Thanks, Michael, for finding that paper. I was able to access it, and I see that the chromatic number bounds of 6 <= # <= 13 quoted below actually apply to a family of balls in 3-space, non-overlapping except for tangencies (NOET), that can have *arbitrary* radii. The chromatic number for NOET spheres of *equal* radii in 3-space is also given bounds in this paper: 5 <= # <= 10 --Dan On 2013-04-30, at 4:20 AM, Michael Kleber wrote:
On Mon, Apr 29, 2013 at 11:15 PM, Dan Asimov <dasimov@earthlink.net> wrote:
OK, well, how about equal spheres in 3-space, non-overlapping except for tangencies, where no two tangent spheres can be the same color.
Is there a maximum chromatic number for such arrangements, and if so, what is it?
According to
"On Configurations of Solid Balls in 3-Space: Chromatic Numbers and Knotted Cycles", Graphs and Combinatorics (June 2007), 23 (1), Supl. 1, pg. 307-320, http://link.springer.com/article/10.1007%2Fs00373-007-0702-7#page-1
the chromatic number for a family of solid balls in 3-space (with mutually tangent balls being different colors) is between 6 and 13. But I can only see the first couple of pages at that URL, and don't have access to the full text.
--Michael
--Dan
On 2013-04-29, at 6:06 PM, Bill Gosper wrote:
Dave Makin>I assume these "countries" didn't include any "holey" ones i.e. rings etc. ? Connectedness is obviously required. I don't see simple-connectedness mattering.
DM> Also given the equivalent assumptions is the answer for 3D just 5 colours ? Infinity, obviously. Scott Kim once showed me that the answer for *convex* 3D countries is "arbitrarily large".
On 29 Apr 2013, at 13:48, Henry Baker wrote:
FYI -- How is it possible that I never knew that (computer scientist) Andrew Appel was Kenneth Appel's son? Obviously, Appel's don't fall far from the tree...
http://www.nytimes.com/2013/04/29/technology/kenneth-i-appel-mathematician-w...
Kenneth I. Appel, Mathematician Who Harnessed Computer Power, Dies at 80
By DENNIS OVERBYE
Published: April 28, 2013
Kenneth I. Appel, who helped usher the venerable mathematical proof into the computer age, solving a longstanding problem concerning colors on a map with the help of an I.B.M. computer making billions of decisions, died on April 19 in Dover, N.H. He was 80.
<snip>
DM>The meaning and purpose of life is to give life purpose and meaning. The instigation of violence indicates a lack of spirituality.
And spirituality instigates jihad. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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