# 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!

# Algorithmic House, Billiards, and Picma

Welcome to this week’s Math Munch!

Check out this beautiful building:

This is the Endesa Pavilion, located in Barcelona, Spain.  It’s also called Solar House 2.0, and that’s because the tops of all of those pyramid-spikes are covered in solar panels.  But that’s not all – this house was designed to best capture sunlight in the exact location it was built using a mathematical algorithm.

To build this house, architect Rodrigo Rubio, who works for the Institute for Advanced Architecture of Catalonia, first tracked the path of the sun over the spot he wanted to build the house.  He then plugged that data into a computer program.  This program is a set of mathematical steps called an algorithm that turns data about the movement of the sun in the sky into a geometric building.  The building it creates is the best – or optimal – building for that spot.

It puts solar panels in locations on the building that get the most sunlight and orients them to get the most exposure.  It places windows of different sizes and overhangs at different angles around the house to get the best ventilation, block sunlight from entering the house, and keep the house cool in the summer and warm in the winter.   And, because it’s an algorithm, it can be used to design the optimal house for any location.  The program then creates a pattern for the wooden pieces that make up the house.  This pattern can be sent to a machine that cuts out the pieces, which builders put together like a puzzle.

In this video, Rodrigo explains how the building was designed, how the design works, and how this design can be used to make eco-friendly houses all over the world.

Next, have you ever played billiards?  Maybe you’ve played pool or watched Donald Duck play billiards.  It’s interesting to see how a pool ball moves around on a rectangular billiards table, which is how the table is usually shaped.  But it’s even more interesting to see how a ball moves around on a triangular, pentagonal, circular, or elliptical billiards table!

Want to try?  Check out this series of applets from Serendip, an exploratory math and science website started by some professors at Bryn Mawr College in Pennsylvania.  Serendip aims to help people ask and answer their own questions about the world we live in.  In these billiards applets, you can explore dynamical systems – mathematical structures in which an object moves according to a rule.   In some situations, the object will move in a predictable way.  But in other situations, the object moves chaotically.  As you play with the applets, see if you can figure out how the shape of the table effects whether the billiard ball will move chaotically or predictably.  These applets also make some beautiful star-like designs!

Finally, here’s a new game: Picma Squared.  In this game, you use logic to figure out how to color the squares in the grid to make a picture.  It starts out simple, but the higher levels are really challenging!  Enjoy!

Look for this game and others on our Games page!

Bon appetit!

# Newroz, a Math Factory, and Flexagons

Welcome to this week’s Math Munch!

You’ve probably seen Venn diagrams before. They’re a great way of picturing the relationships among different sets of objects.

But I bet you’ve never seen a Venn diagram like this one!

Frank Ruskey

That’s because its discovery was announced only a few weeks ago by Frank Ruskey and Khalegh Mamakani of the University of Victoria in Canada. The Venn diagrams at the top of the post are each made of two circles that carve out three regions—four if you include the outside. Frank and Khalegh’s new diagram is made of eleven curves, all identical and symmetrically arranged. In addition—and this is the new wrinkle—the curves only cross in pairs, not three or more at a time. All together their diagram contains 2047 individual regions—or 2048 (that’s 2^11) if you count the outside.

Frank and Khalegh named this Venn diagram “Newroz”, from the Kurdish word for “new day” or “new sun”. Khalegh was born in Iran and taught at the University of Kurdistan before moving to Canada to pursue his Ph.D. under Frank’s direction.

Khalegh Mamakani

“Newroz” to those who speak English sounds like “new rose”, and the diagram does have a nice floral look, don’t you think?

When I asked Frank what it was like to discover Newroz, he said, “It was quite exciting when Khalegh told me that he had found Newroz. Other researchers, some of my grad students and I had previously looked for it, and I had even spent some time trying to prove that it didn’t exist!”

Khalegh concurred. “It was quite exciting. When I first ran the program and got the first result in less than a second I didn’t believe it. I checked it many times to make sure that there was no mistake.”

You can click these links to read more of my interviews with Frank and Khalegh.

I enjoyed reading about the discovery of Newroz in these articles at New Scientist and Physics Central. And check out this gallery of images that build up to Newroz’s discovery. Finally, Frank and Khalegh’s original paper—with its wonderful diagrams and descriptions—can be found here.

 A single closed curve—or “petal”— of Newroz. Eleven of these make up the complete diagram. A Venn diagram made of four identical ellipses. It was discovered by John Venn himself!

For even more wonderful images and facts about Venn diagrams, a whole world awaits you at Frank’s Survey of Venn Diagrams.

