Talk:Turn (angle)
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Others than Hoyle[edit]
I cannot find any other references for this. While Hoyle may have proposed this, he also proposed lots of other things that aren't notable.  Nike 17:38, 29 Dec 2004 (UTC)
Tau symbol in the table[edit]
I find the usage of the unit tau a bit strange. While I like the "Tau = 2*Pi" idea, I think a turn sounds like "1 turn", and so should have the value "1", not "Tau". So imho, the table here as well as in the degrees, radians and grad articles should be:
Units  Values  

Turns  0  1/12  1/10  1/8  1/6  1/5  1/4  1/2  3/4  1 
Degrees  0°  30°  36°  45°  60°  72°  90°  180°  270°  360° 
Radians  0  
Grads  0^{g}  33⅓^{g}  40^{g}  50^{g}  66⅔^{g}  80^{g}  100^{g}  200^{g}  300^{g}  400^{g} 
I mean, if you add the tau = 2pi symbol behind each value in the Turns row, the values are identical to those in the Radians row. And a wheel doing half a turn, is really doing 0.5 turns, not 0.5*2*pi = 3.1415 turns (which would be more than 3 full rotations!). One turn really is a single turn, it's just really incorrect to put a mathematical constant having the value of 2pi there.
Turns is the same as rotations, right? As in "rotations per minute"?
Is this stuff actually officially described anywhere?
Anyone agree?
92.107.35.50 (talk) 13:24, 15 May 2011 (UTC)
 Agreed. If we want a row for τ/12, τ/10, etc, it should be labelled "Radians" (or perhaps "radians in terms of τ"). Or maybe there should be a Radians row which has entries like "π/6 = τ/12" "τ/12 (π/6)" or some such. In the tau manifesto, Hartl uses "πradians" for "radians in terms of π", which I'm not sure I like (it certainly isn't selfexplanatory). As for where this is described, I suspect we cite most of the sources (either directly or by following whatever Hartl cites). Shouldn't be especially complicated: you can use τ whereever you might use 2π. Kingdon (talk) 23:18, 15 May 2011 (UTC)
 I agree too. τ is simply the number of radians in one turn. "πradians" doesn't mean anything and π radians is 180° or half a turn or τ/2 radians. I think the confusion is that radians are dimensionless so strictly speaking (from a dimensional analysis standpoint) the symbol "radians" is equal to one, and simply included for clarity. —Ben FrantzDale (talk) 13:10, 16 May 2011 (UTC)
 Disagree. In the row with degrees we have 180° with a number and a unit. So in the row labelled turns we should have 1/2 turns and not 1/2. If we use τ as a symbol for turn we get 1/2 τ. Since τ may also just be interpreted as a number the turn row and the radian row become identical so the dichotomy between using turns and using radians disappear. Therefore I find the extra row redundant. Entropeter (talk) 18:56, 16 May 2011 (UTC)
 I kinda agree with you. I think to be most pedantic we would remove the degree symbol across that row and the ^{g} symbol across that row. Then the row labels would give the units for the row. I think that would be more confusing than what we have now. I wouldn't mind having "0 turns, 1/12 turn, ..., 1 turn" on the fist row, but to algebraically substitute "τ = 1 turn" in the "turns" row, writing "0 τ, ..., 1 τ", changes that row from units of turns (a dimensionless unit we want ranging from 0 to 1) to units of radians (a different dimensionless unit. I think it's dangerously easy to confuse people here. While τ "is" one turn, it is not algebraically true that τ = 1 turn; Rather, τ = 2&pi = 6.283... radians, and 6.283... radians is one turn. —Ben FrantzDale (talk) 18:59, 17 May 2011 (UTC)
 Whether or not dimensional analysis has units of angle, we need to specify the angular units here because otherwise people can't use the information, and that's true in general. For example Hertz are explicitly cycles per second, not radians per second.
