Tag Archives: computers

Domino Computer, Knitting, and Election MArTH

Welcome to this week’s Math Munch!

First up this week is one of the coolest things I’ve seen in a long time: the world’s largest computer made out of dominoes.  A computer made out of dominoes?! you say.  How??

The Domputer, as it’s been called, was the great idea of mathematician, teacher, and entertainer Matt Parker (see a previous post about Matt here), and he and many volunteers built it at the Manchester Science Festival at the end of October.

Matt and some of his teammates testing domino circuits.

So, what is a domino computer, and how does it work?  As Matt is quoted saying in a podcast that featured the project, “A domino computer is exactly that: a computer made out of chains of dominoes.  Flicking over one domino sends a signal racing along the chain, just like current flows down a wire.  And then interacting lines of dominoes can manipulate the signal exactly the way circuit components do.”

At its very, very basic level, a computer is a machine that does calculations in binary.  You input some sequence of 0s and 1s by flipping signals on and off, and your input starts a chain of electrical communications that results in an output of 0s and 1s.  Most computers do this with electrical circuits.  But it can also be done with dominoes – sending an “on” signal means flipping a domino over, and sending an “off” signal means not flipping a domino, or having a chain of falling dominoes that becomes blocked and stops falling.

Making the domputer.

There are lots of different kinds of commands that you can send by flipping switches on and off and making those signals interact.  For example, suppose you want something to happen only if two switches are on – if the first switch is on AND the second switch is on.  For this you would need to make something called an “AND gate” – an interaction in chains of current that will continue the chain if both switches are on and will stop the chain if either (or both) is off.  How would you do that with dominoes?  In this video, Matt demonstrates how to make an AND gate out of dominoes: Domino AND gate.  Check out this video for OR (the chain continues if one or the other or both are on) and XOR (“exclusive or,” the chain continues if one or the other, but not both, are on) gates:

Matt’s Domputer does something very simple: it adds numbers in binary.  But, as you might imagine, it was extremely complicated to build!  According to the Manchester Science Festival Twitter feed, the Domputer used about 10,000 dominoes and would take about 13,600 years to do what a normal processor could do in a second.  Wow!

Here it is in action.  It messed up on this calculation (9+3), but succeeded in later attempts – and is fascinating to watch nonetheless!

Awesome!

Next up, we’ve written about mathematical knitting before (remember Wooly Thoughts and the prime factorization sweater?), but here’s a great site I recently found made by mathematician, knitter, and dancer Sarah-Marie Belcastro.

This site is full of articles and about and patterns for all kinds of cool mathematical objects – like Klein bottles (which make great hats, by the way)!  In her post about knitted Klein bottles (and all of the other objects she makes), Sarah-Marie not only describes how to knit the objects but a lot of mathematics about them.  I don’t know about you, but I always find mathematical ideas easier to understand when I can make models of them, or at least read about models being made.  Sarah-Marie does a great job of blending mathematical descriptions with how-to-make-it recipes.

Some other patterns that I love are Sarah-Marie’s 8-colored two-hole torus pants and this knitted trefoil knot.

Finally, are you wondering what to do with all those campaign posters you have left over from the election?  Here’s George Hart’s take on what to do with them:

Bon appetit!

Pixel Art, Gothic Circle Patterns, and First Past the Post

Welcome to this week’s Math Munch!

Guess what? Today is Math Munch’s one-year anniversary!

We’re so grateful to everyone who has made this year so much fun: our students and readers; everyone who has spread the word about Math Munch; and especially all the people who do and make the cool mathy things that we so love to find and share.

Speaking of which…

Mathematicians have studied the popular puzzle called Sudoku in numerous ways. They’ve counted the number of solutions. They’ve investigated how few given numbers are required to force a unique solution. But Tiffany C. Inglis came at this puzzle craze from another angle—as a way to encode pixel art!

Tiffany studies computer graphics at the University of Waterloo in Ontario, Canada. She’s a PhD candidate at the Computer Graphics Lab (which seems like an amazing place to work and study—would you check out these mazes!?)

Tiffany C. Inglis, hoisting a buckyball

Tiffany tried to find shading schemes for Sudoku puzzles so that pictures would emerge—like the classic mushroom pictured above. Sudoku puzzles are a pretty restrictive structure, but Tiffany and her collaborators had some success—and even more when they loosened the rules a bit. You can read about (and see!) some of their results on this rad poster and in their paper.

Thinking about making pictures with Sudoku puzzles got Tiffany interested in pixel art more generally. “I did some research on how to create pixel art from generic images such as photographs and realized that it’s an unexplored area of research, which was very exciting!” Soon she started building computer programs—algorithms—to automatically convert smooth line art into blockier pixel art without losing the flavor of the original. You can read more about Tiffany’s pixelization research on this page of her website. You should definitely check out another incredible poster Tiffany made about this research!

To read more of my interview with Tiffany, you can click here.

