# 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.

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!

# Rectangles, Explosions, and Surreals

Welcome to this week’s Math Munch!

What is 3 x 4?   3 x 4 is 12.

Well, yes. That’s true. But something that’s wonderful about mathematics is that seemingly simple objects and problems can contain immense and surprising wonders.

How many squares can you find in this diagram?

As I’ve mentioned before, the part of mathematics that works on counting problems is called combinatorics. Here are a few examples for you to chew on: How many ways can you scramble up the letters of SILENT? (LISTEN?) How many ways can you place two rooks on a chessboard so that they don’t attack each other? And how many squares can you count in a 3×4 grid?

Here’s one combinatorics problem that I ran across a while ago that results in some wonderful images. Instead of asking about squares in a 3×4 grid, a team at the Dubberly Design Office in San Francisco investigated the question: how many of ways can a 3×4 grid can be partitioned—or broken up—into rectangles? Here are a few examples:

How many different ways to do this do you think there are? Here’s the poster that they designed to show the answer that they found! You can also check out this video of their solution.

In their explanation of their project, the team states that “Design tools are becoming more computation-based; designers are working more closely with programmers; and designers are taking up programming.” Designing the layout of a magazine or website requires both structural and creative thinking. It’s useful to have an idea of what all the possible layouts are so that you can pick just the right one—and math can help you to do it!

If you’d like to try creating a few 3×4 rectangle partitions of your own, you can check out www.3x4grid.com. [Sadly, this page no longer works. See an archive of it here. -JL, 10/2016]

Next up, explosions! I could tell you about the math of the game Minesweeper (you can play it here), or about exploding dice. But the kind of explosion I want to share with you today is what’s called a “combinatorial explosion.” Sometimes a problem that appears to be an only slightly harder variation of an easy problem turns out to be way, way harder. Just how BIG and complicated even simple combinatorics problems can get is the subject of this compelling and also somewhat haunting video.

Donald Knuth

Finally, all of this counting got me thinking about big numbers. Previously we’ve linked to Math Cats, and Wendy has a page where you can learn how to say some really big numbers. But thinking about counting also made me remember an experience I had in middle school where I found out just how big numbers could be! I was in seventh grade when I read this article from the December 1995 issue of Discover Magazine. It’s called “Infinity Plus One, and Other Surreal Numbers” and was written by Polly Shulman. I remember my mind being blown by all of the talk of infinitely-spined aliens and up-arrow notation for naming numbers. Here’s an excerpt:

Mathematicians and precocious five-year-olds have long been fascinated by the endlessness of numbers, and they’ve named the endlessness infinity. Infinity isn’t a number like 1, 2, or 3; it’s hard to say what it is, exactly. It’s even harder to imagine what would happen if you tried to manipulate it using the arithmetic operations that work on numbers. For example, what if you divide it in half? What if you multiply it by 2? Is 1 plus infinity greater than, less than, or the same size as infinity plus 1? What happens if you subtract 1 from it?

After I read this article, John Conway and Donald Knuth became heros of mine. (In college, I had the amazing fortune to have breakfast with Conway one day when he was visiting to give a lecture!) Knuth has a book about surreals that’s the friendliest introduction to the surreal numbers that I know of, and in this video, Vi Hart briefly touches on surreal numbers in discussing proofs that .9 = 1. Boy, would I love to see a great video or online resource that simply and beautifully lays out the surreal numbers in all their glory!

It was fun for me to remember that Discover article. I hope that you, too, run across some mathematics that leaves a seventeen-year impression on you!

Bon appetit!

# Squiggles, Spheres, and Taxes

Welcome to this week’s Math Munch!

Check out this cool doodle animation from the blog of Matt Henderson. Matt studied math at Cambridge as an undergrad and now does research on speech and language technology. His idea for a doodle was to start with an equilateral triangle and then encircle it with squiggles until it eventually turned into a square.

 Matt Henderson Matt’s triangle-to-square squiggle

Matt has all kinds of beautiful and intricate mathematical images on his blog, many of them animated using computer code. He made a similar squiggle-doodle that evolves a straight line into a profile of his face; an animation of rolling a ball on a merry-go-round; a million dot generator; and many more!