 Pi is unitless, and Tau also (since it's simply 2 Pi). One turn is clearly Tau radians and not Tau degrees for example. Rememberway (talk) 00:36, 30 June 2011 (UTC)
 I agree. How about something like this though:
Units Values Turns 0 1/12 1/10 1/8 1/6 1/5 1/4 1/2 3/4 1 Radians 0 0 Degrees 0° 30° 36° 45° 60° 72° 90° 180° 270° 360° Grads 0^{g} 33⅓^{g} 40^{g} 50^{g} 66⅔^{g} 80^{g} 100^{g} 200^{g} 300^{g} 400^{g}
 But maybe that's just redundant?  193.84.186.81 (talk) 11:26, 15 February 2012 (UTC)
 Yeah… that doesn't look good. I am in favor of tau but the point is that with tau you don't need to convert between it and turns. It is a turn. It looks weird to put in that half a turn is equal to . The only things I would want to change is the cell, and, in the future once the tau proposal has gotten more traction offsite, the "conversion to radians" text. And also, then, the radian column header can be renamed "radians expressed in terms of pi" or similar. Jikybebna (talk) 09:16, 28 December 2020 (UTC)
 The whole point of tau world is that radians is expressed in terms of turns. Thinking of tau as if it had the same limitations of pi misses out on that beauty. The basic idea is "what if we could use RPM, or turns, for radian angle math, instead of halfturns?" Jikybebna (talk) 09:22, 28 December 2020 (UTC)
The real question here is: "Can we find someone like Euler or or Newton using it in Tau in their work?" If so, the table should look more like:
Units  Values  

Turns  0  
Radians  0  
Degrees  0°  30°  36°  45°  60°  72°  90°  180°  270°  360° 
Grads  0^{g}  33⅓^{g}  40^{g}  50^{g}  66⅔^{g}  80^{g}  100^{g}  200^{g}  300^{g}  400^{g} 
Glas(talk)Nice User skin 03:48, 24 February 2013 (UTC)
 But is only 1 full turn in terms of radians. When you say turns you mean full revolutions, so 1 full turn is just that: 1. Radians divided by tau is equal to the number of full turns, maybe we should put them together? Hence:
Units  Values  

Turns or Radians divided by  0  1/12  1/10  1/8  1/6  1/5  1/4  1/2  3/4  1 
Degrees (°)  0  30  36  45  60  72  90  180  270  360 
Radians  0  
Grads (^{g})  0  33⅓  40  50  66⅔  80  100  200  300  400 
Finbob83 (talk) 14:02, 26 February 2013 (UTC)
 There's no need to add it. τ is just a numeric measure, like π, so most logically would go in the 'radians' row. Except π is used there already. Adding τ there is redundant. Added anywhere else it fits less well and is also redundant. It's not used in mainstream textbooks or teaching, so noone will be helped if it's added, it's just unnecessary clutter.JohnBlackburne^{words}_{deeds} 15:25, 26 February 2013 (UTC)
Not only does τ in 'radians' correspond to it's number of turns, but there is no other symbol for turns. It only feels natural to have one; τ=2π=360°=400^{g}. Do you write 100 or €100? I don't know about you, but I would prefer to receive €100 over c100. Turn is the only row in the table with no symbol, SOMETHING belongs there.Glas(talk)Nice User skin 02:12, 27 February 2013 (UTC)
 I agree with 92.107.35.50 that 1 tau = 1 turn but I think that tau should be expressed in decimal numbers not fractions. Decimals is how we use numbers in calculators etc. This way tau could be used with the SI units.
Joeschmoetwo (talk) 11:49, 28 November 2015 (UTC)
 tau is a numerical constant (like pi), whereas turn is a unit (like degree, radian or gon (aka gradian)).
 A quantity like, for example, an angle consists of two parts, its value and its unit. 1 turn, 360 degrees, 400 gons, or 2*pi radians all describe the same angle. For brevity, most units also have unit symbols, so you could also write 1 tr, 360°, 400^{g} or 2*pi rad.
 You could also express the value of the angle in tau, but it only makes sense for angles given in units of radians. For example, 2*pi radians are equal to 1*tau radians.
