# Tangent Spaces, Transplant Matches, and Golyhedra

Welcome to this week’s Math Munch!

You might remember our post on Tilman Zitzmann’s project called Geometry Daily. If you haven’t seen it before, go check it out now! It will help you to appreciate Lawrie Cape’s work, which both celebrates and extends the Geometry Daily project. Lawrie’s project is called Tangent Spaces. He makes Tilman’s geometry sketches move!

 A box of rays, by Tilman A box of rays, by Lawrie

Not only do Lawrie’s sketches move, they’re also interactive—you can click on them, and they’ll move in response. All kinds of great mathematical questions can come up when you set a diagram in motion. For instance, I’m wondering what moon patterns are possible to make by dragging my mouse around—and if any are impossible. What questions come up for you as you browse Tangent Spaces?

Next up, Dorry Segev and Sommer Gentry are a doctor and a mathematician. They collaborated on a new system to help sick people get kidney transplants. They are also dance partners and husband and wife. This video shares their amazing, mathematical, and very human story.

Dorry and Sommer’s work involves building graphs, kind of like the game that Paul posted about last week. Thinking about the two of them together has been fun for me. You can read more about the life-saving power of Kidney Paired Donation on optimizedmatch.com.

Last up this week, here’s some very fresh math—discovered in the last 24 hours! Joe O’Rourke is one of my favorite mathematicians. (previously) Joe recently asked whether a golyhedron exists. What’s a golyhedron? It’s the 3D version of a golygon. What’s a golygon? Glad you asked. It’s a grid polygon that has side lengths that grow one by one, from 1 up to some number. Here, a diagram will help:

The smallest golygon. It has sides of lengths 1 through 8.

A golyhedron is like this, but in 3D: a grid shape that has one face of each area from 1 up to some number. After tinkering around some with this new shape idea, Joe conjectured that no golyhedra exist. It’s kind of like coming up with the idea of a unicorn, but then deciding that there aren’t any real ones. But Joseph wasn’t sure, so he shared his golyhedron shape idea on the internet at MathOverflow. Adam P. Goucher read the post, and decided to build a golyhedron himself.

And he found one!

The first ever golyhedron, by Adam P. Goucher

Adam wrote all about the process of discovering his golyhedron in this blog post. I recommend it highly.

And the story and the math don’t stop there! New questions arise—is this the smallest golyhedron? Are there types of sequences of face sizes that can’t be constructed—for instance, what about a sequence of odd numbers? Curious and creative people, new discoveries, and new questions—that’s how math grows.

If this story was up your alley, you might enjoy checking out the story of holyhedra in this previous post.

Bon appetit!

# Havel-Hakimi, Temari, and more GIFS

Welcome to this week’s Math Munch!  We’ve got another great game for you, a followup with Temari artist Carolyn Yackel, and some mind-blowing math gifs.

Havel-Hakimi

First up, a nice little graph theory game created by Jacopo Notarstefano.  The game is about whether or not sets of numbers meet the conditions for being “graphical.”  Maybe the best way to understand what that means is to start playing.  If you can beat a level, then the starting number set is graphical.  Go play Havel-Hakimi.

In 1960, mathematicians Paul Erdös and Tibor Gallai proved a theorem about what number sets were graphical.  The name of the game refers to an algorithm you can use to solve the game.  You might figure it out just by playing the game, but here’s a (pretty dry) video explaining how the Havel-Hakimi algorithm works.

Jacopo’s website has a few other nice projects.  See if you can figure out Who Killed the Duke of Densmore, or try Four-Coloring the Dodecahedron.

Carolyn Yackel

Up next, remember Carolyn Yackel.  We wrote about Carolyn and her mathematical art a while back.  Well we finally got around to doing a little Q&A.  Give it a read to learn about Carolyn and her love of math.

Carolyn’s art (which can be seen here) is called temari, the japanese art of embroidered spheres. Since our post about Carolyn we found out that a now 93-year old grandmother posted a lifetime of temari on flickr.  These beautiful objects have symmetry that mimic various polyhedra, which I just love.  Read Carolyn’s Q&A to hear about how you make them.

A Grandmother’s Temari Work

Finally, a while back we shared some mathematical gif animations created by Bees and Bombs.  It’s time once again to look at some amazing animations.  This time they’re created by David Pope.  Here’s the complete archive of animations.  I’ll post some of my very favorites below, but there are dozens of dozens of good animations. (A dozen dozens is gross!)

Have a great week.  Bon appetit!

## Full gallery of mathematical gifs

 Pyramid Rolling Prisms Sphere

Spinning Octahedra

# 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.

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.

There 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.”

Translations 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?

Want 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!

# Math Awareness Month, Hexapawn, and Plane Puzzles

Welcome to this week’s Math Munch!

April is Mathematics Awareness Month. So happy Mathematics Awareness Month! This year’s theme is “Mathematics, Magic, and Mystery”. It’s inspired by the fact that 2014 would have marked Martin Gardner’s 100th birthday.

A few of the mathy morsels that await you this month on mathaware.org!

Each day this month a new piece of magical or mysterious math will be revealed on the MAM site. The mathematical offering for today is a card trick that’s based on the Fibonacci numbers. Dipping into this site from time to time would be a great way for you to have a mathy month.

It is white’s turn to move. Who will win this Hexapawn game?

Speaking of Martin Gardner, I recently ran across a version of Hexapawn made in the programming language Scratch. Hexapawn is a chess mini-game involving—you guessed it—six pawns. Martin invented it and shared it in his Mathematical Games column in 1962. (Here’s the original column.) The object of the game is to get one of your pawns to the other side of the board or to “lock” the position so that your opponent cannot move. The pawns can move by stepping forward one square or capturing diagonally forward. Simple rules, but winning is trickier than you might think!

The program I found was created by a new Scratcher who goes by the handle “puttering”. On the site he explains:

I’m a dad. I was looking for a good way for my daughters to learn programming and I found Scratch. It turns out to be so much fun that I’ve made some projects myself, when I can get the computer…

puttering’s Scratch version of Conway’s Game of Life

Something that’s super cool about puttering’s Hexapawn game is that the program learns from its stratetgy errors and gradually becomes a stronger player as you play more! It’s well worth playing a bunch of games just to see this happen. puttering has other Scratch creations on his page, too—like a solver for the Eight Queens puzzle and a Secret Code Machine. Be sure to check those out, too!

Last up, our friend Nalini Joshi recently travelled to a meeting of the Australian Academy of Science, which led to a little number puzzle.

What unusual ways of describing a number! Trying to learn about these terms led me to an equally unusual calculator, hosted on the Math Celebrity website. The calculator will show you calculations about the factors of a numbers, as well as lots of categories that your number fits into. Derek Orr of Math Year-Round and I figured out that Nalini’s clues fit with multiple numbers, including 185, 191, and 205. So we needed more clues!

Can you find another number that fits Nalini’s clues? What do you think would be some good additional questions we could ask Nalini? Leave your thoughts in the comments!

A result from the Number Property Calculator

I hope this post helps you to kick off a great Mathematics Awareness Month. Bon appetit!