# Polyominoes, Clock Calculator, and Nine Bells

Welcome to this week’s Math Munch!

The first thing I have to share with you comes with a story. One day several years ago, I discovered these cool little shapes made of five squares. Maybe you’ve seen these guys before, but I’d never thought about how many different shapes I could make out of five squares. I was trying to decide if I had all the possible shapes made with five squares and what to call them, when along came Justin. He said, “Oh yeah, pentominoes. There’s so much stuff about those.”

Justin proceeded to show me that I wasn’t alone in discovering pentominoes – or any of their cousins, the polyominoes, made of any number of squares. I spent four happy years learning lots of things about polyominoes. Until one day… one of my students asked an unexpected question. Why squares? What if we used triangles? Or hexagons?

We drew what we called polyhexes (using hexagons) and polygles (using triangles). We were so excited about our discoveries! But were we alone in discovering them? I thought so, until…

A square made with all polyominoes up to heptominoes (seven), involving as many internal squares as possible.

… I found the Poly Pages. This is the polyform site to end all polyform sites. You’ll find information about all kinds of polyforms — whether it be a run-of-the-mill polyomino or an exotic polybolo — on this site. Want to know how many polyominoes have a perimeter of 14? You can find the answer here. Were you wondering if polyominoes made from half-squares are interesting? Read all about polyares.

I’m so excited to have found this site. Even though I have to share credit for my discovery with other people, now I can use my new knowledge to ask even more interesting questions.

Next up, check out this clock arithmetic calculator. This calculator does addition, subtraction, multiplication, and division, and even more exotic things like square roots, on a clock.

What does that mean? Well, a clock only uses the whole numbers 1 through 12. Saying “15 o’clock” doesn’t make a lot of sense (unless you use military time) – but you can figure out what time “15 o’clock” is by determining how much more 15 is than 12. 15 o’clock is 3 hours after 12 – so 15 o’clock is actually 3 o’clock. You can use a similar process to figure out the value of any positive or negative counting number on a 12 clock, or on a clock of any size. This process (called modular arithmetic) can get a bit time consuming (pun time!) – so, give this clock calculator a try!

Finally, here is some wonderful mathemusic by composer Tom Johnson. Tom writes music with underlying mathematics. In this piece (which is almost a dance as well as a piece of music), Tom explores the possible paths between nine bells, hung in a three-by-three square. I think this is an example of mathematical art at its best – it’s interesting both mathematically and artistically. Observe him traveling all of the different paths while listening to the way he uses rhythm and pauses between the phrases to shape the music. Enjoy!

Bon appetit!