[math-fun] Attesting to Atoms
I just watched George Hart's video https://simonsfoundation.org/multimedia/attesting-to-atoms/ and was left with a vexing disquiet about the fact that the macroscopic structure of crystals seems to imply the existence of atoms and yet gives us no information about how big atoms are. If the observed structure of macroscopic crystals is compatible with an infinite range of models of reality, each positing the existence of atoms but at ever-smaller scales, could there be some sort of projective limit of these theories, with "cubes all the way down" but no bottom level? I'm not saying it's a believable physical theory, but it seems like it would give an example of a universe with crystals but without atoms. Or is this idea incoherent in some way? Jim Propp
I don't know how one would guess the size of atoms from crystal structure, but I'd be surprised if there weren't some way to make a reasonable guess. At the opposite extreme, if all scales made equal physical sense, then there would be no way for nature to choose some scale. Also, Occam's Razor would probably complain that that isn't the simplest physical model that fits the evidence. (Not conclusive of course, but suggestive.) On the third hand, I wouldn't be at all surprised if as more is learned about physics, it becomes increasingly clear that there is some kind of structure at scales that are arbitrarily small. Or not. --Dan On 2013-02-08, at 9:05 PM, James Propp wrote:
I just watched George Hart's video https://simonsfoundation.org/multimedia/attesting-to-atoms/ and was left with a vexing disquiet about the fact that the macroscopic structure of crystals seems to imply the existence of atoms and yet gives us no information about how big atoms are. If the observed structure of macroscopic crystals is compatible with an infinite range of models of reality, each positing the existence of atoms but at ever-smaller scales, could there be some sort of projective limit of these theories, with "cubes all the way down" but no bottom level? I'm not saying it's a believable physical theory, but it seems like it would give an example of a universe with crystals but without atoms. Or is this idea incoherent in some way?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
One of the shows in Philip Morrison's great series, "The Ring of Truth", has an experiment that at least suggests the order of magnitude of the size of molecules. He rows out into a pond, and dumps a small vial of light oil onto the surface. It looks like he has about a cubic centimeter of the stuff. It spreads out, and surprisingly it has a visible effect on the pond surface even when it's approaching a monolayer. Morrison roughly measures the area of the patch and calculates its thickness from its known volume. On Sat, Feb 9, 2013 at 12:17 AM, Dan Asimov <dasimov@earthlink.net> wrote:
I don't know how one would guess the size of atoms from crystal structure, but I'd be surprised if there weren't some way to make a reasonable guess.
At the opposite extreme, if all scales made equal physical sense, then there would be no way for nature to choose some scale.
Also, Occam's Razor would probably complain that that isn't the simplest physical model that fits the evidence. (Not conclusive of course, but suggestive.)
On the third hand, I wouldn't be at all surprised if as more is learned about physics, it becomes increasingly clear that there is some kind of structure at scales that are arbitrarily small. Or not.
--Dan
On 2013-02-08, at 9:05 PM, James Propp wrote:
I just watched George Hart's video https://simonsfoundation.org/multimedia/attesting-to-atoms/ and was left with a vexing disquiet about the fact that the macroscopic structure of crystals seems to imply the existence of atoms and yet gives us no information about how big atoms are. If the observed structure of macroscopic crystals is compatible with an infinite range of models of reality, each positing the existence of atoms but at ever-smaller scales, could there be some sort of projective limit of these theories, with "cubes all the way down" but no bottom level? I'm not saying it's a believable physical theory, but it seems like it would give an example of a universe with crystals but without atoms. Or is this idea incoherent in some way?
Jim Propp _______________________________________________ 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
On Sat, Feb 9, 2013 at 12:29 AM, Allan Wechsler <acwacw@gmail.com> wrote:
One of the shows in Philip Morrison's great series, "The Ring of Truth", has an experiment that at least suggests the order of magnitude of the size of molecules. He rows out into a pond, and dumps a small vial of light oil onto the surface. [...]