 While quantities are normally meaningless without specifying both, their value and their unit, angles happen to have a dimension of 1 in the SI system, therefore you can omit the unit (symbol) for as long as the context is known. That's why we're sometimes a bit lax in specifying the units of angles.
 Regarding "using angles with SI prefixes", you can do that already: 0.001 degree are 1 millidegree, 0.001 turn are 1 milliturn, etc.
 Regarding "decimals", the reason why we try to express certain angles as fractions/factors of pi (or tau) when working with radians is because in this context a full angle is defined as 2*pi (or 1*tau), which are irrational numbers, that is, they "never stop" and cannot be expressed exactly as a ratio of integers. If you want to type exact numerical values into your calculator, you are better off working in turns, degrees, or gons.
 The reason, why the "tau proposal" is discussed in this article, is not because "tau is a unit of angle" (it's not), but simply because a full angle is defined as 2*pi radians (or 1*tau radians) or 1 turn, so the value of an angle in turns nicely corresponds with the value of an angle in radians when expressed in terms of tau instead of pi. This makes working in turns and/or expressing radian angles in fractions of tau convenient.
 Matthiaspaul (talk) 15:36, 28 November 2015 (UTC)
"Euler's" identity[edit]
FWIW I didn't explain it very well, but the form of Euler's identity:
Is equivalent to saying that a single turn is an identity operation. Rememberway (talk) 16:42, 30 June 2011 (UTC)
 To me the big problem with the text removed here is that it just throws out a formula. It doesn't explain what complex exponentiation has to do with rotation (or identity), or even that rotation and full turns are involved here. (The +0 is particularly hard to fit into an encyclopedia—even the tau day manifesto calls it "somewhat tongueincheek"). Kingdon (talk) 01:37, 1 July 2011 (UTC)
 It's much more important that for any integer k. The "+0" is unnecessary just like how the "+1=0" part of the original identity was unnecessarily trying to fit more constants into the equation, and it made more sense as just . And anyone who understands Euler's identity should know that it is referring to rotation (as if wasn't explicit enough). The exponential of any integer multiple of the imaginary circle constant is the multiplicative identity in the complex plane. Or in other words, "a rotation by tau is one." Simple and elegant. Important and should definitely be mentioned on this page. — Preceding unsigned comment added by 70.113.56.202 (talk) 23:52, 26 May 2015 (UTC)
Tau proposal revert quote[edit]
"τ is the radian angle measure for one turn of a circle." (below figure 8 http://tauday.com/taumanifesto#figtau_angles )
 (unsigned) 20130914T02:31:44 Reddwarf2956
Circle radians tau.gif[edit]
This [1] animated gif has a critical error that undermines its entire point. I'll go into more detail on it's talk page[2].
Galhalee (talk) 17:09, 19 December 2015 (UTC)
Article disambiguation parenthesis[edit]
Since the turn is a unit of angle, and there seems to be a fair amount of consistency about disambiguating units as "[...] (unit)", I suggest that this article should be moved to "Turn (unit)". Comment? —Quondum 15:56, 14 July 2016 (UTC)
 I was considering this as well, but moved it from "Turn (geometry)" (as disambiguator from "Turn (rational trigonometry)") to "Turn (angle)" now to be consistent in naming with the "Degree (angle)" article (which has multiple meanings as a unit which need to be disambiguated, therefore alternatively renaming this one to "Degree (unit)" was not an option. However, we have a redirect under "Turn (unit)", so this variant is catched as well.
 Matthiaspaul (talk) 11:09, 7 August 2019 (UTC)
Why tau?[edit]
Why are the tau proposals discussed here at all? What evidence is there of WP:NOTABILITY? Why is this not WP:FRINGE?
