February 2020 - mathfun@mailman.xmission.com

Re: [math-fun] Mystery tables
by Steve Witham 08 Feb '20

08 Feb '20

From: "Keith F. Lynch" <kfl(a)KeithLynch.net> Date: 2/6/20, 11:06 PM > For those not familiar with the concept, the xth power mean of a and b > is defined as (a^x + b^x)/2 = m^x. Common examples include: > > -1: Harmonic mean Kewl, now I think I can average two resistors. > 0: Geometric mean I got as far as wondering whether this was something like what happens when integrating x^(-1)... instead of getting (x^0) / 0, and then went to wikipedia... which treats it like a convention: > (See/p/-norm <https://en.wikipedia.org/wiki/Norm_%28mathematics%29#p-norm>). For/p/= 0we set it equal to the geometric mean > (which is the limit of means with exponents approaching zero)... Do these means change smoothly as x crosses zero? This all strikes me as like giving someone honorary citizenship. --Steve

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Re: [math-fun] Mystery tables
by Keith F. Lynch 07 Feb '20

07 Feb '20

Gareth McCaughan <gareth.mccaughan(a)pobox.com> wrote [about my first table]: > I have been thinking half-heartedly (and apparently half-brainedly) > about this one from time to time ... and in the process of writing > down my incoherent thoughts I realise that I have solved it. I often do the same. See https://en.wikipedia.org/wiki/Rubber_duck_debugging > The value in row m, column n is the k such that m^k = 2n^k-1. > For the (1,1) entry any k will do, hence the slightly curious > "all" in that spot. Right. My intention was for k to be the power necessary for the power mean of 1 and m to equal n. For those not familiar with the concept, the xth power mean of a and b is defined as (a^x + b^x)/2 = m^x. Common examples include: -1: Harmonic mean 0: Geometric mean 1: Arithmetic mean 2: Quadratic mean, aka Root Mean Square (RMS) But x can be any real number, not just an integer. For my first table I chose values of x that made the power mean of 1 and an integer (the row number) another integer (the column number). Most such values of x are transcendental numbers. But not all. What rational values can x have? Can it have an irrational algebraic value? (I'm speaking of the row and column both being positive integers, and the column number being less than the row number.) [About my second table:] > But it turns out that the entries in the table satisfy the following > (defining) equation: > if the entry in position (x,y) is z, then x^z + y^z = 2z^z. Right. > It is not obvious to me whether this is actually equivalent to some > sort of generalized mean as defined above. I'd guess not. What of > course _is_ true is that z = ((x^z+y^z)/2)^1/z, so each entry is _a_ > "power mean" of its row and column positions ... but with an exponent > that equals the entry itself :-). Yes, that was my intention. For every set of positive real numbers x<z1<y, z1 is some power mean of x and y. And for every positive real numbers x, y, and real number z2, there's a z2 power mean z1 that is between x and y. Since, given fixed x and y, z1 and z2 are monotonic functions of each other, it necessarily follows that for every x and y there must be a unique z1 that equals z2. I got curious what they looked like. Hence my second table. I assume they're all transcendental except of course where x=y, in which case z=x=y. Can anyone prove this? Or find a counterexample?

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[math-fun] Dragon and Twindragon animations
by Tom Karzes 06 Feb '20

06 Feb '20

I created some dragon animations that might be of interest. The first is a twindragon animation that sweeps through the self-mapping angles, pausing briefly at points of interest (twindragon, tame twindragon, rectangle, reverse tame twindragon, reverse twindragon): https://www.karzes.com/anim_twindragon.gif The second is a Heighway dragon animation that also sweeps through the self-mapping angles (while retaining their difference), pausing briefly at points of interest (Heighway dragon, triangle, reverse Heighway dragon): https://www.karzes.com/anim_dragon.gif Tom

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[math-fun] Mystery tables
by Keith F. Lynch 05 Feb '20

