A Trip to Infinity Inside a Circle
Cut a piece of wire 10 centimeters long and place a mark every centimeter for a scale. Bend the wire into a perfect circle which has circumference 10 cm. Pick a mark to be the start of our trip. Travel 3 cm along the wire in a counterclockwise direction and connect back to the starting point with a straight segment. Continue to move and connect back, and along the circle, you will have traveled
3, 6, 9, 12, and some more, and then finally 30 cm
where the trip ends because 30 is a multiple of 10 as well as 3, and ten steps gets you to land back at the start. We have just constructed the star pattern (10, 3), one of UB's favorites, and a picture follows.
An attractive star, but not an infinite trip as promised. The infinite trip is accomplished by taking advantage of two facts:
increments of the square root of two (√2) can be marked as a scale on the wire, and
√2 is irrational.
Now we repeat the process and create a (10, √2) star starting at point A, the first several segments of which are shown below.
The completed (hah!) star is an infinite trip around this circle. Why? Because if we ever returned to the starting mark, then a whole number n times √2 would equal 10. In algebrese it would look like
n x √2 = 10 and that implies √2 = 10/n which makes it a rational number, which it isn't.
We're taught in geometry that a circle has an infinite number of points, and so an infinite trip is possible, BUT here's a kicker, this infinite trip will never visit all of the circle's points and we can prove it. That being so, this is yet another example of how infinity can be merely a piece of itself.
We will show that, beginning at a common point A, a (10, √3) star and a (10, √6) star, though visiting an infinite number of points on the circle, will never arrive at another common point. [Figure below] It's easy. If they did so, then a whole number of steps (k) times √3 would equal another number of steps (m) times √6. That's impossible because if
k x √3 = m x √6, then k/m, a rational, equals √6/√3, but that equals √2, an irrational. QED as they say.
In the figure two star patterns are initiated. Both take an infinite journey within the circle and they never share another point!
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