Monthly Archives: November 2012

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!

Plate Folding, Birthdays, and Thanksgiving

Welcome to this week’s Math Munch!

Icosahedron made from 4 paper plates. Click for instructions.

Big news this week, but first let’s have a look at some construction projects you can easily do at home using paper plates, paper clips, and some tape. They come to us from wholemovement.com, the website of Bradford Hansen-Smith. It’s not a stretch to say that Bradford is kind of cuckoo for circles, as you can probably tell form this introductory video. Naturally, the website is all about the amazing things you can do and learn from folding circles. Check out his gallery and you’ll see what I mean. Using these instructions and 4 paper plates I made the sculptures in these pictures. Above is an icosahedron with 4 of the 20 triangles left as empty space, and down below you can see the cuboctahedron of sorts. There’s even an instruction video for this one. So grab some cheap plates, fold ’em up, experiment, and send us your pictures.

square face view

triangular face view

Born 11.14.12

OK, now for the big news. Last Wednesday, my daughter was born!!! I’m so so so happy.  In honor of Nora’s 0th birthday (you turn 1 on your 1st birthday, right?), let’s check out some birthday math. Here’s a cool little birthday number trick I found. It’s sort of magical, but it actually works because that tangle of arithmetic actually just multiplies the month by 10,000, the day by 100, and adds those together with the year. Hopefully you can see how this much simpler version works.

Here’s a well-known birthday problem: How many people need to be in a room before it’s likely that two of them share a birthday? If there’s 400 people in a room, then there’s definitely a birthday match, but if there’s 300 it’s almost certain as well. What’s the smallest crowd so that the probability of a match birthday is over 50%? For the answer and analysis, check out this Numberphile video on the subject featuring James Grime or this New York Times article, by Steven Strogatz, a wonderful mathematician and author.

Both of these solutions are actually wrong!  That’s because they make the false assumptions that each day has the same likelihood of being someone’s birthday.  You can see in the graphs above that that’s not true at all! On the left, look how dark the summer months are, and look at how gray the days are around Thanksgiving and Christmas. You can click on the left image for an interactive version, or click on the right for more graphs and analysis.

A Thanksgiving Pie Chart

Finally, I’m incredibly excited for Thanksgiving (my very favorite holiday), and in that spirit, I want to take a few lines to say “thank you” to you, dear reader. THANK YOU! Whether you’re a weekly muncher or a first time reader, it’s great to know you’re out there enjoying the math we share.

Obviously of course, Thanksgiving is also about the food. Delicious delicious food. Yummmmm! So, Vi Hart is making a series of Thanksgiving themed videos to showcase the math of the meal. Enjoy the videos, but be careful. You may get terribly hungry.

Happy Thanksgiving and bon appetit!

Sandpiles, Prime Pages, and Six Dimensions of Color

Welcome to this week’s Math Munch!

Four million grains of sand dropped onto an infinite grid. The colors represent how many grains are at each vertex. From this gallery.

We got our first snowfall of the year this past week, but my most recent mathematical find makes me think of summertime instead. The picture to the right is of a sandpile—or, more formally, an Abelian sandpile model.

If you pour a bucket of sand into a pile a little at a time, it’ll build up for a while. But if it gets too tall, an avalanche will happen and some of the sand will tumble away from the peak. You can check out an applet that models this kind of sand action here.

A mathematical sandpile formalizes this idea. First, take any graph—a small one, a medium sided one, or an infinite grid. Grains of sand will go at each vertex, but we’ll set a maximum amount that each one can contain—the number of edges that connect to the vertex. (Notice that this is four for every vertex of an infinite square grid). If too many grains end up on a given vertex, then one grain avalanches down each edge to a neighboring vertex. This might be the end of the story, but it’s possible that a chain reaction will occur—that the extra grain at a neighboring vertex might cause it to spill over, and so on. For many more technical details, you might check out this article from the AMS Notices.

This video walks through the steps of a sandpile slowly, and it shows with numbers how many grains are in each spot.

A sandpile I made with Sergei’s applet

You can make some really cool images—both still and animated—by tinkering around with sandpiles. Sergei Maslov, who works at Brookhaven National Laboratory in New York, has a great applet on his website where you can make sandpiles of your own.

David Perkinson, a professor at Reed College, maintains a whole website about sandpiles. It contains a gallery of sandpile images and a more advanced sandpile applet.

Hexplode is a game based on sandpiles.

I have a feeling that you might also enjoy playing the sandpile-inspired game Hexplode!

Next up: we’ve shared links about Fibonnaci numbers and prime numbers before—they’re some of our favorite numbers! Here’s an amazing fact that I just found out this week. Some Fibonnaci numbers are prime—like 3, 5, and 13—but no one knows if there are infinitely many Fibonnaci primes, or only finitely many.

A great place to find out more amazing and fun facts like this one is at The Prime Pages. It has a list of the largest known prime numbers, as well as information about the continuing search for bigger ones—and how you can help out! It also has a short list of open questions about prime numbers, including Goldbach’s conjecture.

Be sure to peek at the “Prime Curios” page. It contains intriguing facts about prime numbers both large and small. For instance, did you know that 773 is both the only three-digit iccanobiF prime and the largest three-digit unholey prime? I sure didn’t.

