Tag Archives: computers

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!

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

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

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!

Digital Art, Mastermind, and Pythagoras

Welcome to this week’s Math Munch… on (approximately) Math Munch’s second birthday! Hooray!

Check out this video of mathematical art made by artist Nathan Selikoff:

Cool, right? This piece is called “Beautiful Chaos.” The curves on the screen are made from equations (if you’ve ever graphed a line or a parabola you’ll know what I mean). As the viewer waves her hands around, the equations change– and as the equations change, so do the curves! The result is something that might remind you of the images your computer makes when you play music on it or maybe of something you’d make using a spirograph. All in all, a beautiful and interactive piece of mathematical art.

nathanNathan lives and works as a mathematical artist in Orlando, Florida. As he writes on his website, Nathan uses computer code along with other materials to make art that plays with the mathematical ideas of space, motion, and interaction between objects. To see more of how Nathan does this, check out his giant, interactive marionette or this song that explores the first, second, third, and fourth dimensions:

My school is really lucky to be hosting Nathan this week! We didn’t want any of you, dear readers, to miss out on the excitement, though– so Nathan has kindly agreed to answer your interview questions! Got a question for Nathan? Write it in the box below. He’ll answer seven of your best questions in two weeks!

565px-MastermindNext up, who doesn’t love to play Mastermind? It’s a great combination of logic, patterns, and trickery… but I just hate having to use all those tiny pegs. Well, guess what? You can play it online— no pegs (or opponent) necessary!

As I was playing Mastermind, I started wondering about strategy. What’s the best first guess to make? If I were as smart as a computer, is there a number of guesses in which I could guess any Mastermind code? (This kind of question reminds me of God’s Number and the Rubik’s cube…)

Well, it turns out there is a God’s Number for Mastermind – and that number is five. Just five. If you played perfectly and followed the strategy demonstrated by recreational mathematician Toby Nelson on his website, you could guess ANY Mastermind code in five guesses or less. Toby shares many more interesting questions about Mastermind on his website— I suggest you check it out.

What ARE those irrational numbers, so weird that they get their own bubble??

What ARE those irrational numbers, so weird that they get their own bubble??

Finally, sometime in your mathematical past you may have heard of irrational numbers. These are numbers like the square-root of 2 or pi or e that can’t be written as a fraction– or so people claim. When you start thinking about this claim, however, it may seem strange. There are A LOT of fractions– and none of them equal the square-root of 2? Really? What kind of number is that? It seems like only an irrational person would believe that, at least without proof.

Vi Hart to the rescue! Irrational numbers were encountered long, long ago by the ancient Greek mathematician (and cult leader) Pythagoras– and he didn’t like them much. In this great video, Vi tells all about Pythagoras and the controversial discovery of numbers that aren’t fractions.

If you didn’t follow her explanation of why the square-root of 2 is irrational on your first watch, don’t worry– it’s a complicated idea that’s worth a second (or third or fourth) run-through.

Thanks for a great two years of Math Munch! Bon appetit!

TesselManiac, Zeno’s Paradox, and Platonic Realms

Welcome to this week’s Math Munch!

Before we begin, we’d like to thank all of you who have checked out the site in the past week. Since we’ve kicked off our share campaign, we’ve had so many new visitors and heard from many of them, too! Reading your feedback – whether a recommendation, some praise, a question, or just a brief, “Hello!” – brings us great joy and helps us to know that you all are out there.

Whether you’re a regular reader or visiting the site for the first time, we’d like to ask you for a little favor. If you see some math you like, share it with someone who you think would like it, too! Do you love the burst of excitement that you get from reading about a new mathematical idea, seeing an original piece of math artwork, or trying out a new game? Do you know someone who would love that, too? Then tell them about Math Munch – we’d love to spread the joy.

If you enjoy Math Munch, join in our “share campaign” this week.

You can read more about the share campaign here. There are lots of ways to participate, and you can let us know about your sharing through this form. We’d love to see the share total rise up to 1000 over the course of the next week.

Now for the post!


