Tag Archives: optimization

Functionized Photos, Projective Games, and Traffic

Welcome to this week’s Math Munch!

Have you ever looked in a distorted mirror– one that stretched and squeezed your face so that you looked very, very silly? If you like that, check out this program called the Function Explorer that distorts your picture according to different functions!

Crazy Mikos

My cat under the “fraction” function

To use the program, you’ll have to turn on your webcam. Then, select one of the functions listed– maybe similarity, log, or fraction. Then, watch as the image in front of your webcam twists, expands, and repeats as the function distorts the picture!

What’s going on here? The program treats your picture like it’s on something called the complex plane— which is kind of like the regular two-dimensional plane we’re used to, except that some of the numbers multiply strangely. One of the dimensions on the complex plane is made of regular, normal numbers– which, in this situation, are called the “real numbers”– while the other dimension is made of different numbers, called “imaginary numbers.” These are the numbers that do weird things when you multiply them together. Maybe you’ve heard that you can’t take the square-root of a negative number. Well, on the complex plane you can. And when you do, you get an imaginary number!


Windows, under 1/z

If you’re curious about these crazy creatures called imaginary numbers and how they work to make images go wild on the complex plane, I recommend you check out this site. It gives a great interactive explanation of imaginary numbers (and teaches you about fractals, too!). But I also wouldn’t blame you if you wanted to spend a few hours holding things in front of your webcam and seeing what happens to them under different function transformations!

Gummy bears

Gummy bears! Which function did this?


Meet Donna

Next up, I’d like to share a fun collection of games with you. They’re all made by mathematician Donna Dietz, and they all have to do with a particular kind of math that I find very interesting– projective geometry! You can still enjoy the games even if you know nothing about projective geometry (and you might learn something at the same time).

screen-shot-2016-09-14-at-9-19-29-pmThe rules are pretty simple: Donna gives you a bunch of cards with symbols on them. For example, in the version shown here, you get 13 cards with 4 symbols on them each. There are a bunch of different symbols. Your task is to pick four cards to discard and arrange the remaining nine so that the cards in each row, column, and diagonal share exactly one symbol.

Donna’s projective geometry games page has links to lots more games (if you think the game with cards in three rows and columns is too easy, try one with five) and information about them.

“What does this have to do with geometry?” you might be wondering. These games show a very important property of points and lines in projective geometry. In regular geometry (which you could also call Euclidean geometry), you can have two lines that don’t share any points– meaning that they’d be parallel. But this isn’t possible in projective geometry. All pairs of lines share exactly one point. How is this related to Donna’s games? If lines are rows, columns, and diagonals of cards, and points the symbols on them…

If you’d like to learn more about how and why Donna developed these games, check out this page!

Finally, I’ve been driving a lot lately. I live in the Bay Area, and there is SO MUCH TRAFFIC AAAAAAAA!!! I went searching for solutions, and I came across this great video by our friend CGP Grey (who also made these great videos about voting theory). There’s a lot of math going on here, even if it isn’t immediately apparent. Can you find the math? (Oh, and can you stop causing traffic jams? Thanks.)

Don’t Math Munch and drive, and bon appetit!

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.

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.

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.

trinity-glass2-small sqBox8 wp4f6b3871_0f

Delaunay triangulationLast 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.

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.

 armandmevis-1  fox

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