Hartl's Tau Manifesto is selfpublished. In Google Scholar, it gets a total of 18 citations, almost all from unpublished papers. Macrakis (talk) 22:29, 26 November 2017 (UTC)
 It’s a compromise. There were some who wanted a whole article on the Tau promoted by Hartl etc., others who thought it did not belong and sought its deletion. In the end it was decided to not have a standlone article, but to merge it into other articles where it made sense. This Talk:Tau (proposed mathematical constant)/Archive 3 is I think the most recent discussion.JohnBlackburne^{words}_{deeds} 22:42, 26 November 2017 (UTC)
 Largely it's because the proposal is ultimately correct. I mean the table is stupid when written it Tau because 1/393 turn is tau/393. It simplifies things to the point that it makes the article seem utterly superfluous. While the proposal does need more notoriety it does have being correctness on it's side. Tat (talk) 13:23, 1 September 2018 (UTC)
 There is less here than meets the eye. There is an occasional paper that uses τ, but no mainstream (or reliable) paper discusses τ. — Arthur Rubin (talk) 09:46, 18 October 2018 (UTC)
 One might almost conjecture that τ is less under scrutiny than feet, Farenheit, pound, ... are. Sorry for sucking inappropriate fun from such discussions. Purgy (talk) 10:24, 18 October 2018 (UTC)
 No problem. If a unit or mathematical constant were commonly used, it might be difficult to find an article about it. I don't think this is the problem finding reliable sources about τ. — Arthur Rubin (talk) 15:39, 19 October 2018 (UTC)
 One might almost conjecture that τ is less under scrutiny than feet, Farenheit, pound, ... are. Sorry for sucking inappropriate fun from such discussions. Purgy (talk) 10:24, 18 October 2018 (UTC)
 "the proposal is ultimately correct. I mean the table is stupid when written [with] Tau" ← holy shit yes this, so much. Jikybebna (talk) 19:39, 16 December 2020 (UTC)
 There is less here than meets the eye. There is an occasional paper that uses τ, but no mainstream (or reliable) paper discusses τ. — Arthur Rubin (talk) 09:46, 18 October 2018 (UTC)
To me, τ is an unfortunate choice because it has historically been used (e.g. by H. S. M. Coxeter and, probably following him, Magnus Wenninger) for the golden ratio – nowadays more often φ, but sometimes I need φ for an angle (usually colatitude) and a symbol for the golden number. —Tamfang (talk) 18:18, 30 January 2023 (UTC)
 It is probably one of the reasons it hasn't become popular. As one of the main problems in mathematics is that there are not enough letters in the alphabets, I expect proponents of tau to use pi for something other than tau/2. —Kusma (talk) 19:27, 30 January 2023 (UTC)
 @Tamfang: is used for (e.g.) permutations and the prime counting function. According to your logic, is an unfortunate choice as well. :) A1E6 (talk) 16:54, 7 February 2023 (UTC)
Angles and area[edit]
A radius of √2 is needed for a circular sector area to equal the angle subtended. Circular angle is only one of three types of angles in the plane, and the other two are also associated with a radial length of √2: hyperbolic angle is defined with respect to hyperbola xy=1 for which the shortest radius is √2. Furthermore, in the dual number plane the angle concept is given by difference of slopes, and here the area of a triangle with one side on x=√2 corresponds to the angle. Thus the use of 2 in the expression of the length of the circumference of a circle is not accidental. The tau proposal obscures the connection with these other angles. − Rgdboer (talk) 19:13, 21 December 2020 (UTC)
 Whaddayamean obscures? Please clarify what would change. One radian is still gonna be the same. One full turn is approx 710/113 in both pi world and tau world. All we're saying is that it was frustrating, growing up, to learn trig to have to deal with halfturns all the time. A quarter of a circle is "half a halfturn" or radians, a sixth of a circle is "a third of a halfturn" radians and so on. So what we're asking is for math to be taught in terms of turns rather than halfturns. The halfturn constant doesn't have to go away for applications where it's better suited, i.e. where you're dealing with semicircles, halfwaves and so on.
