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[math-fun] In Praise of toc1955 & others
by Brad Klee 02 Feb '20

02 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 02 Feb '20

02 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 31 Jan '20

31 Jan '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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[math-fun] Non-existence from parity (was Re: Existence from parity)
by Keith F. Lynch 31 Jan '20

31 Jan '20

For certain values of N, it's possible to arrange the first N positive integers such that the sum of each adjacent pair is a square. For instance: 8 1 15 10 6 3 13 12 4 5 11 14 2 7 9 I've been looking into whether the same is true of sequences other than squares. With cubes, I still don't know whether I didn't find any because there aren't any, or whether I just didn't try a large enough N. But with powers of 2, I soon came up with a parity-related proof that there can't be a (non-empty) solution no matter how large N is.

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[math-fun] Who else has noticed the neat spiral effect you get
by Bill Gosper 31 Jan '20

31 Jan '20

from misjoining two Dragon skins <http://gosper.org/dragspir.png>? —rwg

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Re: [math-fun] Cubic equations, ellipses and hyperbolic functions
by Henry Baker 31 Jan '20

31 Jan '20

At 08:35 PM 1/30/2020, Brad Klee wrote: >One issue with your model though: how do you set parameters to get >z^3=1? It sounds like circular Steiner ellipses would have c = 0, implying >only z^3=0. Can f(z) equal a constant? z^3=c^3 is soooo boring -- you can solve it by inspection. But I'd be interested if there is another parameterization that generalizes my parameterization to handle this degenerate case.

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[math-fun] Gimbal lock??
by Henry Baker 31 Jan '20

31 Jan '20

Perhaps it's time to learn quaternions?? >From comp.risks: Date: Fri, 10 Jan 2020 20:24:07 +0000 From: "Clive D.W. Feather" <clive(a)davros.org> Subject: Boeing 737s can't land facing west (FAA) "The FAA received reports earlier this year of three incidents of display electronic unit (DEU) software errors on Model 737 NG airplanes flying into runway PABR in Barrow, Alaska. All six display units (DUs) blanked with a selected instrument approach to a runway with a 270-degree true heading, and all six DUs stayed blank until a different runway was selected. [...] The investigation revealed that the problem occurs when this combination of software is installed and a susceptible runway with a 270-degree true heading is selected for instrument approach. Not all runways with a 270-degree true heading are susceptible; only seven runways worldwide, as identified in this AD, have latitude and longitude values that cause the blanking behavior." (Note that this is all 6 displays on each plane, not 2 displays on each of three planes.) The runways in question are: Runway 26, Pine Bluffs, Wyoming, USA (82V) Runway 28, Wayne County, Ohio, USA (KBJJ) Runway 28, Chippewa County, Michigan, USA (KCIU) Runway 26, Cavern City, New Mexico, USA (KCNM) Runway 25, Barrow, Alaska, USA (PABR) Runway 28, La Mina, La Guajira, Colombia (SKLM) Runway 29, Cheddi Jagan, Georgetown, Guyana (SYCJ) (The numbers are magnetic bearings, whereas the problem is apparently related to true bearing.) Original FAA notice: <http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgad.nsf/0/3948342a978cc…>

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[math-fun] Cubic equations, ellipses and hyperbolic functions
by hbaker1 30 Jan '20

30 Jan '20

Cubic equations, ellipses and hyperbolic functions I contend that the most elegant form for the general cubic equation in the complex plane is the trigonometric form, with *complex* c,A,z: f(z) = (z-c*cos(A))*(z-c*cos(A+2pi/3))*(z-c*cos(A+4pi/3)), with roots located at positions r1,r2,r3 on the complex plane: r1 = c*cos(A) r2 = c*cos(A+2pi/3) r3 = c*cos(A+4pi/3) We assume that the centroid of the roots is at the origin, so if c,A,z are all complex, then this form is completely general. Expanding and simplifying, we get 4*f(z) = 4*z^3 - 3*c^2*z - c^3*cos(3*A) Solving f'(z)=0, we find that z = +-c/2, which are the *foci* of the *Steiner inellipse* of the triangle of the roots, while the roots r1,r2,r3 lie on the *Steiner (circum)ellipse* of the triangle of the roots, which ellipse is twice as big, hence its foci are at z = +-c. So what are the parameters a,b of the Steiner ellipse? a = c*cosh(ip(A)) b = c*sinh(ip(A)) where ip(A) is the *imaginary part* of the complex angle A. We have the familiar equation c^2=a^2-b^2 for this Steiner ellipse, as well as this less familiar equation for its eccentricity: e = sech(ip(A)) Note that the basic properties of the Steiner ellipse are *independent* of the realpart of A, and hence the roots can revolve about the origin along the Steiner elliptical orbit, with the orbital position(s) indicated by the realpart of complex angle A. Indeed, if we keep ip(A) constant, we can *animate* the roots in their Steiner orbit by making complex angle A a function of time t, e.g.: r1 = c*cos(A(t)), A(t) = t + i*ip(A) ("i" is the complex number i^2=-1.) The area of the triangle of the roots is the real number: Area = area(0,r1,r2)+area(0,r2,r3)+area(0,r3,r1) = 3*area(0,r1,r2) (0 is the centroid) = 3 * sqrt(3)/8 * sinh(2*ip(A)) * c*c' (c' = conjugate(c)) = 3^(3/2)/4 * sinh(ip(A))*cosh(ip(A)) * c*c' = 3^(3/2)/4 * c*sinh(ip(A)) * c'*cosh(ip(A)) = 3^(3/2)/4 * a * b' analogous to the formula for the area of an ellipse with semimajor axis a and semiminor axis b, and where 3^(3/2)/4 is the area of an equilateral triangle inscribed within the unit circle. Once again, note that the triangle area is *independent* of the location of the roots within their Steiner elliptical orbit. Since we're talking about polynomials, it is interesting to consider the *discriminant* of our cubic: (r1-r2)^2*(r2-r3)^2*(r3-r1)^2 = (27*sin(3*A)^2*c^6)/16 Note that the discriminant is *complex*, and is obviously *NOT* independent of the realpart of complex angle A. Q: Are there additional interesting parameters computable from the roots which are independent of the realpart of complex angle A ? Q: Which symmetric functions of the roots are independent of the realpart of complex angle A? Clearly, there are some, as r1+r2+r3=0, and r1*r2+r2*r3+r3*r1=-3*c^2/4.

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[math-fun] A Legendre Question
by Brad Klee 30 Jan '20

30 Jan '20

Cross-post from seqfan: http://list.seqfan.eu/pipermail/seqfan/2020-January/020454.html See also: https://pdfs.semanticscholar.org/3a41/3ed6779db174ff01c0debea7f41e12a215d0.… Maybe it is easy to implement a reduction algorithm, but I'm not quite to the point of trying yet. --Brad

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[math-fun] Neat Dragon question
by Bill Gosper 28 Jan '20

28 Jan '20

What is the longest line segment inscribable in the Dragon linking 0+0i to 1 + 0i via (1+i)/2? "Hint": If the segment be t z2 + (1-t)z1, with 0<t<1, then every point on it is (empirically, at least) a triple point of the spacefill whenever t is a dyadic rational! —rwg (This impacts OEIS A260747, etc.)

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