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Friday, March 13, 2009

How I Spent Pi Day

I firmly believe that my school puts on one of the truly great pi day observances. More on that later, when I have some pictures.

In my classes on Pi Day, I have found that kids, understandably, want to know "how they came up with all those digits". (My school has a digit memorizing contest.)

So I spend half the period telling them about the life and death of Archimedes. My material comes from Chapter 4 of Journey Through Genius by William Dunham (a fantastic read). I actually spend a fair amount of time reading to them from the book - I skip some parts, emphasize the cool stuff - they seem to really like it. It is a very well-written book.

I tell them about how he was born in Syracuse. Not the one we live in, but the one our city is named after.

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I tell them about how he was both a mathematician and an engineer. I show them a picture of the Archimedean screw, an invention for moving water uphill, and tell them that it is still in use today.

I tell them about how he was an eccentric genius, who would focus such single-minded attention on a problem that his friends and family had to force him to eat and bathe. About how King Hieron set him to the task of figuring out if his new laurel wreath crown was real gold, or if his goldsmith was cheating him.

And Archimedes puzzled over how to find the volume of the crown, so that he could calculate the density, and finally had a breakthrough while getting into one of his infrequent baths (water displacement = volume), and was so excited that he jumped from the tub and ran through the streets of Syracuse shouting "Eureka!", sans toga.

I tell them about how Hieron was so impressed with this, and with Archimedes' amazing work with levers and pulleys, that he set Archimedes to defending Syracuse from the invading Roman General Marcellus. Archimedes rigged a wall to shoot 1000 arrows simultaneously. Archimedes dropped boulders on their heads. Archimedes picked up their ships by the prow and dumped them into the sea. Archimedes may have used a mirror or lens to turn the sun into a death ray and set their ships on fire. Dunham calls him a "one man military-industrial complex".

Archimedes made the Roman army turn back. Marcellus retreated and instead laid seige to Syracuse. The seige continued until the Syracusans, after too much partying during a Feast honoring the goddess Diana, left a portion of their city wall too lightly defended. The Romans overcame them and entered the city triumphant. It is said that Marcellus wept when he saw the beautiful Syracuse, knowing the devestation his army would wreak.

Marcellus had come to respect Archimedes so much that he ordered him to be captured unharmed. However during the invasion, Archimedes' mind was so lost and absorbed in an interesting problem that he didn't even notice the Roman invasion. When a Roman soldier found him, Archimedes (instead of identifying himself and surrendering), begged the soldier not disturb his work. The soldier, annoyed, ran him through.

Then I start to tell them about the technique Archimedes used to calculate Pi to three decimal places. How it used to drive people like Euclid, Plato, and Pythagoras nuts that they didn't know the exact length of the circumference of a circle, given the diameter. People knew it was a little more than three times as long as the diameter, but hadn't done much better than that. Three generations after Euclid, Archimedes was the first to take a scientific approach to a more precise value.

I pass out patty paper and bullseye compasses (a compass/stright edge hybrid). I have the students construct a circle and inscribe and circumscribe a hexagon. We take the diameter of the circle to = 1 unit. Everyone notices that the perimeter of the inscribed hexagon is 3, and it has to be shorter than the circumference. The older students calculate the perimeter of the circumscribed hexagon. I then have them double the number of sides of both polygons, so they have an inscribed and circumscribed 12-gon. We notice that the perimeter of the inside one got longer, but is still less than the circumference. And the outside one got shorter, but is still greater than the circumference.

I tell them that this is the exact technique Archimedes used. He honed in on the value of pi from both sides. He went up to 96 sides, and found

$3\frac{1}{7}<\pi<3\frac{10}{71}$ And he didn't have the place value decimal system. Or any kind of calculator. The really neat thing is that until the invention of the calculus nearly 2000 years later, people used the exact same technique to get more and more digits' accuracy, they just had better tools. And that's the classroom portion of Pi Day. A lotta history, a little math. Gotta love it. Here are some files. The Smart Notebook of the pictures related to Archimedes, a collection of potential projects students could do around pi day, and a short classroom activity about how pi is related to hat size.