Tag Archives: graphs

Marjorie Rice, Inspired by Math, and Subways

Welcome to this week’s Math Munch!

A few weeks ago, I learned about an amazing woman named Marjorie Rice.  Marjorie is a mathematician – but with a very unusual background.

mrice_picMarjorie had no mathematical education beyond high school.  But, Marjorie was always interested in math.  When her children were all in school, Marjorie began to read about and work on math problems for fun.  Her son had a subscription to Scientific American, and Marjorie enjoyed reading articles by Martin Gardner (of hexaflexagon fame).  One day in 1975, she read an article that Martin Gardner wrote about a new discovery about pentagon tessellations.  Before several years earlier, mathematicians had believed that there were only five different types of pentagons that could tessellate – or cover the entire plane without leaving any gaps.   But, in 1968, three more were discovered, and, in 1975, a fourth was found – which Martin Gardner reported on in his article.

Marjorie's first type of pentagonWhen she read about this, Marjorie became curious about whether she could find her own new type of pentagon that could tile the plane.  So, she got to work.  She came up with her own notation for the relationships between the angles in her pentagons.  Her new notation helped her to see things in ways that professional mathematicians had overlooked.  And, eventually… she found one!  Marjorie wrote to Martin Gardner to tell him about her discovery.  By 1977, Marjorie had discovered three more types of pentagons that tile the plane and her new friend, the mathematician Doris Schattschneider, had published an article about Marjorie’s work  in Mathematics Magazine.

type11There are now fourteen different types of pentagons known to tile the plane… but are there more?  No one knows for sure.  Whether or not there are more types of pentagons that tile the plane is what mathematicians call an open problem.  Maybe you can find a new one – or prove that one can’t be found!

Marjorie has a website called Intriguing Tessellations on which she’s written about her work and posted some of her tessellation artwork.  Here is one of her pentagon tilings transformed into a tessellation of fish.

fishgrid fishsm

By the way, it was Marjorie’s birthday a few weeks ago.  She just turned 90 years old.  Happy Birthday, Marjorie!

wild about math logoNext up, I just ran across a great blog called Wild About Math!  This blog is written by Sol Lederman, who used to work with computers and LOVES math.  My favorite part about this blog is a series of interviews that Sol calls, “Inspired by Math.”  Sol has interviewed about 23 different mathematicians, including Steven Strogatz (who has written two series of columns for the New York Times about mathematics) and Seth Kaplan and Deno Johnson, the producer and writer/director of the Flatland movies.  You can listen to Sol’s podcasts of these interviews by visiting his blog or iTunes.  They’re free – and very interesting!

subway map 2Finally, what New York City resident or visitor isn’t fascinated by the subway system? And what New York City resident or visitor doesn’t spend a good amount of time thinking about the fastest way to get from point A to point B?  Do you stay on the same train for as long as possible and walk a bit?  Or do you transfer, and hope that you don’t miss your train?

chris and matt

Chris and Matt, on the subway.

Well, in 2009, two mathematicians from New York – Chris Solarz and Matt Ferrisi – used a type of mathematics called graph theory to plan out the fastest route to travel the entire New York City subway system, stopping at every station.  They did the whole trip in less than 24 hours, setting a world record!  Graph theory is the branch of mathematics that studies the connections between points or places.  In their planning, Chris and Matt used graph theory to find a route that had the most continuous travel, minimizing transfers, distance, and back-tracking.  You can listen to their fascinating story in an interview with Chris and Matt done by the American Mathematical Society here.

If you’re interested in how graph theory can be used to improve the efficiency of a subway system, check out this article about the Berlin subway system (the U-bahn).  Students and professors from the Technical University Berlin used graph theory to create a schedule that minimized transfer time between trains.  If only someone would do this in New York…

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!