Sunday, 30 January 2011

Math Trivia for the Super Bowl

Here is one to bring out and impress your friends during the Super Bowl commercials.....
Ole Romer is mostly remembered (by those who remember at all) for being the guy who first measured the speed of light. He cleverly used Cassini's observation that the moons of Jupiter seemed to have an irregular pattern and decide to measure the eclipse of Io as the earth approached and receded from Jupiter. The difference led him to the first calculation of the speed of light; "light seems to take about ten to eleven minutes [to cross] a distance equal to the half-diameter of the terrestrial orbit".
He was a man of many interests, and among other things was responsible for the first street lights in Copenhagen and the idea of a Meridian Circle. And while working on the design of gears, he came upon the idea of a shape called the Astroid (star like). Because it can be formed by a point moving on a circle rolling around another circle it is also called a hypocycloid. Steeler's fans will recognize it from the helmet decal. If you are old enough, you may remember drawing these with the Spirograph set you got for Christmas.
File:HypotrochoidOn4.gif


The formula for the astroid is x2/3+y2/3=a^2/3. The actual length of the curve is 6a. In a unit circle, this makes the length of the astroid exactly the same as the perimeter of an inscribed hexagon. The area inside the astroid is 3/8 of the circle area.

The astroid is interesting because if you draw any tangent to the curve and extend it so it cuts both the x and y axis, the length of the tangent segments will all be the same.



And perhaps as it should be, there is an asteroid in space, named for Ole Romer who first came up with the astroid shape. And the difference in spelling??? That's just how it's done. Who can explain spelling? It's not like it's math or something that makes sense.

3 comments:

Steven Colyer said...
This comment has been removed by the author.
Steven Colyer said...

MM-m, spirograph. Many an 8-1/2" x 11" sheet of paper (and colored ink pens) were sacrificed to the great god Spiro back in my/our day, remember? Well of course you remember or you wouldn't have raised the subject. As an extra bonus, it kept the kids (us) occupied and off the streets where trouble (and excitement) lay.

Hard to tell I watched "Nowhere Boy" last night, hmm? Lot's of street action in that movie. Mostly it reminded me of "Little Rascals" long past, that in an age without the internet and other technology, kids found ways to put on shows, to ... start something. What else was there to do? Our culture has always been a bit tough on "the lads." That's changing.

Back to astroids. It's a neat way of turning linear motion into circular motion and vice versa, as every internal-combustion piston and rods camshaft will agree. Pretty neat that from such a simple and circular motion a quantic world can emerge.

Thanks, Pat. I had no idea Romer was involved, but now I do.

I still want the Steelers to lose. Too much of a cheese-head Packer fan, I guess.

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