Along the same “lines” as Matt’s squiggle, Ted Theodosopoulos wrote an article in Peer Points reviewing a research paper by Stanford mathematician Ravi Vakil. The title of Ravi’s paper is “The Mathematics of Doodling.”

Ravi’s doodle

Next up, check out this cool visualization of a sphere.

The title of the video is Spherikal and was created by Ion Lucin, a graphic artist in Spain.

Something neat comes out about Ion’s attitude toward learning and sharing in a comment he makes:

“Thanks for appreciating my work. I was thinking the same, not to reveal my secrets, but then, i to learned from the videos and tutorials of others, i have been working with 3D for a year and a half, and all i know about it i learned it by myself, by seeing tutorials, im from fine arts. In a way a feel i must share , like other did and helped me”

What a great attitude!

Another spherical idea comes from a post on one of my favorite websites: MathOverflow, a question-and-answer site for research-level mathematicians…and anyone else! The question I have in mind was posted by Joe O’Rourke, a mathematician at Smith College and one of my favorite posters on MathOverflow. It’s about a certain kind of random walk on a sphere. Check it out!

For this step distance, it looks like a random walk will fill up the whole sphere. What about other step distances?

Again, such a cool picture is created by translating a mathematical scenario into some computer code!

Since this week is when federal income taxes are due, I’ll leave you with a few links about taxes and the federal budget. First, here’s the IRS’s website for kids. (Yes, for real.)

Next, this infographic lets you examine how President Obama’s 2011 budget proposal divvied up funds to all of the different departments and projects of the federal government. Can you find NASA’s budget?

On a more personal scale, this applet called “Where did my tax dollars go?” does just that—when you give it a yearly personal income, it will calculate how much of it will go toward different ends.

Finally, this applet lets you tinker with the existing tax brackets and see the effect on total revenue generated for the federal government. Can you find a flat tax rate that would keep total tax revenue the same?

Whew! That was a lot; I hope you didn’t find it too taxing. Bon appetit!

# Pentomino Puzzles, Knight’s Tours, and Decimal Maxing

Welcome to this week’s Math Munch!

Have a pentomino tiling problem that’s got you stumped?  Then perhaps the Pentominos Puzzle Solver will be right up your alley! Recently I’ve been thinking a lot about using computer programming and search algorithms to solve mathematical problems, and the Pentomino Puzzle Solver is a great example of the power of coding.  Written by David Eck, a professor of math and computer science at Brandeis University, the solver can find tilings of a variety of shapes.  Watch the application in slow-mo to see how it works; put it into high-gear to see the power of doing mathematics with computers!

Next, here’s a wonderful page about knight’s tours maintained by George Jelliss, a retiree from the UK.  He says on his introductory page, “I have been interested in questions related to the geometry of the knight’s move since the early 1970s.” George has investigated “leapers” or “generalized knights”—pieces that move in other L-shapes than the traditional 2×1—and he even published his own chess puzzle magazine for a number of years.  His webpage includes a great section about the history of knights tours, and I’m a fan of the beautiful catalog of “crosspatch” tours. Great stuff!

Multiplication, addition, division: which gives the biggest result?

Last but not “least”, to the left you’ll find a tiny chunk of a very large table that was constructed and colored by Debra Borkovitz, a math professor at Wheelock College.  Debra describes how, “Students often have poor number sense about multiplication and division with numbers less than one.”  She created an investigation where students decide, for any pair of decimals, which is biggest–multiplying them, adding them, or subtracting them.  For 1.0 and 1.0 the answer is easy–you should add them, so that you get 2.  .5 and 1 is trickier–adding yields 1.5, multiplying gives .5, but dividing 1 by .5 makes 2, since there are two halves in 1. Finding the biggest value possible given some restrictions is called “maximization” in mathematics, and it’s a very popular type of problem with many applications.

This investigation about makes me wonder: what other kinds of tables could I try to make?

Debra mentions that she got the inspiration for this problem from a newsletter put out by the Association of Women in Mathematics.  There’s lots to explore on their website, including an essay contest for middle schoolers, high schoolers, and undergraduates.

I hope you found something here to enjoy.  Bon appetit!

# Number Gossip, Travels, and Topology

Thanksgiving was great, but I hope you saved room for this week’s Math Munch!

