Est. 1998 — 27 years of woodworking knowledge
Friday Puzzler -- pi = 4?
Alex Y
Consider a circle with diameter 1, inscribed in a square of sides =1:
The circumference of the circle is pi, while the path around the square is 4.
If this doesn't convince you that pi = 4, what is wrong with the argument?
Re: Friday Puzzler -- pi = 4?
John Veerkamp
You can never reach infinity.
Re: Friday Puzzler -- pi = 4?
Alex Y
No, but you can get arbitrarily close to the circle.
Re: Friday Puzzler -- pi = 4?
Henry Higginbotham
I still can't see what's wrong with John's answer. That is, since you never get to infinity, even though the little right-angle detours get vanishingly small, they are still there, and their numbers get infinitely large so you never truly get to take that shortcut along the circle.
Correct!
Alex Y
Nice job, John and Henry.
Didn't mean to imply that John was wrong, just pointing out that while you can't reach infinity, the path was getting arbitrarily close to the path around the circle.
The key is what you said, that the number of detours in the line offset the decreasing size of those detours. So while the AREA of the jagged shape approaches the area of the circle as the angular jaggies get infinitely small (never getting there because you can't reach infinity), the length along the jagged path does not converge toward the length of the path around the circle.
I like to look at what happens at the 45* (northeast) point on the circle. As the right angle paths get smaller, the hypotenuses of those right triangles approach the circle, but the path along the jagged line, say from 40* to 50* is darned close to sqrt(2) times as long as the path along the circle between those points, no matter how small you make the triangles.