Lee boxThis beautiful tessellated wooden box was made by computer scientist and mathematical artist Kevin Lee. I met Kevin two weeks ago at the MOVES conference (which Justin and Paul have both written about already). Kevin teaches computer science at Normandale Community College in Minnesota. He makes woodcut tessellations (which won an award for the “Best Textile, Sculpture, or Other Medium” at the Joint Mathematics Meetings art exhibition this year!). He’s also used a combination of his knowledge of computer science and his love of Escher-type tessellations to make software that helps you create tessellations. His new software, TesselManiac!, is due out soon (watch this short movie Kevin made about it for the Bridges conference), but you can download an older version of the software here and play a preview version of The Flipping Tile Game.

tesselmaniac pictures

To play this game, you must fill in an outline of a tessellation with the piece given. You can use any of four symmetry motions – translation (or shift), rotation, reflection, or glide reflection (which reflects the tile and then translates it along a line parallel to the line of reflection). You get points for each correct tile placed (and lose points if you have to delete). Translations are the simplest, and only give you 5 points each. Reflections are the most difficult – you get 20 points for each one used!

dot to dotWhile you’re downloading The Flipping Tile Game, try one of Kevin’s Dot-to-Dot puzzles. These are definitely not your typical dot-to-dot. Only the portion of the image corresponding to one tile in the tessellation is numbered. Once you figure out the shape of that single tile, you have to figure out how to number the rest of the puzzle!

Lucky for us, Kevin has agreed to answer some questions about his life and work as a math artist and computer scientist. Leave a question for Kevin here. (We’ll take questions for the next two weeks.)

tortoiseI’ve recently been thinking about a paradox that has puzzled mathematicians for centuries. Maybe you’ve heard of it. It’s one of the ancient Greek philosopher Zeno‘s paradoxes of motion, and it goes like this: Achilles (a really fast Greek hero) and a tortoise are going to run a race. Achilles agrees to give the tortoise a head-start, because the tortoise is so slow. Achilles then starts to run. But as Achilles catches up with the tortoise, the tortoise moves a little further. So the tortoise is still ahead. And as Achilles moves to catch up again, the tortoise moves even further! Sounds like Achilles will never catch up to the tortoise, let alone pass him… But that doesn’t make sense…

Will Achilles lose the race??? Check out this great video from Numberphile that explains both the paradox and the solution.

logo_PR_225_160While I was looking for information about this paradox, I stumbled across a great math resource site called Platonic Realms. The homepage of this site has a daily historical fact, mathematical quote, and puzzle.

The site hosts a math encyclopedia with explanations of all kinds of math terms and little biographies of famous mathematicians. You can also read “mini-texts” about different mathematical topics, such as this one about M. C. Escher (the inspiration behind the art at the beginning of this post!) or this one about coping with math anxiety.

I hope we here at Math Munch have given you something to tantalize your mathematical taste buds this week! If so, we’d love it if you would pass it along.

Thank you for reading, and bon appetit!

Rush hourP.S. – We’ve posted a new game, suggested to us by one of our readers! It’s an online version of Rush Hour. Check it out!

Circling, Squaring, and Triangulating

Welcome to this week’s Math Munch!

How good are you at drawing circles? To find out, try this circle drawing challenge. There are adorable cat pictures for prizes!

What’s the best score you can get? And hey—what’s the worst score you can get? And how is your score determined? Well, no matter how long the path you draw is, using that length to make a circle would surround the most area. How close your shape gets to that maximum area determines your score.

Do you think this is a good way to measure how circular a shape is? Can you think of a different way?

Dido, Founder and Queen of Carthage.

Dido, Founder and Queen of Carthage.

This idea that a circle is the shape that has the biggest area for a fixed perimeter reminds me of the story of Dido and her famous problem. You can find a retelling of it at Mathematica Ludibunda, a charming website that’s home to all sorts of mathematical stories and puzzles. The whole site is written in the voice of Rapunzel, but there’s a team of authors behind it all. Dido’s story in particular was written by a girl named Christa.