Last but not least, I ran across this article about how a software company has come up with a new solution for mixing colors on a computer screen by using six dimensions rather than the usual three.

Dimensions of colors, you ask?

The arithmetic of colors!

Well, there are actually several ways that computers store colors. Each of them encodes colors using three numbers. For instance, one method builds colors by giving one number each to the primary colors yellow, red, and blue. Another systems assigns a number to each of hue, saturation, and brightness. More on these systems here. In any of these systems, you can picture a given color as sitting within a three-dimensional color cube, based on its three numbers.

A color cube, based on the RGB (red, green, blue) system.

If you numerically average two colors in these systems, you don’t actually end up with the color that you’d get by mixing paint of those two colors. Now, both scientists and artists think about combining colors in two ways—combining colored lights and combining colored pigments, or paints. These are called additive and subtractive color models—more on that here. The breakthrough that the folks at the software company FiftyThree made was to assign six numbers to each color—that is, to use both additive and subtractive ideas at the same time. The six numbers assigned to a given number can be thought of as plotting a point in a six-dimensional space—or inside of a hyper-hyper-hypercube.

I think it’s amazing that using math in this creative way helps to solve a nagging artistic problem. To get a feel for why mixing colors using the usual three-coordinate system is such a problem, you might try your hand at this color matching game. For even more info about the math of color, there’s some interesting stuff on this webpage.

Bon appetit!

Factorization Dance, Vanishing, and Storm Infographics

Welcome to this week’s Math Munch!

Think fast!  How many dots are there in this picture?

This beautiful picture comes to you from Brent Yorgey and Stephen Von Worley.  If you counted the dots, you probably didn’t count them one at a time.  (And, if you did, can you think of another way to count them?)  If you counted them like I did, you noticed that the dots are arranged in rings of five.  Then maybe you noticed that the rings of five are themselves arranged in rings of five.  And then, finally, you may have noticed that those rings are also arranged in rings of five!  How many dots is that?  5x5x5 = 125!

In this blog post, Brent describes how he wrote the computer program that creates these pictures.  The program factors numbers into primes.  Then, starting with the smallest prime factor, the program arranges dots into regular polygons of the appropriate size with dots (or polygons of dots) at the vertices of the polygon.

Here’s how that works for 90.  90’s prime factorization is 2x3x3x5:

As Brent writes in his post, this counting gets much harder to do with numbers that have large prime factors.  For example, here is 183:

From this picture, I can tell that 183 has 3 as a prime factor.  But how many times does 3 go into 183?  It isn’t immediately clear.

When Stephen saw Brent’s creation, he decided the diagrams would be even more awesome if they danced.  And so he created what he calls the Factor Conga.  If you only click on one link today, click that one.  The Factor Conga is beautiful and totally mesmerizing.

For more factor diagrams, check out this post from the Aperiodical.  There’s a link to the factor diagram by Jason Davies that we posted about over the summer.

Next up, a few months ago we posted about the puzzles of Sam Loyd – one of which was a puzzle called Get Off the Earth.  In this puzzle, the Earth spins and – impossibly – one of the men seems to vanish.  This puzzle is a type of illusion called a geometrical vanish.  In a geometrical vanish, an image is chopped into pieces and the pieces are rearranged to make a new image that takes up the same amount of space as the original, but is missing something.

Here’s a video of another geometrical vanish:

No matter the picture, these illusions are baffling for the same reason.  Rearranging the pieces of an image shouldn’t change the image’s area.  And, yet, in these illusions, that’s exactly what seems to happen.

Check out some of these other links to geometrical vanishes.  Print out your own here.  And think about this: Are these illusions math – and, if it so, how?  I came across geometrical vanishes because a friend asked if I thought the Get Off the Earth puzzle was mathematical.  He isn’t convinced.  If you have any ideas that you think can convince him either way, leave them in the comments section!

Finally, the Math Munch team’s home, New York City, (and this writer’s other home, New Jersey) was hit by a hurricane this week.  The city and surrounding areas are still recovering from the storm.  Sandy left millions of people without power and many without homes.  One way people have tried to communicate the magnitude of what happened is to make infographics of the data.  Making a good infographic requires a blend of mathematics, art, and persuasion.  Here some of the most interesting infographics about the storm that I’ve found.  Check out how they use size, placement, and color to communicate information and make comparisons.

This infographic from the New York Times shows the number of power outages in the northeast and their locations in different states. The size of the circle indicates the number of people without power. Why would the makers of this infographic choose circles? Why do you think they chose to place them on a map? What do you think of the overlapping?

This is part of an infographic from the Huffington Post that compares hurricanes Sandy and Katrina. Click on the image to see the rest of the infographic. What conclusions can you draw about the hurricanes from the information?

This is a wind map of the country captured at 10:30 in the morning on October 30th, the day hurricane Sandy hit. The infographic was made by scientist-artists Fernanda Viegas and Martin Wattenberg. It shows how wind is flowing around the United States in real-time. Check out their site (click on this image) to see what the wind is doing right now in your part of the country!

To those in places affected by Hurricane Sandy, be safe.  To all our readers, bon appetit!