# Byrne’s Euclid, Helen Friel, and PolygonJazz

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.

 Persian mathematician Nasir al-Din al-Tusi‘s 13th century arabic translation of Euclid’s proof. A late 14th century English manuscript of Euclid’s “Elements.”

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.

Oliver Byrne’s 1847 color edition.  Click the image for the full proof of the Pythagorean Theorem as presented by Oliver Byrne in 1847.

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.

Paper Engineer Helen Friel

“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.

Click to send us a pic.  Yes, that is a paper camera Helen made.