If you have any trouble drawing circles in the applet, you might try using pencil and paper or a chalkboard. I bet if you practice your circling and get good at it, you might even be able to challenge this fellow:

The simple perfect squared square of smallest order.

The simple perfect squared square
of smallest order.

Next up is squaring and the incredible Squaring.Net. The site is run by Stuart Anderson, who works at the Reserve Bank of Australia and lives in Sydney.

The site gathers together all of the research that’s been done about breaking up squares and rectangles into squares. It’s both a gallery and an encyclopedia. I love getting to look at the timelines of discovery—to see the progress that’s been made over time and how new things have been discovered even this year! Just within the last month or so, Stuart and Lorenz Milla used computers to show that there are 20566 simple perfect squared squares of order 30. Squaring.Net also has a wonderful links page that can connect you to more information about the history of squaring, as well as some of the delightful mathematical art that the subject has inspired.

trinity-glass2-small sqBox8 wp4f6b3871_0f

Delaunay triangulationLast up this week is triangulating. There are lots of ways to chop up a shape into triangles, and so I’ll focus on one particular way known as a Delaunay triangulation. To make one, scatter some points on the plane. Then connect them up into triangles so that each triangle fits snugly into a circle that contains none of the scattered points.

Fun Fact #1: Delaunay triangulations are named for the Soviet mathematician Boris Delaunay. What else is named for him? A mountain! That’s because Boris was a world-class mountain climber.

Fun Fact #2: The idea of Delaunay triangulations has been rediscovered many times and is useful in fields as diverse as computer animation and engineering.

Here are two uses of Delaunay triangulations I’d like to share with you. The first comes from the work of Zachary Forest Johnson, a cartographer who shares his work at indiemaps.com. You can check out a Delaunay triangulation applet that he made and read some background about this Delaunay idea here. To see how Zach uses these triangulations in his map-making, you’ve gotta check out the sequence of images on this page. It’s incredible how just a scattering of local temperature measurements can be extended to one of those full-color national temperature maps. So cool!


Zachary Forest Johnson

A Delaunay triangulation used to help create a weather map.

A Delaunay triangulation used to help create a weather map.

Finally, take a look at these images that Jonathan Puckey created. Jonathan is a graphic artist who lives in Amsterdam and shares his work on his website. In 2008 he invented a graphical process that uses Delaunay triangulations and color averaging to create abstractions of images. You can see more of Jonathan’s Delaunay images here.

 armandmevis-1  fox

I hope you find something to enjoy in these circles, squares, and triangles. Bon appetit!

Maths Ninja, Folding Fractals, and Pi Fun

Welcome to this week’s Math Munch!

ninjaFirst up, have you ever been stuck on a gnarly math problem and wished that a math ninja would swoop in and solve the problem before it knew what hit it?  Have you ever wished that you had a math dojo who would impart wisdom to you in cryptic but, ultimately, extremely timely and useful ways?  Well, meet Colin Beverige, a math (or, as he would say, maths) tutor from England who writes a fun blog called Flying Colours Maths.  On his blog, he publishes a weekly series called, “Secrets of the Mathematical Ninja,” in which the mathematical ninja (maybe Colin himself?  He’s too stealthy to tell)  imparts nuggets of sneaky wisdom to help you take down your staunchest math opponent.

colin_bridgeFor example, you probably know the trick for multiplying by 9 using your fingers – but did you know that there’s a simple trick for dividing by 9, too?  Ever wondered how to express thirteenths as decimals, in your head?  (Probably not, but maybe you’re wondering now!)  Want to know how to simplify fractions like a ninja?  Well, the mathematical ninja has the answers – and some cute stories, too.  Check it out!

A picture of the Julia set.

A picture of a Julia set.

