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Turns are Better than Radians (2022)

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Re: Turns are Better than Radians (2022)

#101

Post by robertlagrant »

ogogmad wrote: post #82
kazinator wrote: post #70

The math is definitely not fine with turns, because your Euler formula e^ix = cos x + i sin x no longer holds. We can use a base other than e , namely B = e^2pi which around 535.4916. This doesn't have the nice e properties like d/dx e^x = e^x. The elegant fact that the base of the natural logarithm, which produces an exponential function that is its own derivative, also shows up as the basis for the above Euler's fo…

In another comment, I asked why people chose to use the symbol τ over just writing turn or " rev (olution)" (defined to be the constant ≈ 6.28318530718 ) given how unambiguous the latter is as a name for 2π. And why not just write sinrev() or sinturn(), and leave the symbols sin() and rev (defined to be ≈ 6.28318530718) alone?

I think it's Tau[0].

[0] https://en.wikipedia.org/wiki/Tau_(mathematics)

Re: Turns are Better than Radians (2022)

#102

Post by thyristan »

simiones wrote: post #95
thyristan wrote: post #57

Earlier quoted context omitted.

In a very awkward way: rad is m/m, which is 1...

Dimensions and units are separate things, though. For example, 1 minute and 1 second are different units of the same time dimension. Similarly, 1 rad and 1 degree are both dimensionless, but they are both different units.

Theoretically, you can define systems of measurement where a lot of seemingly separate things fall on the same units and dimensions. There are "natural units" in physics where you take the fundamental nature of particles and relativity into account and make some convenient choices for some physical constants such as the speed of light c := 1. Then the speed becomes dimensionless, length and time have the same unit and dimension (a length of 1eV is a time period of 1eV), and almost all units are simply derived from a measurement of energy (electron volt, not as basic as people would like, but useful enough).

https://en.wikipedia.org/wiki/Natural_units

Re: Turns are Better than Radians (2022)

#103

Post by WCSTombs »

thrtythreeforty wrote: post #45

Here's another good reason to think in turns: it turns Euler's formula from this Eldritch Terror: e^(i*x) = cos(x) + i*sin(x) into something you can kinda understand by staring at the complex plane: -1^(2x) = cost(x) + i*sint(x) Credit to justinpombrio for this: https://news.ycombinator.com/item?id=32986869

Sorry but this is pretty bogus. (-1)^x is only well defined when x is an integer. This is generally the case for r^x whenever r isn't a positive real number. For example, when x = 0.5, r has two distinct square roots. Sure, you can choose one of them arbitrarily and declare it to be the value of r^0.5 (and math libraries typically do this), but there's unfortunately no good way to make this arbitrary choice consistently for all values of r simultaneously.

Re: Turns are Better than Radians (2022)

#104

Post by lioeters »

srean wrote: post #65
math-man wrote: post #6

Earlier quoted context omitted.

It's because both radians and degrees are are a ratio of a length to another length and are thus dimensionless. No matter how you measure it, all angles are without a unit. It's most obvious with radians but it's also the case with degrees. Using radians, you are guaranteed to not introduce unusual extra terms to rescale angles, if you use any other scale of angle you will have to keep track of extra terms. That may…

It is dimensionless by fiat and convention. It clearly has units such as degrees, grad and radians. Just like other quantities that have units, a specific measurement is expressed as a pure numerical multiple of an unit which may be radians, degrees etc. This is a known wrinkle in dimensional analysis and people have considered making angles a fundamental quantity such as mass, length and time but have not done so be…

That's my rabbit hole of the week.

> The current state of affairs leads inevitably to ghostly appearances and disappearances of the radian in the dimensional analysis of physical equations.

In "A spectral unit", Nature Physics (2020) - https://www.nature.com/articles/s41567-020-0997-3.pdf Giacomo Prando summarizes the troubled history of the radian, a unit with the odd property of appearing and disappearing seemingly at will in dimensional formulas

It led me to reading about "dimensionless quantity".

> There have been periodic proposals to "patch" the SI system to reduce confusion regarding physical dimensions. For example, a 2017 op-ed in Nature argued for formalizing the radian as a physical unit.

SI units need reform to avoid confusion (2017) - https://doi.org/10.1038%2F548135b

> The idea was rebutted on the grounds that such a change would raise inconsistencies for both established dimensionless groups, like the Strouhal number, and for mathematically distinct entities that happen to have the same units, like torque (a vector product) versus energy (a scalar product).

Don't tamper with SI-unit consistency (2017) - https://doi.org/10.1038%2F549160d

---

What's surprising is that these discussions are so recent, considering the wide implications on so much practical work in mechanics and engineering. Maybe people got so used to working around the question, that any proposed solution would be too disruptive - so it's better to keep things backward compatible rather than conceptually simple/clear and explicit.

In another comment someone said, "types" (in programming and type theory, I'd guess) are a poor model of physics units. But I wonder about that, it seems "units" are something like types with associated quantities, like degrees with 360, or meters with the speed of light.

