Date: Apr 7, 1995 6:45 PM Author: Ted Alper Subject: Resend (re: coins)

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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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