The primary usefulness of integer multiples of the radian circular measure is being argued against by Bob Palais. Here are some things I think are worth thinking about, whether the grass really is greener on the other side of the fence, as Mr. Palais sees it, or not.
—————
Principal Values:
Multiples of Pi/2 locate both nodes and antinodes of the sin() and cos() functions. Multiples of Tau/2 do not.
The second derivatives of the sin() and cos() functions for multiples of Pi/2 locate both inflexion points and extrema. With multiples of Tau/2, they locate extrema of the cos() function and nodes of the sin() function, but they do not locate nodes of the cos() function or extrema of the sin() function.
In short, Pi/2 locates nodes, antinodes, and inflexion points, which are at values which have always, for good reason, been called Principal Values of the sin() and cos() functions. The substitution of Tau/4 does not in any way improve upon the present scheme for identifying these values.
—————
Function Values:
The series expansions for calculation of the sin() and cos() functions of an argument x given in radian measure are as follows:
sin x = sum{i=0,1,2,…}((((-1)^(i))/((2i+1)!))((x)^(2i+1)))
cos x = sum{i=0,1,2,…}((((-1)^(i))/((2i)!))((x)^(2i)))
We note that in summation of the series we must exponentiate the argument x to either the power 2i or 2i+1 in forming successive terms of each series. Since the argument x, under the “Tau proposal”, would have to be subject to measurement using some other-than-radian measure, a different series expansion would be needed to provide the correct numeric value for any given argument (unless everyone, every time they want to evaluate a sine or cosine, happily agrees to divide their number by two before substituting it as the argument x in either of these equations).
If I was Bob, I’d rush right out and get a patent on all of the algorithms I could think of which can be used to quickly calculate such a series, because if Mr. Hartl intends to force computer and microprocessor manufacturers to provide FPU support for it, the patent race might leave Bob behind.
Abramowitz and Stegun do not provide a simplified series expansion for one suitable as a math unit approximation function; and one can only surmise that Mr. Hartl has friends at Texas Instruments, Intel, AMD, or Motorola, who are right on top of the production line for chips that provide the new tausin(), taucos(), and tautan() functions right on their output pins.
The fact that using Tau doesn’t excite me any isn’t all that big of a deal. But the idea that some people are worse than pushy when it comes to getting what they want, and the idea that they are willing to foist an un-needed “simplification” onto everyone who uses a computer just to make money for them is actually a very big deal; and I object to it strongly.
So, what about the value of “the angle” itself, knowing the value of the sin() function, for instance? Of course, there’s a well-known series expansion which produces the numeric value, in radian measure, of an angle, given the sine of that angle as a function argument:
arcsin(x) = sum{i=0…}((((x)^(2i+1))/(2i+1))(prod{j=1..i}((2j-1)/(2j))))
This series isn’t going to return a value normalized to Tau, so everyone must happily agree to multiply their function return by 2, or go out looking for Mr. Hartl’s new CPU/FPU combination and pay up front for it. All these afficiondos really need is to multiply or divide the value returned by a floating-point calculation circuit by 2 to get their Tau-normalized value; and in so doing they may be able to avoid a patent infringement case when they use a design for the circuitry containing significant design or invention which someone else already patented and which returns a value normalized to radian measure. Who knows? People have thought of all sorts of absurd schemes to “get around” patent protection measures. Is that what this hullabaloo is all about?
—————
Fourier methods:
For non-periodic functions, we can use half-range expansions, either by reflecting the function about the origin to create an odd function, or about the y-axis to create an even function. In both cases, the period is twice the range of definition (between 0 and L or between -L and 0) of the function; and since the range denoted by Tau is synonymous with the period, half of its range is outside the range of definition of the function we are seeking to expand.
For periodic functions, the obvious simplification from the argument (2*Pi*t*n/L) to (w*t*n) leads to a simpler form for the Fourier series expansions, and if W = 2*Pi/L = Tau/L, we have done nothing but hide the irrational number Pi (and its equally irrational double, Tau) without introducing any other obvious simplification.
—————
Physical methods:
The Planck constant divided by 2*Pi, called h-bar, appears in many calculations in modern physics; and substituting Tau for 2*Pi would oblige us to divide Planck’s constant by Tau instead. But this doesn’t do anything to solve the basic problem, which is that the Planck constant alone lacks a commonly accepted and used measurement system to support it; and that even though h-bar works well enough in the SI system of units, there always exists a renormalization problem when applying equations which are supposed to be correct as-is. The use of Tau does not in any way solve this problem, because the use of h-bar is wrong to begin with; although that fact goes almost unrecognized. And, short of changing from the SI system to some other completely unified system– which the standards organizations haven’t discovered, yet –the accepted values of the Planck time, length, and volume will not have values which are reconcilable with h rather than h-bar.
Whether Bob Palais intends to substitute 1/Tau for the factor of 1/2Pi appearing in the equations for uncertainty, or to relegate integer- and half-integer- spin factors to the garbage heap, I have no idea. There are, however, a significant number of university classrooms in which the question has been raised as to why the factor of 1/2Pi cannot be dispensed with in the calculation of integrals involving expectation values for particle position or momentum; and the most common answer is that both the number e and the value 2Pi will pop up here and there just because of the methods you are using, so there’s nothing you can do about it even if you should deliberately try to get rid of them.
Glenn's space 