Cartoon Tiffany explains what makes a good pixelization. Check out the full comic!

I met Tiffany this past summer at Bridges, where she both exhibited her artwork and gave an awesome talk about circle patterns in Gothic architecture. You may be familiar with Apollonian gaskets; Gothic circle patterns have a similar circle-packing feel to them, but they have some different restrictions. Circles don’t just squeeze in one at a time, but come in rings. It’s especially nice when all of the tangencies—the places where the circles touch—coincide throughout the different layers of the pattern. Tiffany worked on the problem of when this happens and discovered that only a small family has this property. Even so, the less regular circle patterns can still produce pleasing effects. She wrote about this and more in her paper on Gothic circle patterns.

I’m really inspired by how Tiffany finds new ideas in so many place, and how she pursues them and then shares them in amazing ways. I hope you’re inspired, too!

A rose window at the Milan Cathedral, with circle designs highlighted.

A mathematical model similar to the window, which Tiffany created.

An original design by Tiffany. All of these images are from her paper.

Here’s another of Tiffany’s designs. Now try making one of your own!

Using the Mathematica code that Tiffany wrote to build her diagrams, I made an applet where you can try making some circle designs of your own. Check it out! If you make one you really like—and maybe color it in—we’d love to see it! You can send it to us at MathMunchTeam@gmail.com.

(You’ll may have to download a plug-in to view the applet; it’s the same plug-in required to use the Wolfram Demonstrations Project.)

Finally, with Election Day right around the corner, how about a dose of the mathematics of voting?

I’m a fan of this series of videos about voting theory by C.G.P. Grey. Who could resist the charm of learning about the alternative vote from a wallaby, or about gerrymandering from a weasel? Below you’ll find the first video in his series, entitled “The Problems with First Past the Post Voting Explained.” Majority rule isn’t as simple of a concept as you might think, and math can help to explain why. As can jungle animals, of course.

Thanks again for being a part of our Math Munch fun this past year. Here’s to a great second course! Bon appetit!


PS I linked to a bunch of papers in this post. After all, that’s the traditional first anniversary gift!

Turing, Nets, and More Yoshimoto

Welcome to this week’s Math Munch!

The Turing Tenner

What you see there is a 10 pound note. You know, British money. So who’s that guy on there? Must be a president or king or prime minister or something, right? NO! That’s Alan Turing, one of the most important mathematicians of the 20th century. During WWII, he was a codebreaker for the Allies, intercepting German submarine codes. His analysis of the Enigma Machine was a huge turning point in the war. (video explanation)

In England they put the queen on one side of the money, but the other’s used for significant Brits. Charles Darwin is currently on the 10 pound note, but these things change, and there’s a petition to get Turing on the ten. A Turing Tenner, as they call it. It’s all part of Turing’s 100th birthday celebration.

Google’s homage to Alan Turing

Since Turing did some of the earliest work on computing theory and artificial intelligence, Google paid tribute to the computer legend with a recent doodle. It’s a fantastic little puzzle game based on his work. I’ll let you figure it out, but definitely try this one. Click here to play!

In last week’s munch, Justin introduced us to the Yoshimoto Cube, and we’ve kept on thinking about it.  Here’s a couple simple templates for making one cubelet.  (template 1, template 2)  Make 8 of those and hinge them together with some tape.  I made a short video to show you how to connect them.  But it didn’t end there!

A flat template for a 3D model like that is called a net or a mesh.  Do you know any nets for a cube?  There’s actually lots.  Check out this site, where it’s your job to figure out which nets fold up into a cube and which ones don’t.  It’s a lot of fun.  Here’s another net site showing lots of nets for a pyramid, dodecahedron, and a whole bunch of other solid shapes.  How many do you think there are for a tetrahedron?  Can you design one for an octahedron?

The Monster Mesh

I spent some time this week trying to design a better net for the

The Mega-Monster Mesh
A one-sheet model for the Yoshimoto cube.

Yoshimoto cube, and I think I succeeded!  The tape on my hinges kept breaking, so I wanted to try to make paper hinges.  With my first attempt, which I called The Monster Mesh, I was able to design a net for half of the star.  Down from 8 tape hinges to 2 was a big improvement, but last night I got it perfect!  Using my new version, The Mega-Monster Mesh,  you can make the entire cube without any taped hinges!  The model is pretty complicated, so if you want to give it a shot, feel free to email us at MathMunchTeam@gmail.com with any questions.

Finally, something I’m really really proud of.  Justin and I spent most of Sunday afternoon on the floor of my apartment making a stop motion animation of with Yoshimoto Cube models.  It’s called “Yoshimoto Friends,” and we hope you love it as much as we do.  (We used the free iMotionHD app for iPad and iPhone, in case you want to make your own stop motion animation.)

Bon appetit!

Update:

I made another video showing how the mega-monster mesh folds up.  Here it is, acting like a transforming bug!