 The textbook we used when we first learned about pi in middle school introduced ⌀ for the diameter which it (mistakenly) claimed was pronouced phi. And we were drilled that phi × pi = circumference. And then we never ever used the diameter again and only used r from then on. ⌀ was just a throwaway. It would've been awesome to instead use a full turn of radians as the constant. r × tau = circumference. And then when in a context where we do need to use the halfturn constant, they could say "By the way, tau/2 has a pretty famous history, does anyone know what it's called? That's right, pi!" and we could learn a little bit about Zu Chongzhi, Archimedes etc and how while Gregory and Euler first did use the full turn as their constant, Euler after a while switched to using the halfturn (from Mechanica on), like Barrow before him. How people like to memorize digits of pi and so on.
 It's just so nice to be able to deal with full turns, and multiples (including fractional multiples) of full turns when dealing with radians and angles. At the like middle school, high school level. In highlevel math none of this matters; people can define and use any constant. If a particular paper needs to deal with halfturns or thirdturns or quarterturns, it can, and it can define the appropriate constant. A fifth of a turn is pretty messy to define in terms of pi (2/5 pi) but easy in terms of tau. Jikybebna (talk) 09:06, 28 December 2020 (UTC)
Kinematics of turns[edit]
This section Turn_(angle)#Kinematics_of_turns should be removed from this article. Nothing in it has anything to do with the unit of angle that this article is about (nor is any of it actually about kinematics). The first and third paragraphs are about mathematical representations of rotations. The second is about the fact that a certain book uses the word "turn" to refer to points on the unit circle. (Incidentally, the only reference in the section is that book itself.) When I boldly removed the section, I was reverted by Rgdboer. Hence me soliciting opinions here. Danstronger (talk) 04:33, 9 October 2021 (UTC)
"Cyclus (geometry)" listed at Redirects for discussion[edit]
An editor has identified a potential problem with the redirect Cyclus (geometry) and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 March 6#Cyclus (geometry) until a consensus is reached, and readers of this page are welcome to contribute to the discussion. eviolite (talk) 04:23, 6 March 2022 (UTC)
Revolution[edit]
@Quondum: What is the difference between revolution and turn mentioned in your edit summary? I think the two are equivalent because "turns per minute" would mean the same thing as "revolutions per minute". ISO 800003 mentioned in the article describes N, rotation, as "the number (not necessarily an integer) of revolutions, for example, of a rotating body about a given axis" and that it is equal to rotational displacement, φ, divided by 2π, which is how you get the number of turns from an angle. Not sure what your note about frequency and angular frequency means because revolution isn't a measure of frequency. — Eru·tuon 07:17, 4 December 2022 (UTC)
 @Erutuon: From ISO800003:2019, rev = 1 and rad = 1. This article says turn = 2π rad. Hence, turn ≠ rev.
 I am also unhappy with the statement "A turn is also referred to as a cycle (abbreviated cyc. or cyl.), complete rotation (abbreviated rot.) or full circle." These are also almost certainly wrong, for the same reason. I'm not sure why I did not remove them too, and intend to soon. —Quondum 17:12, 4 December 2022 (UTC)
 I am old enough to clearly remember when records playing at 33⅓ and 45 revolutions per minute were very common, and even to just about remember when 78 revolutions per minute were common. Those were not radians. The word "revolution" has a very well established tradition of use to mean full turn (2π radians), and I have no memory whatever of ever having known it in use to mean a radian. JBW (talk) 21:32, 10 December 2022 (UTC)
 Not true. The unit revolution has the wellestablished tradition (and I suspect it was even standardized in ISO 800003:2006) of being a unit of rotation and not of angle. The relationship between these two quantities is akin to that between the radius and circumference of a circle: if you equate them, the world explodes. As defined in this article, the turn is a unit of angle (a.k.a. rotational displacement), not of rotation as the revolution is, and cannot be equated to the revolution any more than the radius and circumference describing the same circle can. —Quondum 22:42, 10 December 2022 (UTC)