Yep, I did that in science class at age 14 ("9th grade" in USA). You put talcum powder on the surface of the water and the oil-drop pushes the particles of powder aside, and forms a lipid monolayer. One can then calculate the thickness of the monolayer, and thus how many "cubes" would be needed to make it. We cheated a bit, the teacher told us the approximate length:width ratio of the molecules. We didn't bother dividing the answer by the number of atoms in the molecule. Surprisingly though, this was not one of the methods used to estimate the Avogadro number 100 years ago. In 1914 Perrin [1] summarized the work up to that point; the methods used were: Viscosity of gases (Maxwell/Clausius); Brownian motion (Einstein/Perrin; 4 methods); Critical opalescence (Smoluchowski); Black-body spectrum (Planck); charged spheres (Millikan); Radioactivity (e.g. Rutherford, Curie/Debierne; 4 methods). The results were all between 6.0e23 and 7.5e23, so they were off by 10%, but still pretty impressive. [1] Jean Perrin, Atoms (translated from the original French by D. Ll. Hammick) D. Van Nostrand (New York), 1916. Page 206. -- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
Rutherford scattering should give the effective size of the nucleus. (Though note that "size" is a squishy term in quantum mechanics.) Quantum electro-dynamics predicts the orbital structure of the electrons, which should give an effective size to any given atom. As to arbitrarily small structures, the Planck scale seems to give the bottom, if you believe quantum gravitation theories. I have always found this idea interesting, since if you quantize gravity then space-time becomes quantized, which means (at least to this naive observer) that the universe is countable. How cool is that? Rowan. On Fri, Feb 8, 2013 at 9:17 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I don't know how one would guess the size of atoms from crystal structure, but I'd be surprised if there weren't some way to make a reasonable guess.
At the opposite extreme, if all scales made equal physical sense, then there would be no way for nature to choose some scale.
Also, Occam's Razor would probably complain that that isn't the simplest physical model that fits the evidence. (Not conclusive of course, but suggestive.)
On the third hand, I wouldn't be at all surprised if as more is learned about physics, it becomes increasingly clear that there is some kind of structure at scales that are arbitrarily small. Or not.
--Dan
On 2013-02-08, at 9:05 PM, James Propp wrote:
I just watched George Hart's video https://simonsfoundation.org/multimedia/attesting-to-atoms/ and was left with a vexing disquiet about the fact that the macroscopic structure of crystals seems to imply the existence of atoms and yet gives us no information about how big atoms are. If the observed structure of macroscopic crystals is compatible with an infinite range of models of reality, each positing the existence of atoms but at ever-smaller scales, could there be some sort of projective limit of these theories, with "cubes all the way down" but no bottom level? I'm not saying it's a believable physical theory, but it seems like it would give an example of a universe with crystals but without atoms. Or is this idea incoherent in some way?
Jim Propp _______________________________________________ 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
There can't be a continuum down there, simply because we can't have enough information to specify it. Information density is limited by the Beckenstein limit on black holes. As Henry points out, the idea that a single real number has an infinite information content means that real numbers are only a smoothed approximation to discrete physical observables. My favorite speculation for structure at the Planck scale is a 4-D crystal with dislocations in the crystal structure giving rise to observable particles. Past is represented with a solidified structure, present is the phase boundary, and quantum weirdness and interference is the attempt to extend the phase boundary in a consistent way. We observe the net growth of the crystal, but it is an epiphenomenon -- a result of the crystal growth. I don't expect anyone else to take this seriously, since there is no experimental support. Can we classify 4-D crystal dislocation types and align them with the known particle spectrum? On Feb 9, 2013, at 12:36 AM, Rowan Hamilton wrote:
Rutherford scattering should give the effective size of the nucleus. (Though note that "size" is a squishy term in quantum mechanics.) Quantum electro-dynamics predicts the orbital structure of the electrons, which should give an effective size to any given atom.
As to arbitrarily small structures, the Planck scale seems to give the bottom, if you believe quantum gravitation theories. I have always found this idea interesting, since if you quantize gravity then space-time becomes quantized, which means (at least to this naive observer) that the universe is countable. How cool is that?
Rowan.
On Fri, Feb 8, 2013 at 9:17 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I don't know how one would guess the size of atoms from crystal structure, but I'd be surprised if there weren't some way to make a reasonable guess.
At the opposite extreme, if all scales made equal physical sense, then there would be no way for nature to choose some scale.
Also, Occam's Razor would probably complain that that isn't the simplest physical model that fits the evidence. (Not conclusive of course, but suggestive.)
On the third hand, I wouldn't be at all surprised if as more is learned about physics, it becomes increasingly clear that there is some kind of structure at scales that are arbitrarily small. Or not.
--Dan
On 2013-02-08, at 9:05 PM, James Propp wrote:
I just watched George Hart's video https://simonsfoundation.org/multimedia/attesting-to-atoms/ and was left with a vexing disquiet about the fact that the macroscopic structure of crystals seems to imply the existence of atoms and yet gives us no information about how big atoms are. If the observed structure of macroscopic crystals is compatible with an infinite range of models of reality, each positing the existence of atoms but at ever-smaller scales, could there be some sort of projective limit of these theories, with "cubes all the way down" but no bottom level? I'm not saying it's a believable physical theory, but it seems like it would give an example of a universe with crystals but without atoms. Or is this idea incoherent in some way?
Jim Propp _______________________________________________ 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
On 2/9/2013 8:23 AM, Tom Knight wrote:
There can't be a continuum down there, simply because we can't have enough information to specify it.
