One of my favorite math sites on the internet is mathpuzzle. It’s written and curated by recreational mathematician Ed Pegg Jr. About once a month, Ed makes a post that shares a ton of awesome math—interesting tilings, tricky puzzles, results about polyhedra and polyominos, and so much more. Below are some of my favorite finds at mathpuzzles. Go to the site to discover much more to explore!
Shapes that three kinds of polyominoes can tile.
Erich Friedman’s 2012 holiday puzzles
A slideable, flexible hypercube you can hold in your hands! Video below.
Next, have you ever wanted to be a movie star? How about a math movie star? Then there are two math video contests that you should know about. The first is for middle schoolers— the Reel Math Challenge. It’s run by MATHCOUNTS, which has for many years run a middle school problem solving contest. (I competed in it when I was in middle school.) This is only the second year for the Reel Math Challenge, but lots of videos have already been created. You can check them out here.
The second contest is for high schoolers and is called Math-O-Vision. The challenge is to make a video that shows “the way Math fills our world.” Math-O-Vision is sponsored by the Dartmouth College Math Department and the Neukom Institute.
Finally, here’s a fun little applet I found called Make-a-Flake. You can use it to make intricate digital snowflake designs.
Two snowflakes from the Make-a-Flake gallery.
Of course, it’s a lot of fun to make non-virtual snowflakes as well—find a pair of scissor and some paper and go for it! For basic instructions, head over to snowflakes.info. And for some inspiration, check out this Flickr group!
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!
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.
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:
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.
The second contest is for high schoolers and is called Math-O-Vision. The challenge is to make a video that shows “the way Math fills our world.” Math-O-Vision is sponsored by the Dartmouth College Math Department and the Neukom Institute.
Finally, here’s a fun little applet I found called Make-a-Flake. You can use it to make intricate digital snowflake designs.
Math Munch 