The World Cup Group Stage, Math at First Sight, and Geokone

Welcome to this week’s Math Munch! We’ve got some World Cup math from a tremendous recreational mathematics blog and a new mathematical art tool. Get ready to dig in!

Brazuca: The 2014 World Cup Ball

Brazuca: The 2014 World Cup Ball

I’ve been meaning to share the really fantastic Puzzle Zapper Blog, because it’s so full of cool ideas, but the timing is perfect, because IT’S WORLD CUP TIME!!! and the most recent post is about the math of the world cup group stage! It’s called “World Cup Group Scores, and “Birthday Paradox” Paradoxes,” and I hope you’ll give it a read. (For some background on the Birthday Paradox, watch this Numberphile video called 23 and Football Birthdays.)

The thing that got me interested in the article was actually just this chart. I think it’s really cool, probably because I always find myself two games through the group stage, thinking of all the possible outcomes. If you do nothing else with this article, come to understand this chart. I was kind of surprised how many possible outcomes there are.

All Possible World Cup Group Stage Results

All Possible World Cup Group Stage Results

Long story short (though you should read the long story), there’s about a 40% chance that all 8 world cup groups will finish with different scores.

Alexandre Owen Muñiz, Author of Puzzle Zapper.  (click for an interview video about Alexandre's interactive fiction)

Alexandre Owen Muñiz, Author of Puzzle Zapper.  (click for an interview video about Alexandre’s interactive fiction)

Puzzle Zapper is the recreational mathematics blog of Alexandre Owen Muñiz. You can also find much of his work on his Math at First Sight site. He has a lot of great stuff with polyominoes and other polyforms (see the nifty pics below). Alexandre is also a writer of interactive fiction, which is basically a sort of text-based video game. Click on Alexandre’s picture to learn more.

The Complete Set of "Hinged Tetriamonds"

The complete set of “hinged tetrominoes”

A lovely family portrait of the hinged tetriamonds.

A lovely, symmetric family portrait of the “hinged tetriamonds”

I hope you’ll poke around Alexandre’s site and find something interesting to learn about.

For our last item this week, I’ve decided to share a new mathematical art tool called Geokone. This app is a recursive, parametric drawing tool. It’s recursive, because it is based on a repeating structure, similar to those exhibited by fractals, and it’s parametric, because the tool bar on the right has a number of parameters that you can change to alter the image. The artistic creation is in playing with the parameter values and deciding what is pleasing. Below are some examples I created and exported.

geokone2 geokone1

geokone3

I have to say, Geokone is not the easiest thing in the world to use, but if you spend some time playing AND thinking, you can almost certainly figure some things out! As always, if you make something cool, please email it to us!

Now go create something!  Click to go to Geokone.net.

I hope you find something tasty this week. Bon appetit!

Girls’ Angle, Spiral Tilings, and Coins

Welcome to this week’s Math Munch!

GirlsAngleCoverGirls’ Angle is a math club for girls. Since 2007 it has helped girls to grow their love of math through classes, events, mentorship, and a vibrant mathematical community. Girls’ Angle is based in Cambridge, Massachusetts, but its ideas and resources reach around the world through the amazing power of the internet. (And don’t you worry, gentlemen—there’s plenty for you to enjoy on the site as well.)

Amazingly, the site contains an archive of every issue of Girls’ Angle Bulletin, a wonderful bimonthly journal to “foster and nurture girls’ interest in mathematics.” In their most recent issue, you’ll find an interview with mathematician Karen E. Smith, along with several articles and puzzles about balance points of shapes.

There’s so much to dig into at Girls’ Angle! In addition to the Bulletins, there are two pages of mathematical videos. The first page shares a host of videos of women in mathematics sharing a piece of math that excited them when they were young. The most recent one is by Bridget Tenner, who shares about Pick’s Theorem. The second page includes several videos produced by Girls’ Angle, including this one called “Summer Vacation”.

Girls’ Angle can even help you buy a math book that you’d like, if you can’t afford it. For so many reasons, I hope you’ll find some time to explore the Girls’ Angle site over your summer break. (And while you’ve got your explorer’s hat on, maybe you’ll tour around Math Munch, too!)