That's a very positivist attitude. I think limited information could be encoded in, for example real intervals. Even integer arithmetic encodes truths than can't be proven.
Information density is limited by the Beckenstein limit on black holes. As Henry points out, the idea that a single real number has an infinite information content means that real numbers are only a smoothed approximation to discrete physical observables.
It would be nice to have physical theories that didn't assume real numbers, and of course in any actual calculation or measurement only rationals are used. But experimental attempts to measure the discreteness of spacetime, as hypothesized by loop quantum gravity for example, have come up empty - even down close to the Planck scale, http://arxiv.org/pdf/1109.5191.pdf Brent Meeker
My favorite speculation for structure at the Planck scale is a 4-D crystal with dislocations in the crystal structure giving rise to observable particles. Past is represented with a solidified structure, present is the phase boundary, and quantum weirdness and interference is the attempt to extend the phase boundary in a consistent way. We observe the net growth of the crystal, but it is an epiphenomenon -- a result of the crystal growth.
I don't expect anyone else to take this seriously, since there is no experimental support.
Can we classify 4-D crystal dislocation types and align them with the known particle spectrum?
On Feb 9, 2013, at 12:36 AM, Rowan Hamilton wrote:
Rutherford scattering should give the effective size of the nucleus. (Though note that "size" is a squishy term in quantum mechanics.) Quantum electro-dynamics predicts the orbital structure of the electrons, which should give an effective size to any given atom.
As to arbitrarily small structures, the Planck scale seems to give the bottom, if you believe quantum gravitation theories. I have always found this idea interesting, since if you quantize gravity then space-time becomes quantized, which means (at least to this naive observer) that the universe is countable. How cool is that?
Rowan.
On Fri, Feb 8, 2013 at 9:17 PM, Dan Asimov<dasimov@earthlink.net> wrote:
I don't know how one would guess the size of atoms from crystal structure, but I'd be surprised if there weren't some way to make a reasonable guess.
At the opposite extreme, if all scales made equal physical sense, then there would be no way for nature to choose some scale.
Also, Occam's Razor would probably complain that that isn't the simplest physical model that fits the evidence. (Not conclusive of course, but suggestive.)
On the third hand, I wouldn't be at all surprised if as more is learned about physics, it becomes increasingly clear that there is some kind of structure at scales that are arbitrarily small. Or not.
--Dan
On 2013-02-08, at 9:05 PM, James Propp wrote:
I just watched George Hart's video https://simonsfoundation.org/multimedia/attesting-to-atoms/ and was left with a vexing disquiet about the fact that the macroscopic structure of crystals seems to imply the existence of atoms and yet gives us no information about how big atoms are. If the observed structure of macroscopic crystals is compatible with an infinite range of models of reality, each positing the existence of atoms but at ever-smaller scales, could there be some sort of projective limit of these theories, with "cubes all the way down" but no bottom level? I'm not saying it's a believable physical theory, but it seems like it would give an example of a universe with crystals but without atoms. Or is this idea incoherent in some way?
Jim Propp _______________________________________________ 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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
----- No virus found in this message. Checked by AVG - www.avg.com Version: 2013.0.2897 / Virus Database: 2639/6085 - Release Date: 02/06/13
I'm a complete ignoramus when it comes to quantum mechanics. But I wonder whether many of its impossibility pronouncements are more about what we can possibly measure, rather than what can be actually there. (Is this generally agreed upon among professional quantum mechanics, or are interpretations like whether QM is a limitation on human knowledge or on what actually may exist still matters of debate today?) I guess positivists -- as Tom Knight says -- would say that anything we can't detect is meaningless to talk about. But I don't hold with that attitude. (E.g., maybe there are other universes that always were and always will be separate from ours in every respect. A positivist might say it's meaningless to speculate about whether intelligent life might exist there. But the positivist humanoids in those other universes are thinking the very same thing about us.) --Dan On 2013-02-09, at 4:27 PM, meekerdb wrote:
On 2/9/2013 8:23 AM, Tom Knight wrote:
There can't be a continuum down there, simply because we can't have enough information to specify it.
That's a very positivist attitude. . . . . . .
Last time I checked the most accurate physical theory in human history was Quantum Electro-Dynamics. The value of the anomalous magnetic moment of the electron has been measured and compared to the QED theory to one part in 10^12. That's pretty impressive. Everything else is coffee house argument, until somebody comes up with a new experiment. Rowan (the ex-theorist and ex-experimentalist). On Sat, Feb 9, 2013 at 4:51 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I'm a complete ignoramus when it comes to quantum mechanics.
But I wonder whether many of its impossibility pronouncements are more about what we can possibly measure, rather than what can be actually there.
