welcome to
Reply
to Robin, Diminishing Circle Argument
Jesse L. Yoder
Subject: Reply to Robin, Diminishing Circle Argument
Author: Jesse Yoder <jesse@flowresearch.com >
Organization: Epigone
Date: 30 Oct 1997 10:56:23 -0500
Robin - Thanks very much for your response to the note I posted on
calculus and geometry. You
wrote:
"I'm intrigued, but how does the diameter equal circumference divided
by 2? It seems you have eliminated Pi from the equation. Perhaps this is
why Pi figures in so importantly when computing curved areas within the
Cartesian coordinate system.
Good luck with your work. Send me more info any time."
Diminishing Circle Argument
I concede that the idea that diameter of a circle = the circumference
divided by 2 is
counterintuitive. This comes about from what I call the "diminishing
circle argument." This argument goes as follows:
Take any circle E and draw two smaller circles inside the larger circle.
The two smaller circles should be equal in area, with one edge
intersecting the outside edge of the circle E (the point where the
diameter intersects the circumference) and the other edge passing through
the radius
of E (this is easier to draw than to describe). This gives you two smaller
circles, A and B, each with a diameter that is half the diameter of the
original circle E. This is because the area of E is (pi x r squared), and
the area of circle B is (pi x (r/2) squared), (where r refers to the
radius of A).
This equals pi x r squared/4, or 1/4 the area of the original circle E.
Since A and B are equal in size, they have the same area, which is equal
to 1/4 the area of E. Hence the areas of A and B together equal 1/2 the
area of E, the large circle. This means that the remaining areas, C and D
(which are also equal, but not circles) are together each equal in area to
the areas of the smaller circles A and B.
The above shows that circle E is divided into four equal areas: the areas
of A, B, C, and D. The next step shows that the circumference is B = the
Circumference of E/2. The circumference of E = 2 x pi x r. Since the
radius of B is 1/2 the radius of E, the radius of B is 2 x pix r/2 (where
r refers to the radius of E). But 2 x pi x r/2 = pi x r, which is 1/2 the
circumference of E. So the circumference
of B (one of the smaller circles) is half the circumference of E. This
means that half way around the circumference of E, which is C/2, equals
the entire circumference of circle B. This also means that halfway around
the circumference of B plus halfway around the circumference of A equals
halfway around the circumference of E (since A and B are equal, their
circumferences are equal).
So going from one end of the diameter of E to the other via the
circumference of E is the same distance as going from one end of the
diameter of E to the other via the two circumferences of the smaller
circles A and B.
This last result is the main result I need to get the argument that the
diameter of a circle equals the circumference divided by 2. Now we have
circle E divided into four equal areas, the areas of A, B, C, and D. Now
imagine further dividing circles A and B each into four smaller areas in
the
same way, by drawing two smaller but equal circles within A and B, each
dividing the areas of A and B into four smaller areas. Again, going
halfway around A is the same distance as traveling along circumferences of
the two smaller circles within A. Continue this process indefinitely. At
the end point, we have the diameter consisting of the smallest allowable
points packed up against one another. But the distance around all these
points still equals the distance around the circumference of E, so the
diameter of circle E will equal half the circumference. The same argument
can be repeated for any circle.
At this last stage of the argument, I used to say "Now imagine this
process completed to infinity. At infinity, the circles will be infinitely
small, thereby equaling the points on a line, and the distance along the
diameter will equal half the circumference." Sandy Norman of the
University of Texas at San Antonio has convinced me of the folly of, on
the one hand, rejecting the notion of infinity and then, on the other
hand, relying on it when it is convenient to do so. But I don't believe
that I need to rely on the notion of infinity to complete the diminishing
circle argument.
In this geometry I am proposing, the points on a line are densely packed,
each with a specifiable area. In other words, they are not
"infinitely small" (whatever that is supposed to mean). How
small the points will vary with what is being measured-but once the units
are specified, they can only be made smaller by re-specifying the unit of
measurement. (This is the only way to avoid
Zen's paradox.) This means that in this geometry, there is a finite number
of points on the number line, not an infinite number, and that I reject
the idea that between any two points, there always lies another.
Of course, I realize that the diminishing circle argument is
counterintuitive, since it certainly seems shorter to walk along the
diameter of a circle than to go along its circumference. The difference is
in how you analyze motion along a diameter. If you analyze it as moving
along the circumferences of a very large number of very small circles (or
points), then the distance is in fact the same as the distance along the
circumference of the circle whose diameter you are traversing. On the
other hand, if you analyze the motion along the diameter as motion along
the path of a moving point, giving you motion along a "straight
line," then the distance is less than traveling along the
circumference of the circle whose diameter you are traversing. But since I
reject straight lines, I can't accept this "straight line"
analysis of motion along a diameter.
I hope that this answers your question (if you've gotten this far). Since
you asked about my background, and what level I'm working at, I might as
well answer that too. I have a PhD in philosophy (from U. of Mass.
Amherst). While my main area of concentration is in philosophy of mind, I
have also done some philosophy of mathematics. My main area of interest in
this subject is to try to understand if the assumptions that underlie
mathematics are valid, and to try to suggest alternatives where I feel
they are not. My view of philosophy is that it mainly consists in
evaluating the assumptions the other fields of study take for granted. In
the case of mathematics, I have always been bothered by the notion of
infinity, and I object to the idea of analyzing what is continuous as
being infinitely many discrete things. So I am trying to come up with a
different way of analyzing continuous phenomena (such as the number
"line") without relying on the notion of infinity.
I am working as a market analyst for a market research company (Flow
Research), writing about industrial applications for flowmeters and
pressure transmitters. I am trying to figure out whether my geometry can
be used to improve or change the ways in which flow rate is calculated,
since the fundamental flow equation is Volumetric Flow = Area x Velocity,
and pipe areas typically are around, thereby invoking pi.
Thanks again for your question, Robin, and I welcome any comments you may
have on my response.
Jesse
To view the original post, go to:
http://forum.swarthmore.edu/epigone/geometry-research/froxpalzhand/ac7ma91i33e4@forum.swarthmore.edu
Jesse Yoder
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