# 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!

### 38 responses »

1. at first i did not understand what they are doing but now i do there using gongs to make nine bells

• Hi Brianna! Yes, it is kind of hard to tell what’s going on in the video because the action is so far away. I like that about it, though. It emphasizes the paths – which I think are very interesting.

2. I didnt get what he was doing at first but then i realized he was going on all the different paths possible. It was really cool. I liked how he used the bells though.

3. I think the video is really interesting. I didn’t get it at first, but I think its cool that he takes different paths to make music.

4. I was kinda confused on what he was trying to do at first but as I watched I saw him retrace the same path and change the order in which he hit the bells. I am still not sure why the bells in the top right and bottom left were not used.

• I wasn’t sure about this either, but then I noticed that the name of the piece is VII – which is seven in Roman numerals! He only uses seven of the bells…

5. I was confused about what was going on so can you give me more information about it because for the hole 4:02 seconds he was going around and hitting bells. When this person wean`t around hitting the bells, was there a reason for that??

• That’s an interesting question. He may have not hit bells every other time around to separate the different paths. If he’d hit bells every time around, all of the paths would have merged together – so we would have a more difficult time distinguishing between them and seeing them as different paths. I think the gaps in the music also sound nice. They allow the sounds of the bells to merge and reverberate in the empty space. What do you think? Do you think the silences make the music better?

• Thank you Anna it makes a little bit more séance. Yes I think it makes the sounds of the music better. 🙂

6. It was cool that he took paths to make music. It is very interesting and I liked how he used the bells.

• I think it’s cool, too. I’m glad you liked it!

7. At first when I watched this I thought to myself that this guy is just running around and hitting bells ,I don’t get it . Then when I read the the little paragraph about the video , and I just thought it was amazing how he did that. 😉

• Haha, awesome! I think it’s amazing, too. It’s got me thinking about other ways to make music using a similar idea.

8. That was so cool! Do you know if this was pre-composed, or did he just make it up as he went along? Either way, this was impressive.

• I’m pretty confident he composed the piece ahead of the performance. He must certainly have worked out his paths in advance to make sure he got everything he wanted. Don’t you think?

• The piece was composed by Tom Johnson (whose compositions are often more mathematical than musical in nature) in the early 80s, and a recording of Johnson performing the piece exists on LP.

9. I thought that was a really unique way of creating music, him running around and hitting the bells. When I was watching it I thought someone was going to say now this is Tom playing this piece of music. I noticed that he didn’t hit the bell in the far right and left corners why was that?

10. I like how he uses the bells, I didn’t really get it at first but as I watched on it noticed that he takes different paths to create music

11. The way he used the bells was very unique but i didn’t really get it at first but after watching him play the bells i saw him walk different paths to make music i think it is really amazing.

12. I’m not quite sure what he is doing in this video. It doesn’t make much sense to me. I don’t understand the concept of him going in different paths each time.

13. This way of creating music by running around and hitting bells was really cool. I liked how he used different paths to make the music, but I don’t get why he didn’t use the bell in the far left and the bell in the far right.

14. When i was watching it i got lost at the beggining an i was wondering why he was hitting the bells but then i knew he was using diffrent paths to make music sounds

15. This video was interesting. When I read the other comments, I thought the same thing. Why did he only ring seven bells when he had nine? Why did he only use seven bells, was it because of the musical pattern?

16. At first I did NOT understand what this man was doing. Once I saw the video for the third time I noticed that he was going in different paths, but why did he only hit seven bells. Why didn’t he his the bells in the top right corner? 🙂

17. I know that you need a good eye .I think that he go in a circle and then start on the last one he hit. when I first put up the volume I said ” is it church time.” it sound like music. I can see the patterns I can see why you choses this video .

18. I didn’t understand it at first but as I watch all of it , I thought it was cool that .

19. I never heard of polyominoes until I came to this site. This site teaches you a number of things that you never heard before (www.recmath.com/PolyPages/index.htm) The shapes remind me of Tetris and I never thought of Tetris as a “Math” game before. I liked how it showed the different areas of different combinations of shapes.

20. This video was really cool. He took so many different paths, and it was really funny because I was trying to guess which bell he would hit and I was wrong almost every time. But still, it was really interesting, and I noticed that he never hit the top right and bottom left bell.

21. the viedo was very cool because you started hiting the different paths and it was getting funny because you were going around in circles

22. When I first watched this video I didn’t really understand what was going on but as I watched I thought it was really cool how he used so many different paths to make music. But I noticed he didn’t bottom left or top right bell. Do you know why he never used these bells?

23. I thought that was a really cool way of using different paths to make music. At first, I had no idea what he was doing until he retraced his steps. A good video and was fun. I wondered, why didn’t he hit the top right bell and the bottom left?

24. okay, i must admit that this video was pretty confusing. i watched the video again and saw what he was doing. I liked that it made me go back and think about what he was doing and see him retrace his steps.

25. Out of the four times I have watched this video I’ve noticed that he only uses seven of the gongs and using different combinations of the gongs. I wonder how many more combinations could be used if he used the other two gongs? also I liked the different patterns he used.

• This is but one movement from the piece, which lasts around an hour (and, according to Johnson, requires walking at a steady pace for about 3 miles), with the full set of bells used accordingly by the end.

26. The Nine Bells video was very interesting. I wonder how many times he had to practice before he got his walking rhythm down. I’m sure he likes music enough to incorporate it into math.