Welcome to this week’s Math Munch– and happy Pi Day!
You probably know some pretty cool things about the number pi. Perhaps you know that pi has quite a lot to do with circles. Maybe you know that the decimal expansion for pi goes on and on, forever and ever, without repeating. Maybe you know that it’s very likely that any string of numbers– your birthday, phone number, all the birthdays of everyone you know listed in a row, followed by all their phone numbers, ANYTHING– can be found in the decimal expansion of pi.
But did you know that pi can be approximated by dropping needles on a piece of paper? Well, it can! If you drop a needle again and again on a lined piece of paper, and the needle is the same length as the distance between the lines, the probably that the needle lands on a line is two divided by pi. This experiment is called Buffon’s needle, after the French naturalist Buffon.
This may seem strange to you– but if you think about how the needle hitting a line has a lot to do with the distance between the middle of the needle and the nearest line and the angle it makes with the lines, maybe you’ll start to think about circles… and then you’ll get a clue about the connection between this experiment and pi. Working out this probability exactly requires some pretty advanced mathematics. (Feeling ambitious? Read about the calculation here.) But, you can get some great experimental results using this Buffon’s needle applet.
I had the applet drop 500 needles. Then, the applet used the fact that the probability of the needle hitting a line should be two divided by pi and the probability it measured to calculate an approximation for pi. It got… well, you can see in the picture. Pretty close, right?
Here’s another thing you might not know: pi is a transcendental number. Sounds trippy– but, like some other famous numbers with letter names, like e, pi can never be the solution to an algebraic equation involving whole numbers. That means that no matter what equation you give me– no matter how large the exponent, how many negatives you toss in, how many times you multiply or divide by a whole number– pi will never, ever be a solution. Maybe this doesn’t sound amazing to you. If not, check out this video from Numberphile about transcendental numbers. Numbers like pi and e don’t do mathematical things we’re used to numbers doing… and it’s pretty weird.
Still curious about transcendental numbers? Here’s a page listing the fifteen most famous transcendental numbers. My favorite? Definitely the fifth, Liouville’s number, which has a 1 in each consecutive factorial numbered place.
Finally, maybe you don’t like pi. Maybe you like cookies instead. Lucky for you, you can do many mathematical things with cookies, too. Like make cookie tessellations! This mathematical artist and baker made cookie cutters in the shapes of tiles from Escher tessellations and used them to make mathematical cookie puzzles. Beautiful, and certainly delicious.
If you happen to have a 3D printer, you can make your own Escher cookie cutters. Here’s a link to print out the lizard cutter. If you don’t have a 3D printer, you could try printing out a 2D image of an Escher tessellation and tracing a tile onto a sheet of paper. Cut out the tile, roll out your dough, and slice around the outside of the tile to make your cookies. If you do it right, you shouldn’t have to waste any dough…
Here’s hoping you eat some pi or cookies on pi day! Bon appetit!
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I was running behind schedule in math (due to weather) so we won’t get to circle stuff until this week. I DID show my fifth graders the video by Vi Hart about how pi “isn’t as special as we say”. They loved it, and now can’t wait to learn ABOUT pi this week. I’ll share some of this with them as well.
Hey Maria! That’s great. I think the video on transcendental numbers is a good follow-up to the pi “isn’t that special” video– because it shows how there are so many other numbers that share some of pi’s properties that are also interesting to study. I think it would be a tricky thing for 5th graders to follow, but I bet they could be coaxed through it with support. Enjoy your classes on pi!
That last video was a great find! Until recently, I had only worked with numbers that were algebraic and I didn’t even know it. But those proofs that were used to prove that e and pi are both transcendental are magnificent. I could only hope to have the patience to be able to do something like that. I found it especially nice how I learned the e^(i)(pi) is equal to -1. Just a cool little tidbit of information for today. Just a quick question: what are the other transcendental numbers that we currently know of? Thanks for the share!