Reply to Harry, Diminishing Circles Argument

Jesse L. Yoder

Subject: Reply to Harry, Diminishing Circles Argument

Author: Jesse Yoder jesse@flowresearch.com

Organization: Flow Research

Date: 31 Oct 1997 11:52:02 ñ0500

Hi Harry ñ

You wrote the following:

"I enjoyed reading this and thinking about it. You have started with a semicircle, divided it into two semicircles, divided each of those into two semicircles, etc., while the total length remained the same. You could have started with an isosceles triangle instead of a semicircle and performed similar operations. Your argument would then say that the diameter of the circle is two times the length of a side of the original triangle. This argument would then assert that the diameter of a triangle is any number greater than half the circumference.

The problem here is the following. If you have a sequence of functions f(n) on an interval (whose length is the length of the diameter) which converge to another function f on the interval (in this case the zero function), convergence either pointwise or uniform, it does not mean that the lengths of the graphs of the f(n)'s converge to the length of the graph of f. To see why this is true one need only to look at the formula for arc length and consider your most interesting example."

Thanks for replying to my diminishing circles argument. I'm not sure, however, that I completely understand your parallel example of the isosceles triangle. First of all, do you mean that I should draw an isosceles triangle inside a semicircle? If I draw one isosceles triangle inside a semicircle, with the tip of the triangle intersecting the circle circumference, the distance "around" the triangle is not equal to the distance around half circumference. Of course, if we are to believe calculus, if we continue to draw isosceles triangles one on top of the other, with the tips touching the circumference of the circle, the limit of these triangles approaches the circle circumference. But it is precisely this way of determining the area of a circle that I object to.

At the end of your first paragraph, do you mean "diameter of any circle" instead of "diameter of any triangle"?

Possibly your point about functions and their lengths comes down to this: The limit of any function as it approaches infinity does not necessarily have the same properties as the function before the limit. If this is your argument, then if the diameter of the circle is the limit of the diminishing circles, the diameter does not necessarily share the same length as the diminishing circles before they reach the diameter. While the diminishing circles have the same "distance around" as the circumference/2 until they reach their limit, at the limit they no longer have this property.

In response to this, please note that I do not rely on the idea that the function "goes to infinity"; instead, the "limiting case" of the circles is when the circles become the points on the number line. Isn't there a problem that the circle has area while a point does not? In my geometry, these points have area, and the unit of measurement that is specified when the circle is created (or the "line" is drawn) determines their area. When the circles become as small as points on the number line (as specified for each line), this is their limit and they actually become identical to the points on the line.

As I said before, however, it is vital to my argument that a line be viewed as "bumpy"; as consisting of a set of points packed up against each other. If a line is instead viewed as the path of one of these points in motion, giving you the effect of a straight line, then my argument will not work. This is because traveling along the diameter will no longer be traveling up and down these points, but instead will be along a "straight line" that intersects these points at the tip of the circumference of the circles that lie on the line.

Thanks again for your challenging comments. Possibly I have strayed from the point of your example, but I hope that you will explain it further. Also, please clarify or elaborate on your comment about the formula for arc length.

Best wishes,

Jesse

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`Jesse L. Yoder`
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