On Frank’s website you can also find his Amazing Mathematical Object Factory! Frank has created applets that will build combinatorial objects to your specifications. “Combinatorial” here means that there are some discrete pieces that are combined in interesting ways. Want an example of a 5×5 magic square? Done! Want to pose your own pentomino puzzle and see a solution to it? No problem! Check out the rubber ducky it helped me to make!

A pentomino rubber ducky!

Finally, Frank mentioned that one of his early mathematical experiences was building hexaflexagons with his father. This led me to browse around for information about these fun objects, and to re-discover the work of Linda van Breemen. Here’s a flexagon video that she made.

And here’s Linda’s page with instructions for how to make one. Online, Linda calls herself dutchpapergirl and has both a website and a YouTube channel. Both are chock-full of intricate and fabulous creations made of paper. Some are origami, while others use scissors and glue.

I can’t wait to try making some of these paper miracles myself!

Bon appetit!

# Music Box, FatFonts, and the Yoshimoto Cube

Welcome to this week’s Math Munch!

 The Whitney Music Box Jim Bumgardner Solar Beat

With the transit of Venus just behind us and the summer solstice just ahead, I’ve got the planets and orbits on my mind. I can’t believe I haven’t yet shared with you all the Whitney Music Box. It’s the brainchild of Jim Bumgardner, a man of many talents and a “senior nerd” at Disney Interactive Labs. His music box is one of my favorite things ever–so simple, yet so mesmerizing.

It’s actually a bunch of different music boxes–variations on a theme. Colored dots orbit in circles, each with a different frequency, and play a tone when they come back to their starting points. In Variation 0, for instance, within the time it takes for the largest dot to orbit the center once, the smallest dot orbits 48 times. There are so many patterns to see–and hear! There are 21 variations in all. Go nuts! In this one, only prime dots are shown. What do you notice?

You can find a more astronomical version of this idea at SolarBeat.

Above you’ll find a list of the numerals from 1 to 9. Or is it 0 to 9?

Where’s the 0 you ask? Well, the idea behind FatFonts is that the visual weight of a number is proportional to its numerical size. That would mean that 0 should be completely white!

FatFonts can also be nested. The first number below is 64. Can you figure out the second?

 This is 64 in FatFonts. What number is this?Click to zoom!

FatFonts was developed by the team of Miguel NacentaUta Hinrichs, and Sheelagh Carpendale. You can see some uses that FatFonts has been put to on their Gallery page, and even download FatFonts to use in your word processor. Move over, Times New Roman!

This past week, Paul pointed me to this cool video by George Hart about interlocking complementary polyhedra that together form a cube. It reminded me of something I saw for the first time a few years ago that just blew me away. You have to see the Yoshimoto Cube to believe it:

In addition to its more obvious charms, something that delights me about the Yoshimoto Cube is how it was found so recently–only in 1971, by Naoki Yoshimoto.  (That other famous cube was invented in 1974 by Ernő Rubik.) How can it be that simple shapes can be so inexhaustible? If you’re feeling inspired, Make Magazine did a short post on the Yoshimoto Cube a couple of years that includes a template for making a Yoshimoto Cube out of paper. Edit: These template and instructions aren’t great. See below for better ones!

Since it’s always helpful to share your goals to help you stick to them, I’ll say that this week I’m going to make a Yoshimoto Cube of my own. Begone, back burner! Later in the week I’ll post some pictures below. If you decide to make one, share it in the comments or email us at

MathMunchTeam@gmail.com

We’d love to hear from you.

Bon appetit!

Update:

Here are the two stellated rhombic dodecahedra that make the Yoshimoto Cube that Paul and I made! Templates, instructions, and video to follow!

Here are two different templates for the Yoshimoto cubelet. You’ll need eight cubelets to make one star.

And here’s how you tape them together:

# A Sweater, Paper Projects, and Math Art Tools

Sondra Eklund and her Prime Factorization Sweater

Welcome to this week’s Math Munch!

Check out Sondra Eklund and her awesome prime factorization sweater! Sondra is a librarian and a writer who writes a blog where she reviews books. She also is a knitter and a lover of math!

Each number from two to one hundred is represented in order on the front of Sondra’s sweater. Each prime number is a square that’s a different color; each composite number has a rectangle for each of the primes in its prime factorization. This number of columns that the numbers are arranged into draws attention to different patterns of color. For instance, you can see a column that has a lot of yellow in it on the front of the sweater–these are all number that contain five as a factor.

You can read more about Sondra and her sweater on her blog. Also, here’s a response and variation to Sondra’s sweater by John Graham-Cumming.

Next up, do you like making origami and other constructions out of paper? Then you’ll love the site made by Laszlo Bardos called CutOutFoldUp.