 My comment was made in response to your "rev = 1 and rad = 1. This article says turn = 2π rad. Hence, turn ≠ rev". The intended meaning of that is far from clear, but it looks as though it is contrasting 1 radian with 2π radians, with 1 relating to "rev", and 2π relating to "turn", which I read as meaning that you think a turn is 2π radians and a revolution 1 radian. If that isn't what you meant then perhaps you can clarify your statement. However, since you have now also raised the rotation/angle distinction, if the word "turn" is not used to refer to rotation, but only to static angles, then I have managed to go through a good many decades totally misunderstanding a huge number of uses of the word. The word "turn" is far more commonly used to refer to a rotation than to an angle, whether you, or the authors of a standard, or anyone else, thinks it should be or not. I am also not at all convinced that your insistence on rigidly separating units of angle from units of rotation is useful. One can have an angle of 180°, and one can have a rotation of 180°; I see no compelling reason why one needs to insist on different units for angles and rotations just because angles and rotations are not the same thing, any more than one is forbidden from having a kilogram of lead and a kilogram of potatoes, just because lead and potatoes are not the same thing. If an object rotates from one orientation to another orientation at an angle of 90° to the original one, then it has rotated 90°; it would be grossly cumbersome to have to use a different unit for the one than for the other. If, then, one is to regard the primary meaning of "turn" as being a unit of angle, as you apparently prefer, then why on earth should one be forbidden to use that unit of angle to measure rotations? The measure of a rotation is the size of the angle through which the rotation occurs. A kilogram is a unit of mass, not a unit of potatoes, and identifying mass and potatoes is at leas as gross an error as identifying rotation with angle, but that is a far different thing from saying that using a kilogram as a unit to measure quantities of potatoes. Also, your comparison with the radius and circumference of a circle is way off the point, because the sizes of both of them are lengths, and both are measured in the same units. The fact that in a particular circle they are different lengths is irrelevant. JBW (talk) 18:51, 11 December 2022 (UTC)
 Not true. The unit revolution has the wellestablished tradition (and I suspect it was even standardized in ISO 800003:2006) of being a unit of rotation and not of angle. The relationship between these two quantities is akin to that between the radius and circumference of a circle: if you equate them, the world explodes. As defined in this article, the turn is a unit of angle (a.k.a. rotational displacement), not of rotation as the revolution is, and cannot be equated to the revolution any more than the radius and circumference describing the same circle can. —Quondum 22:42, 10 December 2022 (UTC)
 I am old enough to clearly remember when records playing at 33⅓ and 45 revolutions per minute were very common, and even to just about remember when 78 revolutions per minute were common. Those were not radians. The word "revolution" has a very well established tradition of use to mean full turn (2π radians), and I have no memory whatever of ever having known it in use to mean a radian. JBW (talk) 21:32, 10 December 2022 (UTC)
 The word "turn" does not seem to be defined as a unit of angle at all, except by a few nonauthoritative sources such as this article. This suggests that this article should be deleted.
 My response may have been unduly cryptic. ISO 800003:2019 defines "rotation" as a count in a way that is incompatible with it being an angle in the SI sense, and ISO 800003:2006 (recently Ήsuperseded, but the best we have) defined "name: one  symbol: 1  conversion factors and remarks: The special name revolution, symbol r, for this unit is widely used in specifications on rotating machines." This implies mathematically what I said above ("rev = 1 = rad"). A statement that is correct in terms of this standardsbased definition would be "An angle of π [radian] corresponds to a rotation of 0.5 [revolution]", where the bracketed unit is optional. Note that we do not say "is equal to". Your use of "revolution" is incompatible with this, though (if you ignore the superseded standard) it is reasonable to interpret a revolution as either a unit of count or of angle (but not both).
