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Resend (re: coins)
Posted:
Apr 7, 1995 6:45 PM
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Date: Fri, 7 Apr 1995 10:40:57 -0700
From: Ted Alper <alper>
Message-Id: <199504071740.KAA06259@Ockham.Stanford.EDU>
To: josborn@genesee.freenet.org
Subject: Re: how many coins?
Cc: nctm-l@forum.Stanford.EDU, tad@midget.towson.edu
Jim, are you sure?
After all, with the original, each circle contributes pi/4 of area and
lies in a square of area 1... so the limiting density of the circles
is pi/4
On the other hand, with the model you describe,
the distance from point A to point B
.. ..
. . . .
A. . .. B. .
.. . . ..
.. . . ..
. . .. . .
. . . .
.. ..
(that is, the distance from the point tangent to the wall
to the corresponding point in the next similarly situated circle)
may be shown to be sqrt(3) (easy, since the angle formed by connectin
the middle circle to the two upper circles is 120 degrees)
Anyhow, the three circles
.. ..
. . . .
. 1 . .. . .
.. . . ..
.. . 2 . ..
. . .. . .
. 3 . . .
.. ..
which form the pattern which is iterated contribute 3pi/4 area, but
the rectangle they inhabit has dimensions 2 by sqrt(3)
(to be precise: circle 2 pokes out of the rectangle a bit -- but aside
from the patterns closest to the left wall, this is compensated for by
the middle circle in the previous pattern poking into the rectangle
from the left -- besides, counting it when it pokes out of the
rectangle would only INCREASE the density)
so the limiting density of the circles is
3pi/(8sqrt(3)) = sqrt(3)pi/8 which is LESS than pi/4.
Just to confirm it, I took eighteen pennies from my jar and put them
on the desk: the two-by nine square was about 6.75 inches, but doing
them the way you describe (and forcing them not to drift apart -- it
must be only two pennies from top to bottom!) took nearly eight
inches....
Ted Alper
alper@epgy.stanford.edu
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