Wednesday, December 17, 2008

Wednesday Math, Vol. 52: The Tomb of Archimedes

Last week, we left the story of Archimedes with the great man being killed by a Roman soldier. The end, right?

Not quite. Marcellus, the Roman general who conquered the Sicilian city-state of Syracuse, was famous enough in his day to warrant a biography, and in his biography the story of the death of Archimedes is recounted. Marcellus had commanded his troops to find Archimedes alive and bring him back to Marcellus, so that the general might get a chance to learn some of the secret weapons Archimedes had designed.

Marcellus felt bad about his troops killing Archimedes, and took it upon himself to make sure Archimedes was given a proper burial and a grand tomb. Archimedes had a request that one of his discoveries be the crowning glory of his tomb, a sphere fit snugly inside a cylinder that is the same height as the sphere, as is pictured above. Think of a racquetball inside a can where the ball has no wiggle room and the can is exactly the height of the ball.

(We know Marcellus did this not just because it is recounted in his biography, but another historian Cicero has a story of visiting Syracuse centuries later and searching for Archimedes' tomb, only to find it neglected and in need of repair.)

Here are the three things Archimedes figured out about the ball and the cylinder.

1. The ball has exactly two thirds the volume of the cylinder, if the cylinder were empty and closed on top and bottom.

2. The ball and the inside of the cylinder have exactly the same surface area.

3. If you slice the cylinder and ball parallel to the flat surface on which the cylinder stands, making a new shorter cylinder and some sliced section of a ball, the inside of the new shorter cylinder and the outside of the ball slice also have exactly the same surface area.

Last night, I had a little Eureka! moment working on a math problem and I blogged about it. This thing Archimedes did was also a moment of discovery, but far more brilliant than mine. All I did was use some knowledge I had already learned years ago and applied it correctly to a new problem. The methods Archimedes used to figure out all these things were entirely his own, and those ideas were the same ideas Isaac Newton would re-discover and turn into the calculus nearly 2,000 years later.

If Archimedes had been the leader of an academy like Euclid or Pythagoras, human civilization might be thousands of years more advanced than it is, because that's how far ahead of his time he was.


dguzman said...

It's these mathematicians' moments of sheer brilliance that really blow my mind. I just cannot even fathom getting an idea like this, a math-related idea. That these people figured these things out, on their own, while doodling or whatever--it blows my mind.

John V said...

In #2 I think you left out the word "same".

Just to be clear, by "the inside of the cylinder" you mean the surface area excluding the top and bottom, right?

Matty Boy said...

Thanks for the correction, John. Yes, the inside of a cylinder excludes top and bottom parts that would make it an enclosed object like a can.