Re: [math-fun] surfaces of revolution & differential equations
Veit>On May 31, 2011, at 11:35 AM, Allan Wechsler wrote: acw> Blah, I answered this just to Henry. I thought of the p-orbitals of> electrons, and some vibrational modes of a circular drum, and was sure there> were simpler examples of symmetry-breaking but couldn't construct one off> the top of my head. ve>Those are actually not examples of "symmetry-breaking" . The degeneracy (in energy/frequency) is just a manifestation of the symmetry, and indicates the system is especially sensitive to perturbations (that typically "break" the symmetry). Here's an example of true symmetry breaking: the ground state of the lithium trimer (three lithium atoms). If we neglect quantum motion of the nuclei, then the ground state shape is an isosceles triangle. Veit Is your five disk packing of the unit disk maximizing sum(radii) (http://gosper.org/HTMLFiles/5disks.gif) an example of either of these asymmetries? How about your semisecret fourteen disk solution, which has no symmetry at all? (http://milou.msc.cornell.edu/images/ seems to be down.) --rwg
Bill, The server is back up and yes, the solutions for 4, 5, 7, 8, ... disks are examples of symmetry breaking (the radii are not equal in the configurations that maximize the radius sum). Fourteen appears to be the first instance where all the radii are distinct. Since fourteen is the first number where symmetry is completely broken, that would make it my favorite number. My place of work was founded on a principle expressed in fourteen words. Veit On Jun 1, 2011, at 5:30 AM, Bill Gosper wrote:
Veit
Is your five disk packing of the unit disk maximizing sum(radii)
(http://gosper.org/HTMLFiles/5disks.gif) an example of either of
these asymmetries? How about your semisecret fourteen disk solution,
which has no symmetry at all? (http://milou.msc.cornell.edu/images/
seems to be down.)
--rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Hey, neat! Fourteen is also the answer to the closure-complement-count problem: find the maximum over all subsets S of the real line of "How many different sets is it possible to obtain from S as a result of arbitrary applications of the "closure" and "complement" operators?" --Michael On Thu, Jun 2, 2011 at 4:16 PM, Veit Elser <ve10@cornell.edu> wrote:
Bill,
The server is back up and yes, the solutions for 4, 5, 7, 8, ... disks are examples of symmetry breaking (the radii are not equal in the configurations that maximize the radius sum). Fourteen appears to be the first instance where all the radii are distinct.
Since fourteen is the first number where symmetry is completely broken, that would make it my favorite number.
My place of work was founded on a principle expressed in fourteen words.
Veit
On Jun 1, 2011, at 5:30 AM, Bill Gosper wrote:
Veit
Is your five disk packing of the unit disk maximizing sum(radii)
(http://gosper.org/HTMLFiles/5disks.gif) an example of either of
these asymmetries? How about your semisecret fourteen disk solution,
which has no symmetry at all? (http://milou.msc.cornell.edu/images/
seems to be down.)
--rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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-- Forewarned is worth an octopus in the bush.
On Thu, Jun 2, 2011 at 1:16 PM, Veit Elser <ve10@cornell.edu> wrote:
My place of work was founded on a principle expressed in fourteen words.
Mine was founded on the first principle, ordered lexicographically, that takes more than fourteen English words to express. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
"Wait till you see where I'm going to put it." is only ten words. Rich --------- Quoting Veit Elser <ve10@cornell.edu>:
Bill,
The server is back up and yes, the solutions for 4, 5, 7, 8, ... disks are examples of symmetry breaking (the radii are not equal in the configurations that maximize the radius sum). Fourteen appears to be the first instance where all the radii are distinct.
Since fourteen is the first number where symmetry is completely broken, that would make it my favorite number.
My place of work was founded on a principle expressed in fourteen words.
Veit
On Jun 1, 2011, at 5:30 AM, Bill Gosper wrote:
Veit
Is your five disk packing of the unit disk maximizing sum(radii)
(http://gosper.org/HTMLFiles/5disks.gif) an example of either of
these asymmetries? How about your semisecret fourteen disk solution,
which has no symmetry at all? (http://milou.msc.cornell.edu/images/
seems to be down.)
--rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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participants (5)
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Bill Gosper -
Michael Kleber -
Mike Stay -
rcs＠xmission.com -
Veit Elser