Monthly Archives: April 2012

Polyominoes, Rubix, and Emmy Noether

Welcome to this week’s Math Munch!

Check out the Pentomino Project, a website devoted to all things about polyominoes by students and teachers from the K. S. O. Glorieux Ronse school in Belgium.

Their site is full of lots of useful information about polyominoes, such as what the different polyominoes look like and how they are formed.

In this puzzle, place the twelve pentominoes as "islands in a sea" so that the area of the sea is a small as possible. The pentominoes can't touch, even at corners. Here's a possible solution.

Even more awesome, though, is their collection of polyomino puzzles – about dissections, congruent pieces, tilings, and more!  They have a contest every year  – and people from around the world are encouraged to participate!  If you solve a puzzle, you can send them your solution and they might post it on their site.

Next, have you ever thought to yourself, “Gee, I wonder if I can make my own Rubix Cube?”  Well, sixth grader August did just that.  And, after several days of searching for patterns and working hard with paper, scissors, string, and tape, August succeeded!  His 2-by-2 Rubix Cube works just like any other, is fun to play with, and – even better – was fun to make.

Try it yourself:

Finally, ever heard of Emmy Noether?  It’s not surprising if you haven’t, because, according a New York Times article about her, “few can match in the depths of her perverse and unmerited obscurity….”  But, she was one of the most influential mathematicians and scientists of the 20th century – and was named by Albert Einstein the most “significant” and “creative” woman mathematician of all time.  You can read about Emmy’s influential theorem, and her struggles to become accepted in the mathematical community as a Jewish woman, in this article.

Want to learn more about women mathematicians throughout history?  Check out this site of biographies from Agnes Scott College.

Bon appetit!

Rice, Rectangles, and Mathmagicland

Welcome to this week’s Math Munch!

Want to practice your math facts?  Want to help feed hungry people around the world?  Well, with Free Rice you can do both at once!  Every time you answer a question correctly, the website donates 10 grains of rice through the UN’s World Food Programme.  You can work on multiplication or pre-algebra, as well as vocabulary, flags of the world, and other subjects.  It’s good practice for a good cause!  What do you say?  Will you help?

Up next, meet Edmund Harriss.

I found him through his fantastic math blog, Maxwell’s Demon, but he’s also a visiting professor at the University of Arkansas and a mathematical artist to boot.  We’re going to take a look at his recent blog post, “the 2×1 rectangle and domes.”  I seriously encourage you to read the entire thing, but I’ll share a few highlights.  The 2×1 rectangle is called a domino, and when you cut one in half along the diagonal, you get a lovely triangle with nifty tiling capabilities!

Also, standard plywood comes in the same proportion (8’x4′), and they can be easily combined to make several types of domes, as you can see below.  Click here to see how a hexayurt is built.  Edmund goes on to talk about the truncated octahedron, and how we can use its shape to design these domes.  How amazingly clever!

Finally, let’s take a look at a classic Disney film, from 1959 – Donald in Mathmagicland.  Donald Duck, on some sort of hunt, finds himself in a very strange place, surrounded by numbers, shapes, and patterns.  The trees even have square roots!  Mr. Duck meets “The True Spirit of Invention,” a mysterious voice that leads him (and us) on an adventurous trip through Mathmagicland.  If you skip to 16:48 in the video, you can learn about Billiards, a game played on the 2×1 rectangle!  How fitting!

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 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?

2011budget

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!

Impossible, Impossible, Impossible

Welcome to this week’s Math Munch!

The Penrose Triangle is an “impossible figure” – or so claim many reputable mathematics sources.  It’s a triangle made of square beams that all meet a right angles – which does sound pretty impossible.  Penrose polygons features in some of M. C. Escher’s most confounding artwork, like this picture:

But, little do these mathematicians know… you can build your own Penrose Triangle out of paper!  Check out these instructions and confound your friends.

Want more optical illusions?  Check out these awesome ones by scientist Michael Bach.

Mathematicians also seem pretty sure that .99999999…. = 1.  Well, trust Vi Hart to show them what’s-what.  Here’s a video in which she tells us all that, in fact, .99999999999… is NOT 1.

Finally, did you know that 13×7=28?  Well, it does.  And here’s the proof:

Bon appetit!  Oh – and April Fools!