From: James Buddenhagen <jbuddenh@texas.net>
Subject: Re: Circle packings on a circle
Date: Fri, 07 May 1999 08:21:21 -0500
Newsgroups: sci.math

Best known results up to 65 circles in a circle are in the paper "Dense
packings of congruent circles in a circle", by
Graham,Lubachevsky,Nurmela,Ostergard, in:  Discrete Mathematics 181
(1998) 139-154  (there have been one or two changes since then).

Pictures up to 13 circles are at
http://www.stetson.edu/~efriedma/packing.html

There is some info on line at  http://www.frii.com/~dboll/packing.html
However, many of the results there are not best possible.

--Jim Buddenhagen
==============================================================================

From: Kari Nurmela <kjnu@orbit.tcs.hut.fi>
Subject: Re: Help with circles
Date: 09 Jun 1999 12:00:25 +0300
Newsgroups: sci.math.research

>>>>> "JB" == Josh Berg <jaberg@midway.uchicago.edu> writes:
JB> Can anyone supply me with a formula for the following condition:
JB> We have a circle whose diameter is a fixed at some number X.  Inside of
JB> this circle, we want to place as many smaller circles of diamteter Y. 
JB> Is there a function which allows for an easy computation of the number
JB> of 'Y circles' given X and Y?  How about if X=.033, and Y=.009?

It seems to me that you want to pack as many non-overlapping equal
circles as possible within a larger circle. Unfortunately, finding the
best such packings is an unsolved problem and optimal solutions are
only known for small number of circles.

Information and references can be found for example in the following
works:

  Dense packings of congruent circles in a circle (R. L. Graham, B. D.
  Lubachevsky, K. J. Nurmela, and P. R. J. Ostergard (Östergård))
  Discrete Mathematics 181 (1998), pp. 139--154.

  Packing and Covering with Circles, Hans Melissen, PhD Thesis,
  Universiteit Utrecht, the Netherlands (1997).

Based on hastily calculation, it would appear that with your X and Y
values the largest possible number of circles is 9 (check it with the
references mentioned above).

Kari Nurmela
==============================================================================

From: Hop David <hopspage@tabletoptelephone.com>
Subject: Re: Tiling the Circle
Date: Wed, 15 Dec 1999 12:34:54 -0700
Newsgroups: sci.math

Matthew,

Here's a website you might like:
http://www.stetson.edu/~efriedma/packing.html

He has a page on packing circles into circles. Some of his polygon packings
could be described as tiling, I think.

Regards,

Hop

Matthew Montchalin wrote:

>
> I have a big circle with a known radius.
>
> I also have one or more smaller circles that can fit inside the big
> circle, and none of them are allowed to intersect with each other.
>
> My goal is to fill the big circle with as many of the smaller circles as
> possible, arranging them in any way whatsoever, but I do not wish to have
> any of them overlap one another.  Tangential contact is permitted,
> however.
>
> R = Radius of the big circle
> r = radius of any of the smaller circles
> r < R
> ________________________
>
> n = number of small circles satisfying the requirements above.
>
> ------------------------