 12 by 12 square– and the challenge is to make as many different squares as possible using all of the pieces. No one is totally sure who invented the Stomachion puzzle, but it’s definite that Archimedes, one of the most famous Ancient Greek mathematicians, had a lot of fun working on it.

StomaAnimSometimes Archimedes used the Stomachion pieces to make fun shapes, like elephants and flying birds. (If you think that sounds like fun, check out this page of Stomachion critters to try making and this lesson about the Stomachion puzzle from NCTM.) But his favorite thing to do with the Stomachion pieces was to arrange them into squares!

It’s clear that you can arrange the Stomachion pieces into a square in at least one way– because that’s how they start before you cut them out. But is there another way to do it? And, if there’s a second way, is there a third? How about a fourth? Because Archimedes was wondering about how many ways there are to make a square with Stomachion pieces, some mathematicians give him credit for being an inventor of combinatorics, the branch of math that studies counting things.

Ostomachion536Solutions_850It turns out that there are many, many ways to make squares (the picture above shows all of them– click on it for greater detail)– and Archimedes didn’t find them all. But someone else did, over 2,000 years later! He used a computer to solve the problem– something Archimedes could never have done– but mathematician Bill Cutler found that there are 536 ways to make a square with Stomachion pieces! That’s a lot! If you’ve tried to make squares with the pieces, you might be particularly surprised– it’s pretty tricky to arrange them into one unique square, let alone 536. This finding was such a big deal that it made it into the New York Times. (Though you may notice that the number reported in the article is different– that’s how many ways there are to make a square if you include all of the solutions that are symmetrically the same.)

Other mathematicians have worked on finding the number of ways to arrange the Stomachion pieces into other shapes– such as triangles and diamonds. Given that it took until 2003 for someone to find the solution for squares, there are many, many open questions about the Stomachion puzzle just waiting to be solved! Who knows– if you play with the Stomachion long enough, maybe you’ll discover something new!

Next up, the mathematicians over at Numberphile have worked out a solution to a problem that plagued me a few weeks ago while I was camping– choosing the best outdoor toilet to use without checking all of them for grossness first. Is there a way to ensure that you won’t end up using the most disgusting toilet without having to look in every single one of them? Turns out there is! Watch this video to learn how:

Finally, a little blast from the past. Almost two years ago I share with you a video of something really awesome– a computer made entirely out of dominoes! Well, this year, some students and I finally got the chance to make one of our own! It very challenging and completely exhausting, but well worth the effort. Our domino computer recently made its debut on the mathematical internet, so I thought I’d share it with all of you! Enjoy!

Bon appetit!





Origami Stars, Tessellation Stars, and Chaotic Stars

Welcome to this week’s star-studded Math Munch!

downloadModular origami stars have taken the school I teach in by storm in recent months! We love making them so much that I thought I’d share some instructional videos with you. My personal favorite is this transforming eight-pointed star. It slides between a disk with a hole the middle (great for throwing) and a gorgeous, pinwheel-like eight-pointed star. Here’s how you make one:

Another favorite is this lovely sixteen-pointed star. You can make it larger or smaller by adding or removing pieces. It’s quite impressive when completed and not that hard to make. Give it a try:

type6thContinuing on our theme of stars, check out these beautiful star tessellations. They come from a site made by Jim McNeil featuring oh-so-many things you can do with polygons and polyhedra. On this page, Jim tells you all about tessellations, focusing on a category of tessellations called star and retrograde tessellations.

type3b400px-Tiling_Semiregular_3-12-12_Truncated_Hexagonal.svgTake, for example, this beautiful star tessellation that he calls the Type 3. Jim describes how one way to make this tessellation is to replace the dodecagons in a tessellation called the 12.12.3 tessellation (shown to the left) with twelve-pointed stars. He uses the 12/5 star, which is made by connecting every fifth dot in a ring of twelve dots. Another way to make this tessellation is in the way shown above. In this tessellation, four polygons are arranged around a single point– a 12/5 star, followed by a dodecagon, followed by a 12/7 star (how is this different from a 12/5 star?), and, finally, a 12/11-gon– which is exactly the same as a dodecagon, just drawn in a different way.