 We should take care to avoid providing definitions from our own experience, especially when this is contradicted by historical standards. This means that we must not claim that 360° = 1 revolution, only that they correspond. The situation with "cycle" is similar. It is rather a pity that we no longer have any standardsbased terms for "cycle" (in either sense of count or angle), "revolution" (ditto), and even "rotation" in the current ISO standard is a horrible name for the meaning that it is given. —Quondum 23:08, 11 December 2022 (UTC)
 Dear Quondum, not all unitless quantities are equivalent. For example, 1mg/kg is not equivalent to 1mm/km, even tough both simplify to 1ppm. Similarly, radians are not equivalent to turns, despite both also being unitless. The radian is specified in the SI as a derived unit in terms of the base unit of metre as a ratio of lengths, rad:=1m/m=1, which is remarked in ISO 800003 as follows: "The radian is the angle between two radii of a circle that cuts off on the circumference an arc equal in length to the radius of the circle". ISO 800003 defines "rotations" as a ratio of angles, N=ϕ/2π, where ϕ is plane angle and the radian unit is omitted in the denominator; it also has the following remark: "N is equal to the number (not necessarily integer) of turns [emphasis added], e.g. of a rotating body or in a coil." ISO 800003 also mentions "cycles" as part of the definition of (wave) "period" (inverse of ordinary frequency), where it says: "duration of one cycle [emphasis added]". Finally, ISO 800003 remarks with regard to unit name "one" (symbol 1): "The special name revolution, symbol r, for this unit is widely used in specifications on rotating machines." However, as stated in the beginning, not all unitless or dimensionless quantities are equivalent. So, an arbitrary number of rotations, turns, cycles, or revolutions can be related to the corresponding angle measure by means of a constant of proportionality or scaling factor, equal to the measure of a fullcircle angle (de:Vollwinkel); it is given by ISO 800003 in N=ϕ/2π. It's also restated elsewhere in ISO 800003, such as in regards to rotational frequency dN/dt=ω/2π, where ω is angular speed. If that's not enough, we have other authoritative sources stating that 1 rev corresponds to 2π rad, such as the IET and the European units of measurement directives. Admittedly, sometimes folks trip over notation and oversimplify the correspondence as an equation "1 rev = 2π rad", but that's just abuse of notation, acceptable for most physicists and for all engineers. There are also countless textbooks teaching one revolution equals 2 pi radians (or one revolution equals 360), where "equals" could have been stated more correctly as "corresponds to". Can we settle this issue once and for all, please? fgnievinski (talk) 07:58, 8 May 2023 (UTC)
 I do not think that "corresponds to" is more correct than "equals". If you do the dimensional analysis properly by giving angles a dimension then turns and revolutions both have a dimension of angle and a magnitude of 2 pi radians, hence are equal. That's the normal meaning of equality of dimensional quantities  it's not an abuse of notation at all. Also "equals" is the phrasing used in the sources. It's like you're pedantically insisting on saying "1+2 corresponds to 3", even though nobody else says that. Mathnerd314159 (talk) 23:57, 9 May 2023 (UTC)
 I suppose you all feel that mathematical consistency is a prerequisite in any system of units and its underlying system of quantities. Mathnerd314159, since you seem to think you understand this, (a) do you consider your statement to be consistent with the SI and ISQ? If you say 'no' (and I suspect not, since angle is dimensionless in the SI), then what system of units and quantities are you referring to? If 1 cycle/s = 1 Hz (I'm not sure you agree with this), do you regard this as a unit of frequency, and if so, what is the relationship between the angular velocity (or speed) ω and frequency f of a system? (Let's ignore for a moment that there is currently no generally accepted system of units or quantities that regards angle as dimensional, much as I wish there was, and argue the proposition on its merits for the moment). —Quondum 02:32, 10 May 2023 (UTC)