This line of inquiry also led me to "dimensionless physical constants". https://en.wikipedia.org/wiki/Dimensionless_physical_constan...

Certainly pi (and e) must be one of those fundamental constants regardless of any unit of measurement. A "turn", on the other hand - well, if we consider 1 as a dimensionless constant that means a "whole"..

How Many Fundamental Constants Are There? (2011) John Baez https://math.ucr.edu/home/baez/constants.html

Re: Turns are Better than Radians (2022)

#105

Post by beeforpork »

Well, \tau vs \pi is a question of taste, but 1 vs. \tau (or \pi) is not. Because you don't get rid of these weird constants, because \pi (or \tau) is, as a fact, in the circumference and area of circles and in surface and volume of spheres, and in other places. There jus is a weird constant.

And for APIs, you could reasonably well have turns or radians or degrees or even percentage of turns, whatever -- it depend on the context what is 'better'. What's really missing, I think, is the support of units in programming languages (in the type system) so that you cannot mess up when invoking sin()/cos(), because you would be forced to provide a unit.

Re: Turns are Better than Radians (2022)

#106

Post by xg15 »

simiones wrote: post #92
cubefox wrote: post #77

Earlier quoted context omitted.

> You generally can’t apply functions to dimensional units. Perhaps not in mathematics, but in programming that's clearly possible. I guess programming is more general than mathematics.

This is precisely why (programming language) types are poor model of physics units, despite often being touted for this exact use case. 3m is not the same thing as "the value 3 of type meter". It is the multiplication of the dimensionless scalar 3 with the special "m" constant for meters. That's why pow(3m, 2) = 9 m^2, and not `the value 9 of type meter`. Of course, you can define the type `square meter` as well, and…

> with the special "m" constant for meters.

Isn't the "special constant" exactly "value 1, type meters", defined as equal to "value type atoms" etc?

If not, then what would be the result of the multiplication of 3 with "m"?

> Of course, you can define the type `square meter` as well, and define `pow -> meters -> positive integer -> square meters`

As long as your power is an integer, you can reduce it to multiplication. So what you'd really want to define is the result of " * ", "( * ) * " etc.

What this gets you in the end is a type algebra, but that is also not exactly a new concept.

Re: Turns are Better than Radians (2022)

#107

Post by otikik »

chabska wrote: post #4

The problem is that trigonometric functions are used in many more fields beyond geometry. The input is not always an angle around a point in euclidean space, it could be phase angle of a periodic signal. You can make an alternative set of trig functions that take turns, but you will anger a lot of people if you mess with the vanilla trig functions.

Functions are free. Create new ones. Sin1 instead of Sin, Cos1 instead of Cos.

Re: Turns are Better than Radians (2022)

#108

Post by pwdisswordfishq »

hasley wrote: post #53
jameshart wrote: post #26

Earlier quoted context omitted.

You generally can’t apply functions to dimensional units. The only thing units can do is be multiplied or divided together. So I can multiply a mass by a distance or divide a distance by a speed, and I can multiply the result by a scalar; but I can’t take the sine of a distance or the logarithm of a time or exponentiate a mass. Those are things I can only do to scalars. ‘But wait!’ You may cry: ‘the formula for a tra…

I basically agree, at least for standard functions like sin, cos, tan, exp etc. It is even possible to see mistakes in equations just by checking that all the units to standard functions cancel out making the arguments dimensionless. On the other hand I am still unhappy with calling the ratio of two quantities, that happen to have the same units, "dimensionless". This way, you could create any two "dimensionless" qua…

> This way, you could create any two "dimensionless" quantities and try to compare them or use one in place of the other which might be meaningless.

Case in point:

https://trac.ffmpeg.org/ticket/11279

https://trac.ffmpeg.org/ticket/11284

Re: Turns are Better than Radians (2022)

#110

Post by srean »

lioeters wrote: post #104
srean wrote: post #65

Earlier quoted context omitted.

It is dimensionless by fiat and convention. It clearly has units such as degrees, grad and radians. Just like other quantities that have units, a specific measurement is expressed as a pure numerical multiple of an unit which may be radians, degrees etc. This is a known wrinkle in dimensional analysis and people have considered making angles a fundamental quantity such as mass, length and time but have not done so be…

That's my rabbit hole of the week. > The current state of affairs leads inevitably to ghostly appearances and disappearances of the radian in the dimensional analysis of physical equations. In "A spectral unit", Nature Physics (2020) - https://www.nature.com/articles/s41567-020-0997-3.pdf Giacomo Prando summarizes the troubled history of the radian, a unit with the odd property of appearing and disappearing seemingly…

Ensuring units agree is indeed a form of type checking. A more thorough procedure for the former is dimensional analysis.

I think the fact that action and angular momentum have the same units played an illuminating role in making sense in Bohr's model regarding why only certain orbits are allowed.

Not sure how that would play out once angle is considered a fundamental entity.

This sure is a rabbit hole.

Thanks for your submission

https://news.ycombinator.com/item?id=49372847

hope it gets picked up.

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