[math-fun] superconductivity at 203K alleged in rotten eggs
http://arxiv.org/abs/1506.08190 hydrogen sulfide allegedly is superconductive at 203K, by far a new record high temperature, but the catch is you need to pressurize it over 90 GPa in a diamond anvil.
Let a C^oo closed curve in R^3 be called a "curly loop" if its curvature is nowhere vanishing. Two curly loops are for instance A) the unit circle in the plane and B) the unit circle in the plane traversed twice around. PUZZLE: Can A and B be continuously deformed one into the other — through curly loops — so that at all stages of the deformation the tangent directions vary continuously? —Dan
On 2015-07-10 16:07, Dan Asimov wrote:
Let a C^oo closed curve in R^3 be called a "curly loop" if its curvature is nowhere vanishing.
Two curly loops are for instance A) the unit circle in the plane and B) the unit circle in the plane traversed twice around.
PUZZLE: Can A and B be continuously deformed one into the other — through curly loops — so that at all stages of the deformation the tangent directions vary continuously?
—Dan Can't you just untwist it while confined to the surface of a sphere? --rwg
You cannot just untwist it while confined to the surface of a sphere, but maybe my phrasing: "at all stages of the deformation the tangent directions vary continuously" was less than clear. I mean that as the deformation proceeds in "time", the tangent directions of the various curves of the deformation evolve continuously. So for instance you can't pull a kink in a curve tight to make it disappear, since that would create a discontinuity of the tangent directions at the last moment of time. —Dan
On Jul 10, 2015, at 8:52 PM, rwg <rwg@sdf.org> wrote:
On 2015-07-10 16:07, Dan Asimov wrote:
Let a C^oo closed curve in R^3 be called a "curly loop" if its curvature is nowhere vanishing. Two curly loops are for instance A) the unit circle in the plane and B) the unit circle in the plane traversed twice around. PUZZLE: Can A and B be continuously deformed one into the other — through curly loops — so that at all stages of the deformation the tangent directions vary continuously? —Dan Can't you just untwist it while confined to the surface of a sphere? --rwg
One solution is this: Each C^oo curly loop (which we assume is parametrized by arclength) alpha: R/(2pi R) -> R^3 has a well-defined continuously varying ordered right-handed triple of orthonormal vectors at each of its points, hence a closed curve in the rotation group of 3-space, SO(3). (Since, SO(3) is the same space as the space of all such right-handed orthonormal triples. Topologically SO(3) is the real projective 3-space P^3.) The triple is the Frenet frame, (T(s),N(s),B(s)) for the point alpha(s). Any closed curve in SO(3) = P^3 belongs to one of two homotopy classes; loops in the different classes cannot be continuously deformed into one another. (This result is famous in physics; perhaps Veit or Gene can elaborate.) It's easy to check that the circle and the twice-traversed circle give distinct homotopy classes of closed curves in SO(3). And a regular homotopy between any two curly loops (through curly loops) would result in a homotopy between the closed curves in SO(3), implying that the two closed curves are in the same homotopy class, which they aren't. Hence the two curly loops mentioned cannot be regularly homotopic through curly loops. —Dan
On Jul 10, 2015, at 4:07 PM, Dan Asimov <asimov@msri.org> wrote:
Let a C^oo closed curve in R^3 be called a "curly loop" if its curvature is nowhere vanishing.
Two curly loops are for instance A) the unit circle in the plane and B) the unit circle in the plane traversed twice around.
PUZZLE: Can A and B be continuously deformed one into the other — through curly loops — so that at all stages of the deformation the tangent directions vary continuously?
This is why, when you fold a bandsaw blade, you end up with a 3-loop solution: https://www.youtube.com/watch?v=2KYKOd2Gk5E On 07/11/2015 04:58 PM, Dan Asimov wrote:
... Any closed curve in SO(3) = P^3 belongs to one of two homotopy classes; loops in the different classes cannot be continuously deformed into one another. (This result is famous in physics; perhaps Veit or Gene can elaborate.)
I now realize that this is not correct, since it's actually easy to create counterexamples. That's because "regular homotopy" is required to be only once-differentiable (C^1). It's a bit tricky, but not really hard, to find a C^1 1-parameter family of smooth curves, each of which has nowhere vanishing curvature, but whose normal vector does not vary continuously at all points. From this, it is in fact true that the once- and twice-traversed circle ARE regularly homotopic to each other through curly curves. Fooled myself there. —Dan
On Jul 11, 2015, at 1:58 PM, Dan Asimov <asimov@msri.org> wrote:
One solution is this: Each C^oo curly loop (which we assume is parametrized by arclength)
alpha: R/(2pi Z) -> R^3
has a well-defined continuously varying ordered right-handed triple of orthonormal vectors at each of its points, hence a closed curve in the rotation group of 3-space, SO(3). (Since, SO(3) is the same space as the space of all such right-handed orthonormal triples. Topologically SO(3) is the real projective 3-space P^3.)
The triple is the Frenet frame, (T(s),N(s),B(s)) for the point alpha(s).
Any closed curve in SO(3) = P^3 belongs to one of two homotopy classes; loops in the different classes cannot be continuously deformed into one another. (This result is famous in physics; perhaps Veit or Gene can elaborate.)
It's easy to check that the circle and the twice-traversed circle give distinct homotopy classes of closed curves in SO(3).
And a regular homotopy between any two curly loops (through curly loops) would result in a homotopy between the closed curves in SO(3), implying that the two closed curves are in the same homotopy class, which they aren't.
Hence the two curly loops mentioned cannot be regularly homotopic through curly loops.
—Dan
On Jul 10, 2015, at 4:07 PM, Dan Asimov <asimov@msri.org> wrote:
Let a C^oo closed curve in R^3 be called a "curly loop" if its curvature is nowhere vanishing.
Two curly loops are for instance A) the unit circle in the plane and B) the unit circle in the plane traversed twice around.
PUZZLE: Can A and B be continuously deformed one into the other — through curly loops — so that at all stages of the deformation the tangent directions vary continuously?
On 2015-07-10 13:22, Warren D Smith wrote:
http://arxiv.org/abs/1506.08190
hydrogen sulfide allegedly is superconductive at 203K, by far a new record high temperature, but the catch is you need to pressurize it over 90 GPa in a diamond anvil.
A lot of us have gpa's over 90. I've seen tiny hydraulic tubes hold huge pressure. What kind of pressures can you get by drawing out a thin sealed tube? Maybe made of some special Chinese finger trap molecular structure. In the limit of thinness, there might be a single strand of H2S polymer, held unwillingly in a lattice. --rwg
Hmmm... Fiber optics are made by stretching out glass rods by huge amounts. You can build a fair amount of (2D) structure into the glass rod, and then thin it down to very small dimensions by pulling it (+ heat). It would be very "cool" if one could incorporate a superconductor into a glass fiber in this manner. At 01:51 PM 7/13/2015, rwg wrote:
On 2015-07-10 13:22, Warren D Smith wrote:
http://arxiv.org/abs/1506.08190 hydrogen sulfide allegedly is superconductive at 203K, by far a new record high temperature, but the catch is you need to pressurize it over 90 GPa in a diamond anvil.
A lot of us have gpa's over 90.
I've seen tiny hydraulic tubes hold huge pressure. What kind of pressures can you get by drawing out a thin sealed tube? Maybe made of some special Chinese finger trap molecular structure.
In the limit of thinness, there might be a single strand of H2S polymer, held unwillingly in a lattice. --rwg
participants (5)
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Dan Asimov -
Henry Baker -
John Aspinall -
rwg -
Warren D Smith