Welcome to this week’s Math Munch! We’ve got geometry galore, starting with a series of historical math diagrams and a color update to Euclid’s Elements. Then it’s onto modern day paper artist Helen Friel, and finally a nifty new app that makes music from polygons. Let’s get into it.

Euclid’s “Elements” was written around 300BC. It was the first great compilation of geometric knowledge, broken into 13 books, and it is one of the most influential books of all time. Euclid’s proof of the Pythagorean Theorem may be his most famous proof from the book (and all of mathematics for that matter), and in the pictures below you can see three diagrams of the proof, spanning seven centuries.

The idea in each picture is that the area of the top two squares adds up exactly to the area of the bottom square. In the picture below, we see the big square broken up into blue and yellow pieces, whose areas are the same as the squares above them.

This color version comes from Oliver Byrne’s 1847 edition, “The First Six Books of the Elements of Euclid, with Coloured Diagrams and Symbols.” (completely available online). I find the diagrams really beautiful and charming. There’s something extremely modern about them (see De Stijl) though they’re more than 150 years old now. See if you can follow his Oliver Byrne’s version of Euclid’s proof. It’s quite short.

“They’re an absolutely beautiful piece of work and far ahead of their time,” said paper engineer Helen Friel. Helen lives in London, and and as part of a charity project, she designed paper sculptures of Oliver Byrne’s diagrams.

In an interview, she explained, “It’s a more visual and intriguing way to describe the geometry. I love anything that simplifies. I find it very appealing!” In the interview, Helen also talks a little about her attraction to math. “There’s order in straight lines and geometry. Although my job is creative, I use as much logical progression as possible in my work.”

It’s also cool to see Helen’s work side by side with Oliver Byrne‘s, so click for that.

Perhaps the best part in all of this, though, is that you can download Helen’s Pythagaorean Theorem model and make your own! There are plain white version as well as color. If you end up making one, definitely email us a picture, and we’ll show it off here on Math Munch.

Oh, and here’s a quick video documenting the many versions Helen decided not to use. So cool.

Now, on to our final bite.

Recently, John Miller sent me an email showing off his new iPad app called PolygonJazz. In the app, you control the starting direction for a ball inside a polygon. Once you start it moving, the ball bounces off the walls, making a sound every time it hits a side. Check out the video below. I noticed something about the speed of the ball. Can you spot it? (PolygonJazz is available for $0.99 on the iTunes store.)

Speaking of bouncing around, here‘s a previous Math Munch featuring some billiards, and here‘s another bouncy post that features one of my favorite juggling routines. Michael Moschen built a gigantic equilateral triangle and juggles silicon balls inside and off of it. As with the app, Michael is utilizing the sound and geometry of the collisions to make something beautiful. It’s quite mesmerizing.

Have a bouncy week, and bon appetit!

Very keen observation about the speed of the ball in PolygonJazz! I think the ball is going a little too fast in the video to observe what’s going on. When running on the iPad, you can slow the tempo down, then the speed thing is way more evident. I should post other “songs”. (Purposely not spoiling the answer here for anyone.) –JM

Readers interested in Polygonal Billiards might like to munch on PolygonFlux, an iPad app that can draw thousands of lines of geodesic flux inside regular polygons. The gallery on http://polygonjazz.com can be freely enjoyed without the app. The (FREE) app PolygonTrix is a “game-ified” version of Flux. Check it out on polygontrix.com. BTW, I’ve tried to notify Michael Moschen of my apps, but have not gotten past his agent evidently. I really wish he could see it!

Its amazing how precise his throwing angles are in order for him to perform this act.