# Fullerenes, Fibonacci Walks, and a Fourier Toy

Welcome to this week’s Math Munch!

Stan and James

Earlier this month, neuroscientists Stan Schein and James Gayed announced the discovery of a new class of polyhedra. We’ve often posted about Platonic solids here on Math Munch. The shapes that Stan and James found have the same symmetries as the icosahedron and dodecahedron, and they also have all equal edge lengths.

One of Stan and James’s shapes, made of equilateral pentagons and hexagons.

These new shapes are examples of fullerenes, a kind of shape named after the geometer, architect, and thinker Buckminster Fuller. In the 1980s, chemists discovered that molecules made of carbon can occur in polyhedral shapes, both in the lab and in nature. Stan and James’s new fullerenes are modifications of some existing shapes first described in 1937 by Michael Goldberg. The faces of Goldberg’s shapes were warped, not flat, and Stan and James showed that flattening can be achieved—thus turning Goldberg’s shapes into true polyhedra—while also having all equal edge lengths. There’s great coverage of Stan and James’s discovery in this article at Science News and a fascinating survey of the media’s coverage of the discovery by Adam Lore on his blog. Adam’s post includes an interview with Stan!

Next up—how much fun is it to find a fractal that’s new to you? That happened to me recently when I ran across the Fibonacci word fractal.

A portion of a Fibonacci word curve.

Fibonacci “words”—really just strings of 0’s and 1’s—are constructed kind of like the numbers in the Fibonacci sequence. Instead of adding numbers previous numbers to get new ones, we link up—or “concatenate”—previous words. The first few Fibonacci words are 1, 0, 01, 010, 01001, and 01001010. Do you see how new words are made out of the two previous ones?

Here’s a variety of images of Fibonacci word fractals, and you can find more details about the fractal in this article. The infinite Fibonacci word has an entry at the OEIS, and you can find a Fibonacci word necklace on Etsy. Dale Gerdemann, a linguist at the University of Tübingen, has a whole series of videos that show off patterns created out of Fibonacci words. Here is one of my favorites:

Last but not least this week, check out this groovy applet!

Lucas’s applet showing the relationship between epicycles and Fourier series

A basic layout of Ptolemy’s model, including epicycles.

Sometime around the year 200 AD, the astronomer Ptolemy proposed a way to describe the motion of the sun, moon, and planets. Here’s a video about his ideas. Ptolemy relied on many years of observations, a new geometrical tool we call “trigonometry”, and a lot of ingenuity. He said that the sun, moon, and planets move around the earth in circles that moved around on other circles—not just cycles, but epicycles. Ptolemy’s model of the universe was incredibly accurate and was state-of-the-art for centuries.

Joseph Fourier

In 1807, Joseph Fourier turned the mathematical world on its head. He showed that periodic functions—curves with a repeated pattern—can be built by adding together a very simple class of curves. Not only this, but he showed that curves created in this way could have breaks and gaps even though they are built out of continuous curves called “sine” and “cosine”. (Sine and cosine are a part of the same trigonometry that Ptolemy helped to found.) Fourier series soon became a powerful tool in mathematics and physics.

A Fourier series that converges to a discontinuous function.

And then in the early 21st century Lucas Vieira created an applet that combines and sets side-by-side the ideas of Ptolemy and Fourier. And it’s a toy, so you can play with it! What cool designs can you create? We’ve featured some of Lucas’s work in the past. Here is Lucas’s short post about his Fourier toy, including some details about how to use it.

Bon appetit!

# Talk Like a Computer, Infinite Hotel, and Video Contest

01010111 01100101 01101100 01100011 01101111 01101101 01100101 00100000 01110100 01101111 00100000 01110100 01101000 01101001 01110011 00100000 01110111 01100101 01100101 01101011 00100111 01110011 00100000 01001101 01100001 01110100 01101000 00100000 01001101 01110101 01101110 01100011 01101000 00100001

Or, if you don’t speak binary, welcome to this week’s Math Munch!

Looking at that really, really long string of 0s and 1s, you might think that binary is a really inefficient way to encode letters, numbers, and symbols. I mean, the single line of text, “Welcome to this week’s Math Munch!” turns into six lines of digits that make you dizzy to look at. But, suppose you were a computer. You wouldn’t be able to talk, listen, or write. But you would be made up of lots of little electric signals that can be either on or off. To communicate, you’d want to use the power of being able to turn signals on and off. So, the best way to communicate would be to use a code that associated patterns of on and off signals with important pieces of information– like letters, numbers, and other symbols.

That’s how binary works to encode information. Computer scientists have developed a code called ASCII, which stands for American Standard Code for Information Interchange, that matches important symbols and typing communication commands (like tab and backspace) with numbers.

To use in computing, those numbers are converted into binary. How do you do that? Well, as you probably already know, the numbers we regularly use are written using place-value in base 10. That means that each digit in a number has a different value based on its spot in the number, and the places get 10 times larger as you move to the left in the number. In binary, however, the places have different values. Instead of growing 10 times larger, each place in a binary number is twice as large as the one to its right. The only digits you can use in binary are 0 and 1– which correspond to turning a signal on or leaving it off.

But if you want to write in binary, you don’t have to do all the conversions yourself. Just use this handy translator, and you’ll be writing in binary 01101001 01101110 00100000 01101110 01101111 00100000 01110100 01101001 01101101 01100101 00101110

Next up, check out this video about a classic number problem: the Infinite Hotel Paradox. If you find infinity baffling, as many mathematicians do, this video may help you understand it a little better. (Or add to the bafflingness– which is just how infinity works, I guess.)

I especially like how despite how many more people get rooms at the hotel (so long as the number of people is countable!), the hotel manager doesn’t make more money…

Speaking of videos, how about a math video contest? MATHCOUNTS is hosting a video contest for 6th-8th grade students. To participate, teams of four students and their teacher coach choose a problem from the MATHCOUNTS School Handbook and write a screenplay based on that problem. Then, make a video and post it to the contest website. The winning video is selected by a combination of students and adult judges– and each member of the winning team receives a college scholarship!

Here’s last year’s first place video.

01000010 01101111 01101110 00100000 01100001 01110000 01110000 01100101 01110100 01101001 01110100 00100001  (That means, Bon appetit!)

# 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.