05 Feb '20

There have been no posts in response to either of my mystery tables for over a week. Is anyone still working on them, or shall I reveal what they are tables of? Here they are again: 1 2 3 4 5 6 7 8 1: all 2: -inf +inf 3: -inf 1.000000 +inf 4: -inf 0.000000 2.261406 +inf 5: -inf -0.349081 1.000000 3.074601 +inf 6: -inf -0.523305 0.523305 1.564328 3.795684 +inf 7: -inf -0.626499 0.268479 1.000000 2.000000 4.495526 +inf 8: -inf -0.694242 0.109189 0.694242 1.350067 2.385122 5.190739 +inf 9: -inf -0.741892 0.000000 0.500000 1.000000 1.643796 2.748616 5.884929 10: -inf -0.777116 -0.079600 0.364842 0.777116 1.249756 1.908997 3.102748 11: -inf -0.804148 -0.140255 0.265003 0.621297 1.000000 1.469217 2.158924 12: -inf -0.825509 -0.188046 0.188046 0.505587 0.825509 1.192579 1.671024 13: -inf -0.842792 -0.226700 0.126799 0.415933 0.695796 1.000000 1.366735 14: -inf -0.857046 -0.258628 0.076822 0.344232 0.595108 0.857046 1.155985 15: -inf -0.868994 -0.285461 0.035215 0.285461 0.514414 0.746112 1.000000 16: -inf -0.879146 -0.308343 0.000000 0.236333 0.448129 0.657173 0.879146 1 2 3 4 5 6 7 8 1: 1.000000 1.545362 2.293678 3.240906 4.248364 5.259193 6.267074 7.272747 2: 1.545362 2.000000 2.576905 3.343752 4.270644 5.262645 6.267506 7.272793 3: 2.293678 2.576905 3.000000 3.589622 4.367036 5.288115 6.272582 7.273628 4: 3.240906 3.343752 3.589622 4.000000 4.596536 5.380654 6.299598 7.279833 5: 4.248364 4.270644 4.367036 4.596536 5.000000 5.600888 6.389624 7.307724 6: 5.259193 5.262645 5.288115 5.380654 5.600888 6.000000 6.603882 7.395990 7: 6.267074 6.267506 6.272582 6.299598 6.389624 6.603882 7.000000 7.606068 8: 7.272747 7.272793 7.273628 7.279833 7.307724 7.395990 7.606068 8.000000

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Re: [math-fun] Dimensional analysis (was Re: Gimbal lock??)
by Keith F. Lynch 05 Feb '20

05 Feb '20

Brad Klee <bradklee(a)gmail.com> wrote: > If you haven't gone to college, shouldn't you be more inclined to > argue with the entitlement crowd? What gives them the right to vote > and not you? Good question. Thanks for asking it. My answer is complicated, and has little to do with math or even physics, so I will respond off-list, probably this weekend. (Anyone else who wants to be CCd on my reply, please email me. Thanks.) > The gist of Ralston's argument is that classical quantities energy > and mass are not useful in the quantum regime, and that they should > not appear in formal equations of quantum mechanics, such as > Schroedinger equation. My understanding is that mass and energy are equally useful concepts at every scale. The only major issues I can think of offhand are whether mass and rest mass should be distinguished from each other, and whether the energy of the vacuum can really be usefully defined as zero. > Now if we measure photoelectric nu2 on a Hydrogen standard nu1, we > get photoelectric band gap E2 relative to hydrogen standard E1, > *without mentioning Planck's constant*. For any concept, no matter how useful, we can always work around it, using circumlocutions. In some cases, which things are considered fundamental is completely arbitrary. In other cases, working around a lack would be as awkward as abolishing money and using barter instead. I think the Planck constant is somewhere between those two extremes. > That is to say, the photo-electric effect is another version of the > Rydberg Hydrogen experiment (with a hell of a lot more uncontrolled > variables and uncertainties. Also with less of an intelligible > level structure.). We could certainly use the Rydberg constant in place of the Planck constant, but most equations would be more unwieldy. > Once the hangup with photoelectric effect is sidestepped, there is > still often an objection about electron mass. This is a concept > that makes some amount of sense in terms of the oil drop experiment. Are you sure? I thought Millikan was balancing the force from the electron's *charge* in a known electric field against the force from the *oil drop's* mass in a known gravitational field. You can't get the electron's mass from that. Measurement of how much a moving isolated electron's trajectory is bent by a known electric or magnetic field gives you the electron's charge-to-mass ratio. (And, as a side effect, invents television.) And if you already know the charge, it gives you the electron's mass. Assuming those two experiments were the only ways of measuring those quantities, the electron's charge-to-mass ratio would always be known to a higher precision than its mass. Similarly, each planet's GM (product of the gravitational constant and the planet's mass) is known to a much higher precision than either G or M alone. I assume NASA uses each planet's GM for navigation. But that's no reason to deprecate the concept of a planet's mass. Or of an electron's rest mass. > Again, this is not a direct measurement of electron mass, it is > an inference of electron mass from classical physics involving > gravitational and electric fields. Thus the extracted parameter > is a mess of other constants besides m_e. I didn't know there was any issue about an electron's rest mass (other than, as with every measured quantity, the measurement having some error). There is an issue about its charge being larger close up, apparently without limit. It can be modeled as an extreme negative charge surrounded by a cloud of positive charge which almost but not quite cancels it out. I can understand objecting to the concept of the classical electron radius, which comes from assuming the electron's rest mass "comes from" the energy it would classically take to assemble infinitely many infinitesimal like charges (whose total charge sums to that of an electron) from an infinite distance into a small radius. There's a unique value for which the numbers balance, and that's the classical electron radius. Very likely it has no physical significance whatsoever. > Section 2 of the paper does an admirable job showing that Planck's > constant (1900) was over-promoted relative to the fine structure > constant (Sommerfeld 1916), and and the Compton wavelength (1923). > It all drives to the conclusion of equation (24) that: including > Planck's constant and electron mass introduces an extra, perhaps > delusional, degree of freedom. I'm not convinced that it's meaningful to count degrees of freedom in a system in which things can't vary. > In electron experiments, we only need Compton wavelength to > set scale, and fine structure constant to search through the > perturbation hierarchy (alpha as an expansion parameter is > discussed in most QM textbooks). Perhaps we can do without the Compton wavelength too, and only count dimensionless parameters as degrees of freedom. > This analysis of quantum pre-history doesn't exactly answer the > question about generality. It would certainly be nice if we could contact another advanced civilization and compare notes. I would bet that they would also have the concept of the Planck constant. And the concept of prime numbers. I'd love to be proven wrong. I'm reminded of the Hugo and Nebula winning "Story of Your Life," a 1998 science fiction story by Ted Chiang. A race of aliens turn out to be familiar with variational principles in physics, but completely ignorant of cause-and-effect explanations, not just in physics, but also in daily life. (In 2016 it was made into a movie, "Arrival.") > PS. QM has "Symplectic symmetry", so position and momentum are > treated as fully equivalent variables. They're equivalent in some senses, but not in the sense of being the same thing. Whether a person is guilty of trespassing depends entirely on his position, and not at all on his momentum. > The Schroedinger EQ itself can be written in either position or > momentum basis, and nothing much changes. The uncertainty in one is the Fourier transform of the uncertainty in the other (if you ignore signs -- the FT of an FT is not the original function, but its negative). Similarly with looking at radio signals in the time domain vs. the frequency domain. An FT simply rotates a waterfall display by 90 degrees. But that doesn't mean that time and frequency are the same thing. >> Anyhow, the Planck constant would still exist, though it would >> become a mere conversion factor, the number of hertz in a joule, ... Of course I meant the number of joules in a hertz. I'll agree that it's just as technically accurate to say the Titanic broadcast its distress call at 500 kilohertz or at 0.00033 yoctojoules. (Sorry, there's no prefix smaller than yocto.) Actually, both are ahistorical, as they called it 600 meters. Or rather metres, as it was a British ship. Sorry I'm rambling. It's past my bedtime.