First up, meet Tanya Khovonova, a mathematician and blogger who works at MIT.  Number Gossip is a website of hers where you can find the mysterious facts behind your favorite numbers.  For instance, did you know that the opposite sides of a die add to 7, or that 7 is the only prime number followed by a cube (8=23)? Speaking of 7, I also found this cool test for divisibility by 7 on Tanya’s website.

 Tanya Khovonova Is that divisible by 7? Let's take a walk.

Read about how to use it here, but basically you follow that diagram a certain way, and if you land back at the white dot, then you’re number is divisible by 7. I’m amazed and trying to figure out how it works!

Infographic - Holiday Travel Patterns

Next up, I wanted to share this incredible picture I found today.  It’s an infographic showing travel patterns in the US during the holiday season.  The picture must represent millions of little pieces of data, so I’ve spent a lot of time staring and analyzing it.  Did you notice the bumps in the bottom?  Why is that happening?  Why are the blue lines different from the white lines? There are so many good things to be seen.

Finally, take a look at these pictures!  They’re from Kenneth Baker’s Sketches of Topology blog.  Kenneth makes images demonstrating ideas in topology, one of the most visually appealing branches of mathematics.  Some of it is tough to understand, but the pictures certainly are fascinating.

On a related point, have you taken a look at the Math Munch page of math games? (You can always find the link at the top of the column to the right.)  I just added a topology game, the Four Color Game, and I’m kind of loving it.  It’s based on a famous math result about only needing 4 colors to nicely color any flat map.  This is called the Four Color Theorem, and it’s a part of topology.

Bon appetit!

# Balloons, Numbers, and Mathemusic

Welcome to this week’s Math Munch!  We’ve got a full plate for you.

Vi Hart and Balloon Art

Vi Hart is a “recreational mathemusician,” which means she spends a lot of her free time making math, music, and art of all kinds.  She is best known for her “doodling in math class” videos, but her website is full of cool and creative projects.  This week we’re featuring Vi’s balloon art. There are lots of cool pictures and instructions to make your own balloon creations!

Landon Curt Noll

Next up, Landon Curt Noll is a number theorist, computer scientist, and astronomer who does and makes all kinds of cool things.  Three different times, he discovered the largest prime numbers anyone had ever found!  Here’s a link to his list of curious patterns in the prime numbers.  In another venture, Landon wrote a neat little program that tells you the English name of a number.  How do you pronounce 1,213,141,516,171,819?  Give it a try.  I know million, billion, trillion, quadrillion, and quintillion, but what’s after that?  Check it out: Landon lists the first 10,000 powers of ten!

Finally, the connections between math and music often inspire awesome creations.  Here’s a beautiful video by Michael John Blake in which he converts the digits of pi to notes, and we get to hear what pi sounds like.

Here’s a similar video by Lars Erickson who wrote an entire symphony based on the idea.  “The Pi Symphony” also includes the sound of e, another important math number which is about 2.71828…

Bon appetit!

# Pennies, Knights, and Origami Mazes

Welcome to this week’s Math Munch!

How many pennies do you think this is? Click to find out.

Big numbers are sometimes hard to get a feel for.  A billion is a lot, but so is a million.  The MegaPenny Project is a cool attempt at making the difference between large numbers easier to grasp.  Would 1,000,000 pennies fill a football field or would you need a billion pennies for that?  MegaPenny can help you figure it out.

The first kixote puzzle

Next up, we have kixote, a puzzle in the spirit of Sudoku and Ken-Ken, but involving knight’s moves.  Dan Mackinnon–its creator–has a blog called mathrecreation that he says, “helps me go a little further in my mathematical recreations, helps me understand things better, and sometimes connects me to other people who share similar interests. I hope that it might encourage you to play with math too.”  I’m sure we’ll be linking to more of Dan’s posts in the future!

Finally, since the mazes and paper-folding were so popular last week, we thought that this week we would share some paper-folding mazes! Here is a clip of MIT professor Erik Demaine talking about how he has created origami mazes, preceded by a discussion of origami robots that fold themselves!  The clip is a part of a lecture about origami that Erik gave last spring in New York City for the Math Encounters series put on by the Museum of Mathematics.  You can watch Erik’s entire origami lecture from the beginning by clicking here.

Eric Demaine with a sheet of origami cubes

You can also check out Erik’s Maze Folder applet–but if you try it out, take his warning and start with a small maze!

Bon appetit!