I think it’s interesting that the same pattern can be constructed in different ways, and that allowing for cool shapes like stars and different ways of attaching them can open up crazy new worlds of tessellations! Maybe you’ll want to try drawing some star tessellations of your own after seeing some of these.

Screenshot 2014-05-12 10.48.46Finally, to finish off our week of everything stars, check out the star I made with this double pendulum simulator.  What’s so cool about the double pendulum? It’s a pendulum– a weight attached to a string suspended from a point– with a second weight hung off the bottom of the first. Sounds simple, right? Well, the double pendulum actually traces a chaotic path for most sizes of the weights, lengths of the strings, and angles at which you drop them. This means that very small changes in the initial conditions cause enormous changes in the path of the pendulum, and that the path of the pendulum is not a predictable pattern.

Using the simulator, you can set the values of the weights, lengths, and angles and watch the path traced on the screen. If you select “star” under the geometric settings, the simulator will set the parameters so that the pendulum traces this beautiful star pattern. Watch what happens if you wiggle the settings just a little bit from the star parameters– you’ll hardly recognize the path. Chaos at work!

Happy star-gazing, and bon appetit!

Truchet, Truchet, Truchet!

Welcome to this week’s Math Munch!

Why all the excitement about Truchet? And what (or who?) is Truchet, anyway? Great questions, both. I only recently learned about the fascinating world of Truchet and his tiles. I first got hooked on the beautiful patterns you can make with Truchet tiles. So, read on– and maybe you’ll get hooked, too.

Truchet tiling 1

This beautiful pattern is made out of Truchet tiles, deceptively simple square pieces that can fit together to make patterns with enormous complexity. There’s really just one Truchet tile– the square with a triangular half of it colored it– but it can be oriented in four ways.

Truchet tileThere are ten different ways to put two next to each other– or, at least, that’s what Sebastien Truchet discovered when he started working with these tiles way back in the late 1600s. Truchet was a scientist and inventor, but he also dabbled in math. He became interested in the ways of combining these simple tiles while looking at decorations made from ceramic tiles. He started trying to figure out all the different patterns he could make– first with two tiles, then with sequences of them– and soon realized that he was the first person to study this! So, he wrote a little book about his discoveries, which he called “Memoir sur les Combinasions.”

Truchet's chartTranslations of this book are hard to come by (unless you read French), but I found a great site that shows all the images Truchet included in the book. One of my favorites is this chart, in which Truchet shows all the possible ways of combining two tiles. Maybe you’ve noticed that there are way more than ten different combinations on this chart. That’s because this chart is just Truchet brainstorming– drawing everything he can think of. In a later chart, Truchet groups pairs of tiles that he thinks are the same in some really basic way.

Can you see how Truchet grouped the tiles in this chart? Each row corresponds to one of his ten different tiles. Do his categories make sense?

Truchet's equivalences 2Truchet tiling 5Want to try making a beautiful Truchet tiling of your own? You could just start drawing (maybe graph paper would be useful). Or you could try this great Scratch program! You can have it make a random Truchet tiling. Or, you can use a four-digit code, using only the digits 1 through 4, to tell it what pattern to make. Each digit corresponds to a tile with a dark triangle in a different corner. I had fun thinking of a code and then trying to guess what pattern the program would make. Or, you could try to figure out the code for one of your favorite of Truchet’s patterns!

In recent years, mathematicians have starting experimenting with Truchet tiles in new ways. The mathematician and scientist Cyril Stanley Smith was one of the first to do this. He began to introduce curved lines into the tiles and to place them randomly, instead of according to a pattern. These changes make some really interesting results– like this maze made from Truchet-like tiles.

Want to make your own maze from tiles inspired by Truchet? Check out this site with instructions for how to make Truchet mazes! You can use a computer or a more low-tech tool to create an intricate, unique maze.

Bon appetit!