# Circling, Squaring, and Triangulating

Welcome to this week’s Math Munch!

How good are you at drawing circles? To find out, try this circle drawing challenge. There are adorable cat pictures for prizes!

What’s the best score you can get? And hey—what’s the worst score you can get? And how is your score determined? Well, no matter how long the path you draw is, using that length to make a circle would surround the most area. How close your shape gets to that maximum area determines your score.

Do you think this is a good way to measure how circular a shape is? Can you think of a different way?

Dido, Founder and Queen of Carthage.

This idea that a circle is the shape that has the biggest area for a fixed perimeter reminds me of the story of Dido and her famous problem. You can find a retelling of it at Mathematica Ludibunda, a charming website that’s home to all sorts of mathematical stories and puzzles. The whole site is written in the voice of Rapunzel, but there’s a team of authors behind it all. Dido’s story in particular was written by a girl named Christa.

If you have any trouble drawing circles in the applet, you might try using pencil and paper or a chalkboard. I bet if you practice your circling and get good at it, you might even be able to challenge this fellow:

The simple perfect squared square
of smallest order.

Next up is squaring and the incredible Squaring.Net. The site is run by Stuart Anderson, who works at the Reserve Bank of Australia and lives in Sydney.

The site gathers together all of the research that’s been done about breaking up squares and rectangles into squares. It’s both a gallery and an encyclopedia. I love getting to look at the timelines of discovery—to see the progress that’s been made over time and how new things have been discovered even this year! Just within the last month or so, Stuart and Lorenz Milla used computers to show that there are 20566 simple perfect squared squares of order 30. Squaring.Net also has a wonderful links page that can connect you to more information about the history of squaring, as well as some of the delightful mathematical art that the subject has inspired.

Last up this week is triangulating. There are lots of ways to chop up a shape into triangles, and so I’ll focus on one particular way known as a Delaunay triangulation. To make one, scatter some points on the plane. Then connect them up into triangles so that each triangle fits snugly into a circle that contains none of the scattered points.

Fun Fact #1: Delaunay triangulations are named for the Soviet mathematician Boris Delaunay. What else is named for him? A mountain! That’s because Boris was a world-class mountain climber.

Fun Fact #2: The idea of Delaunay triangulations has been rediscovered many times and is useful in fields as diverse as computer animation and engineering.

Here are two uses of Delaunay triangulations I’d like to share with you. The first comes from the work of Zachary Forest Johnson, a cartographer who shares his work at indiemaps.com. You can check out a Delaunay triangulation applet that he made and read some background about this Delaunay idea here. To see how Zach uses these triangulations in his map-making, you’ve gotta check out the sequence of images on this page. It’s incredible how just a scattering of local temperature measurements can be extended to one of those full-color national temperature maps. So cool!

 Zachary Forest Johnson A Delaunay triangulation used to help create a weather map.

Finally, take a look at these images that Jonathan Puckey created. Jonathan is a graphic artist who lives in Amsterdam and shares his work on his website. In 2008 he invented a graphical process that uses Delaunay triangulations and color averaging to create abstractions of images. You can see more of Jonathan’s Delaunay images here.

I hope you find something to enjoy in these circles, squares, and triangles. Bon appetit!

# We Use Math, Integermania, and Best-of-Seven

Welcome to this week’s Math Munch!

“When will I use math?” Have you ever asked this question? Well, then you are in for a treat, because the good people of We Use Math have some answers for you! This site was created by the Math Department at Brigham Young University to help share information about career paths that are opened up by studying mathematics. Here’s their introductory video:

The We Use Math site shares write-ups about a wide range of career opportunities that involve doing mathematics. I was glad to learn more about less-familiar mathy careers like technical writing and cost estimation. Also, my brother has studied some operations management in college, so it was great to read the overview of that line of work. In addition, the We Use Math site has pages about recent math discoveries and about unsolved math problems. Check them out!

Next up is one of my long-time favorite websites: Integermania!

Perhaps you’ve heard of the four 4’s problem before. Using four 4’s and some arithmetic operations, can you make the numbers from 1 to 20? Or even higher? Some numbers are easy to make, like 16. It’s 4+4+4+4. Some are sneakier, like 1. One way it can be created is (4+4)/(4+4). But what about 7? Or 19? This is a very common type of problem in mathematics—which math objects of a certain type can be built with limited tools?

Steven J. Wilson

Integermania is a website where people from around the world have submitted number creations made of four small numbers and operations. It’s run by Steven J. Wilson, a math professor at Johnson County Community College in Kansas. (Steven has even more great math resources at his website Milefoot.com)

There are many challenges at Integermania: four 4’s, the first four prime numbers, the first four odds, and even the digits of Ramanujan’s famous taxicab number (1729).

Here are some number creations made of the first four prime numbers.
Can you make some of your own?

One of my favorite aspects of Integermania is the way it rates number creations by “exquisiteness level“. If a number creation is made using only simple operations—like addition or multiplication—then it’s regarded as more exquisite than if it uses operations like square roots or percentages. I also love how Integermania provides an opportunity for anyone to make their mark in the big world of mathematical research—it’s like scrawling a mathematical “I wuz here!” After years of visiting the site, I just submitted for the first time some number creations of my own. I’ll let you know how it goes, and I’d love to hear about it if you decide to submit, too.

Here are recaps of all the World Series since 1903 from MLB.com

Now coming to the plate: my final link of the week! Monday was the first day of the new Major League Baseball season. I want to share with you a New York Times article from last December. It’s called Keeping Score: Over in Four About a Fifth of the Time. The article digs into the outcomes of all of the World Series championships—not so much who won as how they won. It takes four victories to win a seven-game series, and there are 35 different ways that a best-of-seven series can play out, put in terms of wins and losses for the overall winner. For instance, a clean sweep would go WWWW, while another sequence would be WWLLWW. The article examines which of these win-loss sequences have been the most common in the World Series.

(Can you figure out why there are 35 possible win-loss sequences in a seven-game series? What about for a best-of-five series? And what if we tried to model the outcome of a series by assuming each team has a fixed chance of winning each game?)

A clip of the stats that are displayed in the Times article. Click through to see it all.

I was curious to know if the same results held true in other competitions. Are certain win-loss sequences rare across different sports? Are “sweeps” the most common outcome? After sifting through Wikipedia for a while, I was able to compile the statistics about win-loss sequences for hockey’s Stanley Cup Finals. This has been a best-of-seven series since 1939, and it has been played 73 times since then. (It didn’t happen in 2005 because of a lockout.) You can see the results of my research in this document. Two takeaways: sweeps are also the most common result in hockey, but baseball more frequently requires the full seven games to determine a winner.

It could be a fun project to look at other best-of-seven series, like the MLB’s League Championship Series or basketball’s NBA Finals. If you pull that data together, let us know in the comments!

Batter up, and bon appetit!

******

UPDATE (4/4/13): My first set of five number creations was accepted and are now posted on the Ramanujan challenge page. Here are the three small ones! Can you find a more exquisite way of writing 47 than I did?

# 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!