Next, I find fractals fascinating, but – I’ll admit it – I don’t know much about them.  I do know a little about the number line and graphing, though.  And that was enough to learn a lot more about fractals from this excellent post on the blog Hackery, Math, and Design by Steven Wittens.  In the post How to Fold a Julia Fractal, Steven describes how the key to understanding fractals is understanding complex numbers, which are the numbers we get when we combine our normal numbers with imaginary numbers.

complex multiplicationNow, I think imaginary numbers are some of the most interesting numbers in mathematics – not only because they have the enticing name “imaginary,” but because they do really cool things and have some fascinating history behind them.  Steven does a really great job of telling their history and showing the cool things they do in this post.  One of the awesome things that imaginary numbers do is rotate.  Normal numbers can be drawn on a line – and multiplying by a negative number can be thought of as changing directions along the number line.  Steven uses pictures and videos to show how multiplying by an imaginary number can be thought of as rotating around a point on a plane.

here comes the julia set

A Julia set in the making.

The Julia set fractal is generated by taking complex number points and applying a function to them that squares each point and adds some number to it.  The fractal is the set of points that don’t get infinitely larger and larger as the function is applied again and again.  Steven shows how this works in a series of images.  You can watch the complex plane twist around on itself to make the cool curves and figures of the Julia set fractal.

Steven’s blog has many more interesting posts.  Check out another of my favorites, To Infinity… and Beyond! for an exploration of another fascinating, but confusing, topic – infinity.

Finally, a Pi Day doesn’t go by without the mathematicians and mathematical artists of the world putting out some new Pi Day videos!  Pi Day was last Thursday (3/14, of course).  Here’s a video from Numberphile in which Matt Parker calculates pi using pies!

In this video, also from Numberphile, shows how you only need 39 digits of pi to make really, really accurate measurements for the circumference of the observable universe:

Finally, it wouldn’t be Pi Day without a pi video from Vi Hart.  Here’s her contribution for this year:

Bon appetit!

Folds, GIMPS, and More Billiards

Welcome to this week’s Math Munch!

First up, we’ve often featured mathematical constructions made of origami. (Here are some of those posts.) Origami has a careful and peaceful feel to it—a far cry from, say, the quick reflexes often associated with video games. I mean, can you imagine an origami video game?


One of Fold’s many origami puzzles.

Well, guess what—you don’t have to, because Folds is just that! Folds is the creation of Bryce Summer, a 21-year-old game designer from California. It’s so cool. The goal of each level of its levels is simple: to take a square piece of paper and fold it into a given shape. The catch is that you’re only allowed a limited number of folds, so you have to be creative and plan ahead so that there aren’t any loose ends sticking out. As I’ve noted before, my favorite games often require a combo of visual intuition and careful thinking, and Folds certainly fits the bill. Give it a go!

Once you’re hooked, you can find out more about Bryce and how he came to make Folds in this awesome Q&A. Thanks so much, Bryce!

gimpsNext up, did you know that a new largest prime number was discovered less than a month ago? It’s very large—over 17 million digits long! (How many pages would that take to print or write out?) That makes it way larger than the previous record holder, which was “only” about 13 million digits long. Here is an article published on the GIMPS website about the new prime number and about the GIMPS project in general.

What’s GIMPS you ask? GIMPS—the Great Internet Mersenne Primes Search—is an example of what’s called “distributed computing”. Testing whether a number is prime is a simple task that any computer can do, but to check many or large numbers can take a lot of computing time. Even a supercomputer would be overwhelmed by the task all on its own, and that’s if you could even get dedicated time on it. Distributed computing is the idea that a lot of processing can be accomplished by having a lot of computers each do a small amount of work. You can even sign up to help with the project on your own computer. What other tasks might distributed computing be useful for? Searching for aliens, perhaps?

GIMPS searches only for a special kind of prime called Mersenne primes. These primes are one less than a power of two. For instance, 7 is a Mersenne prime, because it’s one less that 8, which is the third power of 2. For more on Mersenne primes, check out this video by Numberphile.

Finally, we’ve previously shared some resources about the math of billiards on Math Munch. Below you’ll find another take on bouncing paths as Michael Moschen combines the math of billiards with the art of juggling.

So lovely. For more on this theme, here’s a second video to check out.

Bon appetit!