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Re: [math-fun] New (?) conjecture
by Keith F. Lynch 05 Feb '20

05 Feb '20

Allan Wechsler <acwacw(a)gmail.com> wrote: > Gareth McCaughan <gareth.mccaughan(a)pobox.com> wrote: >> Keith F. Lynch wrote: >>> If a regular polygon with N sides is constructable with compass >>> and straight-edge alone, then N expressed in binary is a >>> palindrome followed by zero or more zeros. >> Gauss proved that N has to be a product of distinct primes 2^2^n+1 >> times some power of 2. >> The power of 2 gives you zeros at the end. The other factors give >> you something palindromic: if some product using n<N is palindromic >> then multiplying by 2^2^N+1 gives you another palindrome because >> we're taking two copies of its bits and they are separated by just >> enough 0s that they can't collide. > John Conway and Neil L. White have independently made a similar > observation, pointing out that the orders of odd constructible > regular polygons can all be found in the sequence A001317, the > rows of Fermat's triangle modulo 2 interpreted as binary integers. > Apparently nobody has any clue whether there are any Fermat > primes other than the classic 3, 5, 17, 257, 65537. A very naive > probabilistic argument suggests that this list is exhaustive. If > there are no more, then the constructible orders are just the first > 31 entries in A001317. Thanks. I thought this might be original since neither A003401 nor A045544 mentioned palindromes. Perhaps I should add such a mention to the comments, at least for the latter sequence.

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[math-fun] New (?) conjecture
by Keith F. Lynch 03 Feb '20

03 Feb '20

If a regular polygon with N sides is constructable with compass and straight-edge alone, then N expressed in binary is a palindrome followed by zero or more zeros.

