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Resend (re: coins)
Posted:
Apr 7, 1995 6:45 PM


From MAILERDAEMON Fri Apr 7 15:36:24 1995 Received: from localhost (localhost) by Ockham.Stanford.EDU (8.6.11/8.6.11) with internal id PAA07391; Fri, 7 Apr 1995 15:36:24 0700 Date: Fri, 7 Apr 1995 15:36:24 0700 From: Mail Delivery Subsystem <MAILERDAEMON> Subject: Returned mail: warning: cannot send message for 4 hours MessageId: <199504072236.PAA07391@Ockham.Stanford.EDU> To: alper MIMEVersion: 1.0 ContentType: multipart/mixed; boundary="PAA07391.797294184/Ockham.Stanford.EDU" Status: RO
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ReturnPath: alper Received: (from alper@localhost) by Ockham.Stanford.EDU (8.6.11/8.6.11) id KAA06259; Fri, 7 Apr 1995 10:40:57 0700 Date: Fri, 7 Apr 1995 10:40:57 0700 From: Ted Alper <alper> MessageId: <199504071740.KAA06259@Ockham.Stanford.EDU> To: josborn@genesee.freenet.org Subject: Re: how many coins? Cc: nctml@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 twoby 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
PAA07391.797294184/Ockham.Stanford.EDU



