# A Periodic Table, Linkages, and Dance Squared

Welcome to this week’s Math Munch!

I like finding new ways of organizing information. That’s part of why I enjoy this Periodic Table of Mathematicians.

The letters in the table are the abbreviations of the chemical elements—like gold, helium, and iron—that are found on the usual periodic table. With a little creativity, they can also be abbreviations for the names of a bunch of celebrated mathematicians. Clicking on a square brings up the mathematician’s biography. I like guessing who might pop up!

The table was created by Erich Friedman, a mathematician who works at Stetson University in Florida. We’ve previously shared Erich’s holiday puzzles (here) and weight puzzles (here) and monthly research contest (here), but there’s even more to explore on his site. I’m partial to his Packing Center, which shows the best ways that have been found to pack shapes inside of other shapes. You might also enjoy his extensive listing of What’s Special About This Number?—a project in the same spirit as Tanya Khovanova’s Number Gossip.

 A dense packing of 26 squares within a square that Erich discovered. I wonder what a multiplicative persistence is?

Next up, another Erik—Erik Demaine, whose work we’ve also often featured. What does he have for us this time? Some fantastic uncurling linkages, that’s what!

In 2000, Erik worked with Robert Connelly and Günter Rote to show that any wound-up 2D shape made of hinged sticks can be unwound without breaking, crossing, or lifting out of the plane. In the end, the shape must be convex, so that it doesn’t have any dents in it. For a while Erik and his colleagues thought that some linkages might be “locked” and unwinding some of the examples they created took months. You can find some great animations shared on the webpage that describes their result that locked linkages don’t in fact exist.

One thing that amazes me about Erik’s mathematical work is how young the problems are that he works on and solves. You might think a problem that can be put in terms of such simple ideas would have been around for a while, but in fact this problem of unwinding linkages was first posed only in the 1970s! It just goes to show that there are new simple math problems just waiting to be invented all the time.

Finally, I was so glad to run across this short film called Dance Squared. It was made by René Jodoin, a Canadian director and producer. Check out how much René expresses with just a simple square!

There’s a wonderful celebration of René titled When I Grow Up I Want To Be René Jodoin—written back in 2000 when René was “only” 80 years old. Now here’s 92! Making math is for people of all ages. You might also enjoy watching René’s Notes on a Triangle.

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.

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!