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[math-fun] In Praise of toc1955 & others
by Brad Klee 03 Feb '20

03 Feb '20

I am impressed and humbled. Has anyone else built an interferometer in their garage? The phototube measurement is great, but please see also: https://m.youtube.com/watch?v=uphPKYR2i5M Per my knowledge, The only other reference close to this level of quality is David Prutchi: http://www.prutchi.com/quantum-physics/ The Hardware Hackers Won! Cheers, —Brad

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Re: [math-fun] Dimensional analysis (was Re: Gimbal lock??)
by Keith F. Lynch 03 Feb '20

03 Feb '20

Brad Klee <bradklee(a)gmail.com> wrote: > I alluded to metrology of SI by mentioning Caesium hyperfine. This > is a justification for projective formalism: there is an agreed > upon standard for how to set absolute value 1 in each dimension. Right. Time is really several similar but subtly different concepts. The cesium clock pins down which one we're using. Also, the time of day is given, not just by any cesium clock, but only by one on the geoid (roughly speaking, at mean sea level). In principle, time as measured by each of the four forces could be different, in which case the cesium clock implies we're using time based on the electromagnetic force. Similarly, the original definition of the gram implied that "pure water" is well defined. Actually, it can have a varying isotopic composition, hence a range of densities. > As for time dependence of fine structure, Ralston discusses this in > his article on Planck?s constant: > https://arxiv.org/abs/1203.5557 > ( worth the time to read ) > He was against the recent decision to fix Planck?s constant. I could make no sense of that paper. I can't figure out what he means by abolishing the Planck constant. Setting it to 1? The Planck system of units does exactly that, as it sets it, c, G, and k-sub-B to 1. But its usefulness is limited by the low precision to which G is known, and by the inconvenient sizes of most units. Obviously the Planck constant can't be set to 1 within SI, as that would radically change the sizes of lots of units. Perhaps he means it should be regarded as dimensionless? That would make action and angular momentum dimensionless, meaning that energy and frequency would be considered the same thing. And so would distance and momentum. That doesn't sound useful to me. Anyhow, the Planck constant would still exist, though it would become a mere conversion factor, the number of hertz in a joule, like the number of feet in a mile. If he's arguing that its numerical value is arbitrary, a mere artifact of our system of units, I don't know of anyone who disagrees. He argues that if it was abolished, other constants could be determined better. But that's exactly what the 2019 SI changes did, and a large part of why they were adopted. And you say he was against those changes. Not having gone to college, I'm reluctant to argue with people with doctorates, especially doctorates in hard sciences. Does the fact that he misspells Rydberg and calls Kelvins "degrees Kelvin" count against him, or does it just mean I have the soul of a proofreader? So, is he confused, or am I? If the latter, can someone explain his central thesis to me? Thanks.

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[math-fun] Dimensional analysis (was Re: Gimbal lock??)
by Keith F. Lynch 01 Feb '20

01 Feb '20

There is no one correct form of dimensional analysis. For instance is vertical distance the same kind of thing as horizontal distance? It depends on your application. For interior design, no, since placing an item of furniture on its side violates the rules of furniture arranging. In the cgs system, unlike the SI system, capacitance has the dimension of length; the unit of capacitance is the centimeter. That's because the cgs unit of capacitance is the capacitance of an isolated sphere with a radius of 1 centimeter. In SI, capacitance is ampere-seconds per volt, which makes a nice symmetry with inductance, which is volt- seconds per ampere. Multiply a capacitance by an inductance and you get seconds squared. Take the reciprocal of its square root, and you get a frequency, which is indeed the resonant frequency of the resulting circuit. (Just remember that that frequency is in radians per second, not hertz.) There was a major change in the foundations of SI last year. It didn't (intentionally) change the size of any unit, but it did subtly change what they were units *of*, and which physics equations were conjectures and which were true by definition. Consider the dimensionless fine structure constant, alpha. The speed of light in a vacuum switched from a measured value to a defined value (exactly 299792458 meters per second) in 1982. When I learned that the charge of the electron and the Planck constant were to also change from measured values to defined values, I complained to David Newell <https://www.nist.gov/people/david-b-newell> that this would implicitly make alpha a defined value too, and that that was cheating, since it's dimensionless. He patiently explained that in SI, unlike in cgs, alpha also depends on the permeability of free space, aka the magnetic constant, mu-nought. And that its value, then defined, would become a measured value with the 2019 change. (In cgs, mu-nought doesn't exist, i.e. it's a dimensionless unity.) That means that if alpha turns out not to be constant after all, but to vary over time or location, we're constrained to regard that as a change in mu-nought, rather than in the speed of light, the Planck constant, or the charge of the electron. But it's really all rather arbitrary. George Hart pointed out that torque and energy have the same dimensions, but aren't the same thing. The usual explanation is that the former is a vector and the latter is a scalar. Another way to look at it is that one newton-meter of torque means applying one newton of force at right angles to the center of rotation which is one meter away, while one newton-meter of energy means applying one newton of force over a distance of one meter in the *direction* of that one meter rather than at right angles to it. Torque and energy having the same dimensions implies that so do angular momentum and action. Angular momentum and action are unusual in that both are quantized. The same Planck constant applies to both.

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