Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » Education » math-teach

Topic: Resend (re: coins)
Replies: 0  

Advanced Search

Back to Topic List Back to Topic List  
Ted Alper

Posts: 51
Registered: 12/6/04
Resend (re: coins)
Posted: Apr 7, 1995 6:45 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply


From MAILER-DAEMON 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 <MAILER-DAEMON>
Subject: Returned mail: warning: cannot send message for 4 hours
Message-Id: <199504072236.PAA07391@Ockham.Stanford.EDU>
To: alper
MIME-Version: 1.0
Content-Type: multipart/mixed; boundary="PAA07391.797294184/Ockham.Stanford.EDU"
Status: RO

This is a MIME-encapsulated message

--PAA07391.797294184/Ockham.Stanford.EDU

**********************************************
** THIS IS A WARNING MESSAGE ONLY **
** YOU DO NOT NEED TO RESEND YOUR MESSAGE **
**********************************************

The original message was received at Fri, 7 Apr 1995 10:40:57 -0700
from alper@localhost

----- The following addresses had delivery problems -----
nctm-l@forum (transient failure)

----- Transcript of session follows -----
nctm-l@forum... Deferred: Connection refused by forum.stanford.edu.
Warning: message still undelivered after 4 hours
Will keep trying until message is 5 days old

----- Original message follows -----

--PAA07391.797294184/Ockham.Stanford.EDU
Content-Type: message/rfc822

Return-Path: 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>
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

--PAA07391.797294184/Ockham.Stanford.EDU--







Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.