Re: [math-fun] Kenneth I. Appel, Mathematician Who Harnessed Computer Power, Dies at 80
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.
On 30 Apr 2013, at 02:06, 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".
Oh, that's interesting, I'll have to consider it more carefully.... Thanks. The meaning and purpose of life is to give life purpose and meaning. The instigation of violence indicates a lack of spirituality.
To put it another way, I thought that any connected surface morphs to a triangle (object of minimum sides) and assume that a collection of such objects (such as a map) is morphable to a collection of (arbitrary) triangles, therefore the maximum colours required is 4. Hence 5 for volumes based on morphing to irregular tetrahedra. On 30 Apr 2013, at 03:55, David Makin wrote:
On 30 Apr 2013, at 02:06, 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".
Oh, that's interesting, I'll have to consider it more carefully....
Thanks.
The meaning and purpose of life is to give life purpose and meaning. The instigation of violence indicates a lack of spirituality.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
The meaning and purpose of life is to give life purpose and meaning. The instigation of violence indicates a lack of spirituality.
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 numbers for such arrangements, and if so, what is it? --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
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 numbers 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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush.
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
That cannot be correct (or at least, reasonable): "equal" is a special case of "arbitrary", is it not? WFL On 4/30/13, Dan Asimov <dasimov@earthlink.net> wrote:
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
5 leq 6, and 10 leq 13, so the inequalities go the right way, I think. The more you constrain the class of graphs you are looking at, the fewer(-or-equal) colors you need. Jim Propp On Tuesday, April 30, 2013, Fred lunnon <fred.lunnon@gmail.com> wrote:
That cannot be correct (or at least, reasonable): "equal" is a special case of "arbitrary", is it not? WFL
On 4/30/13, Dan Asimov <dasimov@earthlink.net> wrote:
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
It seems reasonable to me, but I may be speaking out of ignorance: the chromatic number (unknown, but somewhere between 6 & 13) for arbitrary radii should be greater than or equal to the chromatic number (also unknown, but between 5 and 10) for the special case of equal radii. Lower bound of the 1st case is >= lower bound of 2nd case, ditto for upper bounds. ----- Message from fred.lunnon@gmail.com --------- Date: Tue, 30 Apr 2013 17:05:43 +0100 From: Fred lunnon <fred.lunnon@gmail.com> Reply-To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Kenneth I. Appel, Mathematician Who Harnessed Computer Power, Dies at 80 To: math-fun <math-fun@mailman.xmission.com>
That cannot be correct (or at least, reasonable): "equal" is a special case of "arbitrary", is it not? WFL
On 4/30/13, Dan Asimov <dasimov@earthlink.net> wrote:
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
----- End message from fred.lunnon@gmail.com -----
participants (7)
-
Bill Gosper -
Dan Asimov -
David Makin -
Fred lunnon -
James Propp -
mbgreen＠cis.upenn.edu -
Michael Kleber