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Tuesday, April 3, 2012

"Polygon Circumscribing"

 From Clifford Pickover's "The Math Book":
"Draw a circle, with a radius equal to 1 inch (about 2.5 centimeters). Next, circumscribe (surround) the circle with an equilateral triangle. Next, circumscribe the triangle with another circle. Then circumscribe this second circle with a square. Continue with a third circle, circumscribing the square. Circumscribe this circle with a regular pentagon. Continue this procedure indefinitely, each time increasing the number of sides of the regular polygon by one. Every other shape used is a circle that grows continually in size as it encloses the assembly of predecessors. If you were to repeat this process, always adding larger circles at the rate of a circle a minute, how long would it take for the largest circle to have a radius equal to the radius of our solar system?"
That's the question Cliff Pickover posed in the c.1940 entry of his book. As he then notes, it might seem that the circle radii would continually grow larger and larger toward infinity... however, it ain't so! As he says, "the circles initially grow very quickly in size," but then slow down and approach a limiting value, far short of a solar system [given by the infinite product: R = 1/(cos(π/3) x cos(π/4) x cos(π/5)….]
That limiting value, as calculated in 1965, turns out to be ~8.7000, but incredibly, before that calculation was made it was believed erroneously (for around 20 prior years according to Pickover) to be about 12.

This quickie video from WolframAlpha gives a sense of how rapidly the limit is approached:

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