 Laszlo Bardos A Rhombic Spirallohedron A decagon slide-together

Laszlo is a high school math teacher and has enjoyed making mathematical models since he was a kid. On CutOutFoldUp you’ll find gobs of projects to try out, including printable templates. I’ve made some slide-togethers before, but I’m really excited to try making the rhombic spirallohedron pictured above! What is your favorite model on the site?

Last up, Paul recently discovered a great mathematical art applet called Recursive Drawing. The tools are extremely simple. You can make circles and squares. You can stretch these around. But most importantly, you can insert a copy of one of your drawings into itself. And of course then that copy has a copy inside of it, and on and on. With a very simple interface and very simple tools, incredible complexity and beauty can be created.

Recursive Drawing was created by Toby Schachman, an artist and programmer who graduated from MIT and now lives in New York City and attends NYU.  You can watch a demo video below.

Recursive Drawing is one of the first applets on our new Math Art Tools page. We’ll be adding more soon. Any suggestions? Leave them in the comments!

Bon appetit!

# Polyominoes, Rubix, and Emmy Noether

Welcome to this week’s Math Munch!

Check out the Pentomino Project, a website devoted to all things about polyominoes by students and teachers from the K. S. O. Glorieux Ronse school in Belgium.

Their site is full of lots of useful information about polyominoes, such as what the different polyominoes look like and how they are formed.

In this puzzle, place the twelve pentominoes as "islands in a sea" so that the area of the sea is a small as possible. The pentominoes can't touch, even at corners. Here's a possible solution.

Even more awesome, though, is their collection of polyomino puzzles – about dissections, congruent pieces, tilings, and more!  They have a contest every year  – and people from around the world are encouraged to participate!  If you solve a puzzle, you can send them your solution and they might post it on their site.

Next, have you ever thought to yourself, “Gee, I wonder if I can make my own Rubix Cube?”  Well, sixth grader August did just that.  And, after several days of searching for patterns and working hard with paper, scissors, string, and tape, August succeeded!  His 2-by-2 Rubix Cube works just like any other, is fun to play with, and – even better – was fun to make.

Try it yourself:

Finally, ever heard of Emmy Noether?  It’s not surprising if you haven’t, because, according a New York Times article about her, “few can match in the depths of her perverse and unmerited obscurity….”  But, she was one of the most influential mathematicians and scientists of the 20th century – and was named by Albert Einstein the most “significant” and “creative” woman mathematician of all time.  You can read about Emmy’s influential theorem, and her struggles to become accepted in the mathematical community as a Jewish woman, in this article.

Want to learn more about women mathematicians throughout history?  Check out this site of biographies from Agnes Scott College.

Bon appetit!

# (Beat, Beat, Beat…)

Welcome to this week’s Math Munch!

What could techno rhythms, square-pieces dissections, and windshield wipers have in common?

## The Euclidean Algorithm!

Say what?  The Euclidean Algorithm is all about our good friend long division and is a great way of finding the greatest common factor of two numbers. It relies on the fact that if a number goes into two other numbers evenly, then it also goes into their difference evenly.  For example, 5 goes into both 60 and 85–so it also goes into their difference, 25.  Breaking up big objects into smaller common pieces is a big idea in mathematics, and the way this plays out with numbers has lots of awesome aural and visual consequences.

Here’s the link that prompted this post: a cool applet where you can create your own unique rhythms by playing different beats against each other.  It’s called “Euclidean Rhythms” and was created by Wouter Hisschemöller, a computer and audio programmer from the Netherlands.

(Something that I like about Wouter’s post is that it’s actually a correction to his original posting of his applet.  He explains the mistake he made, gives credit to the person who pointed it out to him, and then gives a thorough account of how he fixed it.  That’s a really cool and helpful way that he shared his ideas and experiences.  Think about that the next time you’re writing up some math!)

For your listening pleasure, here’s a techno piece that Wouter composed (not using his applet, but with clear influences!)

Breathing Pavement

Here’s an applet that demonstrates the geometry of the Euclidean Algorithm.  If you make a rectangle with whole-number length sides and continue to chop off the biggest (non-slanty) square that you can, you’ll eventually finish.  The smallest square that you’ll chop will be the greatest common factor of the two original numbers.  See it in action in the applet for any number pair from 1 to 100, with thanks to Brown mathematics professor Richard Evan Schwartz, who maintains a great website.

Holyhedron, layer three

One more thing, on an entirely different note: Holyhedron! A polyhedron where every face contains a hole. The story is given briefly here. Pictures and further details can be found on the website of Don Hatch, finder of the smallest known holyhedron.  It’s a mathematical discovery less than a decade old–in fact, no one had even asked the question until John Conway did so in the 1990s!

Have a great week! Bon appétit!