(Is this generally agreed upon among professional quantum mechanics, or are interpretations like whether QM is a limitation on human knowledge or on what actually may exist still matters of debate today?)
I guess positivists -- as Tom Knight says -- would say that anything we can't detect is meaningless to talk about.
But I don't hold with that attitude. (E.g., maybe there are other universes that always were and always will be separate from ours in every respect. A positivist might say it's meaningless to speculate about whether intelligent life might exist there. But the positivist humanoids in those other universes are thinking the very same thing about us.)
--Dan
On 2013-02-09, at 4:27 PM, meekerdb wrote:
On 2/9/2013 8:23 AM, Tom Knight wrote:
There can't be a continuum down there, simply because we can't have enough information to specify it.
That's a very positivist attitude. . . . . . .
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
It only takes 2-3 orders of magnitude to get a very impressive rotational symmetry from a cubic lattice; e.g., do a high-quality anti-aliased rotation of a hi-def image on your computer screen. Many PDE's are successfully simulated on cubic lattices and faithfully reproduce isotropy "more or less". But experiments of this type suggested in the arxiv paper are wonderful; we need as many of these experiments as we can get. I am certain that discrepancies will eventually appear, but perhaps not from a simplistic cubic lattice model of space. I'm quite intrigued by the holographic model, because it starts getting at quantum issues like entanglement, which have the possibility of saving bits; there aren't nearly as many bits or degrees of freedom in a holographic model as a simple cubic lattice model might suggest. Taking inspiration from Maxwell/Boltzmann/Planck/Bekenstein, it should be possible to push the entropy/information equivalence much harder to better characterize where information resides and flows in the holographic model. At 04:27 PM 2/9/2013, meekerdb wrote: It would be nice to have physical theories that didn't assume real numbers, and of course in any actual calculation or measurement only rationals are used. But experimental attempts to measure the discreteness of spacetime, as hypothesized by loop quantum gravity for example, have come up empty - even down close to the Planck scale, http://arxiv.org/pdf/1109.5191.pdf
Tom Knight:
My favorite speculation for structure at the Planck scale is a 4-D crystal with dislocations in the crystal structure giving rise to observable particles. Past is represented with a solidified structure, present is the phase boundary, and quantum weirdness and interference is the attempt to extend the phase boundary in a consistent way. We observe the net growth of the crystal, but it is an epiphenomenon -- a result of the crystal growth.
:) In the late 1980s, I used an HP75C to obsessively print out evolutions of one-dimensional cellular automaton rule #193. Abstracting some of the behaviors, I wrote a short, unpublished article wherein I likened some of the cyclical behaviors arising out of this rule to cracks in a crystal composed of (a very specific pattern of) staggered size-3 'triangles' that evolve naturally out of random startups under the rule. Each crack had a positive integer associated with it that was the offset of the two crystal regimes between which it was caught. When two cracks collided, they naturally preserved the sum (modulus the crystal 'length') of their offsets. I spent some time just trying to chart the elementary cracks/particles. To me, the implication was that the stuff we now consider the real part of our Universe (i.e., elementary particles) might also be defects in some small-dimensional sea of periodic structure.
On Feb 10, 2013, at 11:27 AM, Hans Havermann wrote:
Tom Knight:
My favorite speculation for structure at the Planck scale is a 4-D crystal with dislocations in the crystal structure giving rise to observable particles. Past is represented with a solidified structure, present is the phase boundary, and quantum weirdness and interference is the attempt to extend the phase boundary in a consistent way. We observe the net growth of the crystal, but it is an epiphenomenon -- a result of the crystal growth.
:) In the late 1980s, I used an HP75C to obsessively print out evolutions of one-dimensional cellular automaton rule #193. Abstracting some of the behaviors, I wrote a short, unpublished article wherein I likened some of the cyclical behaviors arising out of this rule to cracks in a crystal composed of (a very specific pattern of) staggered size-3 'triangles' that evolve naturally out of random startups under the rule. Each crack had a positive integer associated with it that was the offset of the two crystal regimes between which it was caught. When two cracks collided, they naturally preserved the sum (modulus the crystal 'length') of their offsets. I spent some time just trying to chart the elementary cracks/particles. To me, the implication was that the stuff we now consider the real part of our Universe (i.e., elementary particles) might also be defects in some small-dimensional sea of periodic structure.
Exactly. Dislocations have some interesting behavior -- for example, they cannot be created or destroyed, except in pairs of "anti-" dislocations. Conservation laws arise naturally in this context, from the algebra of the Burgers vector description of dislocations.
participants (10)
-
Allan Wechsler -
Dan Asimov -
Hans Havermann -
Henry Baker -
James Propp -
meekerdb -
Robert Munafo -
Rowan Hamilton -
Thomas Knight -
Tom Knight