I did a Google search recently for “regular tilings.” I needed a few quick pictures of the usual triangle, square, and hexagon tilings for a presentation I was making. As I scrolled along, this image jumped out at me:

hexspiral

What is that?! It certainly is a tiling, and all the tiles are the “same”—even if they are different sizes. Neat!

Clicking on the image, I found myself transported to a page all about spiral tilings at the Geometry Junkyard. The site is a whole heap of geometrical odds and ends—and a place that I’ve stumbled across many times over the years. Here are a few places to get started. I’m sure you’ll enjoy poking around the site to find some favorite “junk” of your own.

Spirals

Spirals

Circles and spheres

Circles & spheres

Coloring

Coloring

Last up this week, you may have seen this coin puzzle before. Can you make the triangle point downwards by moving just three pennies?triangleflip

There are lots of variants of this puzzle. You can find some in an online puzzle game called Coins. In the game you have to make arrangements of coins, but the twist is that you can only move a coin to a spot where would it touch at least two other coins. I’m enjoying playing Coins—give it a try!

I solved this Coins puzzle in four moves. Can you? Can you do better?

I solved this Coins puzzle in four moves. Can you? Can you do better?

That’s it for this week’s Math Munch. Bon appetit!

 

Halving Fun, Self-Tiling Tile Sets, and Doodal

Welcome to this week’s Math Munch!

Print out two copies of this pattern, cut them out, and fold each along the dotted lines, making two identical solids. Then fit these two pieces together to make a regular tetrahedron.

Print out two copies of this pattern, cut them out, and fold each along the dotted lines, making two identical solids. Then fit these two pieces together to make a regular tetrahedron.

Our first bit of fun comes from a blog called Futility Closet (previously featured). It’s a neat little cut-and-fold puzzle. The shape to the right can be folded up to make a solid with 5 sides. Two of them can be combined to make a solid with only 4 sides, the regular tetrahedron. If you’d like, you can use our printable version, which has two copies on one sheet.

What do you know, I also found our second item on Futility Closet! Check out the cool family of tiles below. What do you notice?

A family of self-tiling tiles

A family of self-tiling tiles

Did you notice that the four shapes in the middle are the same as the four larger shapes on the outside? The four tiles in the middle can combine to create larger versions of themselves! They can make any and all of the original four!!

Lee Sallows

Recreational Mathematician, Lee Sallows

Naturally, I was reminded of the geomagic squares we featured a while back (more at geomagicsquares.com), and then I came to realize they were designed by the same person, the incredible Lee Sallows! (For another amazing one of Lee Sallows creations, give this incredible sentence a read.) You can also visit his website, leesallows.com.

reptile3

A family of 6 self-tiling tiles

For more self-tiling tiles (and there are many more amazing sets) click here. I have to point out one more in particular. It’s like a geomagic square, but not quite. It’s just wonderful. Maybe it ought to be called a “self-tiling latin square.”

And for a final item this week, we have a powerful drawing tool. It’s a website that reminds me a lot of recursive drawing, but it’s got a different feel and some excellent features. It’s called Doodal. Basically, whatever you draw inside of the big orange frame will be copied into the blue frames.  So if there’s a blue frame inside of an orange frame, that blue frame gets copied inside of itself… and then that copy gets copied… and then that copy…!!!

To start, why don’t you check out this amazing video showing off some examples of what you can create. They go fast, so it’s not really a tutorial, but it made me want to figure more things out about the program.

I like to use the “delete frame” button to start off with just one frame. It’s easier for me to understand if its simpler. You can also find instructions on the bottom. Oh, and try using the shift key when you move the blue frames. If you make something you like, save it, email it to us, and we’ll add it to our readers’ gallery.

Start doodaling!

Make something you love. Bon appetit!

A fractal Math Munch Doodal

A fractal Math Munch Doodal

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!

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 Tilman

A box of rays, by Lawrie.

A box of rays, by Lawrie

409 66 498

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.

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

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

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.

Temari 1 Temari 2 Temari 3
Carolyn Yackel

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.

Grandmother's temari work

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

Pyramid

Rolling Prisms

Rolling Prisms

Sphere

Sphere

Spinning Octahedra

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.

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!