 @Mathnerd314159: Not sure if you've read the discussion above my comment. Essentially, Quondum was arguing 1rev=1rad, which could follow from a literal interpretation of the remark given in ISO 800003 with regard to unit name "one" (symbol 1). I'm trying to get this article and others to better reflect the consensus that 1rev~2πrad (or 1rev=2πrad, if you insist). There's almost an edit war going in the recent history of Turn (angle), Cycle per second, Revolutions per minute, Cycle (unit), Cycle, Spatial frequency, Rotational frequency, Angular velocity, Inverse second, etc. My comment about equality vs. correspondence may seem pure math pedantry but it was aimed at recognizing ISO 800003 defines rotations (or turns, cycles, revolutions) as a ratio of angles, thus a count not an angle; the formula N=ϕ/2π implies strictly 1rev=2π/2π and is consistent with angular frequency, 2π*dN/dt=dϕ/dt. If, instead, you insist on the seemingly trivial equation 1rev=2πrad, then it'll blow up in your face when you substitute it in 2π*dN/dt, resulting in 4π^2*dϕ/dt. Notice the SI Brochure (9th ed.) words the relationship carefully to avoid equality: "For periodic phenomena, the phase angle increases by [emphasis added] 2π rad in one period." fgnievinski (talk) 03:48, 11 May 2023 (UTC)
 @Quondum: The many revisions of ISO 800003 and the SI Brochure are a testament to the continuous evolution of the ISQ and the SI. Despite the common abuse of notation recognized above, I hope to convince you that 1rev=1rad cannot be reasonably attributed to any reliable source. If we don't find a common ground, we'll have to ask for a third opinion in WP Physics. fgnievinski (talk) 03:48, 11 May 2023 (UTC)
 @Quondum Well, I consider SI and ISQ to be flawed, insofar as that they define a lot of quantities to be 1 even though this isn't consistent with their usage (if they were actually 1, why use them at all?). I would say I'm referring to their customary "informal" usage, although it has been formalized in various places, e.g. the Frink units file has turn = revolution = 2 pi radian. That Frink document also has a long comment complaining about the SI definition of Hz vs. rad/s; I agree with it that 1 rad/s is not equal to 1 Hz. In a dimensional angle system, I would say they are different dimensions  Hz is just s^1, while rad/s has a radian unit as well. Similarly we can say 1 cycle/s = 2 pi rad/s but 1 cycle/s != 1 Hz because the dimensions don't match. This is discussed a bit in [3] (published: [4]). Per that the proper formula relating angular frequency ω and frequency f is ω = f*(2π rad).
 @Fgnievinski: You have to be careful about identifying what's an angle, when the equations haven't been formulated with a dimension of angle. Generally you have to go back and investigate the physical interpretation of the formula. In this case though, the formula N=ϕ/2π is not really physical at all. Although your version is N (dimensionless)=ϕ rad/(2π rad), it can be interpreted more usefully as a changeofunits formula N cycles=ϕ rad/(2π rad / cycle), using 1 cycle = 2 pi rad. Then the relationship (2π rad/cycle)*dN/dt=dϕ/dt falls out naturally as a similar changeofunits formula. The 4 pi^2 stuff only happens when you mix up radianspecific equations with the "complete" (unitindependent) equations. Regarding phase angle, I would say the SI's statement is worded carefully so as to avoid conflating periods with radians, similar to the distinction between Hz and rad/s. The formula is (phase angle)=(2 pi rad)/(1 period) * (# periods), and since there is the non1 constant (2 pi rad)/(1 period) this is not just a change of units. If we arbitrarily say 1 period takes 1 second then we see the constant is similar to an angular frequency. Mathnerd314159 (talk) 16:57, 11 May 2023 (UTC)
 I suppose you all feel that mathematical consistency is a prerequisite in any system of units and its underlying system of quantities. Mathnerd314159, since you seem to think you understand this, (a) do you consider your statement to be consistent with the SI and ISQ? If you say 'no' (and I suspect not, since angle is dimensionless in the SI), then what system of units and quantities are you referring to? If 1 cycle/s = 1 Hz (I'm not sure you agree with this), do you regard this as a unit of frequency, and if so, what is the relationship between the angular velocity (or speed) ω and frequency f of a system? (Let's ignore for a moment that there is currently no generally accepted system of units or quantities that regards angle as dimensional, much as I wish there was, and argue the proposition on its merits for the moment). —Quondum 02:32, 10 May 2023 (UTC)
 I do not think that "corresponds to" is more correct than "equals". If you do the dimensional analysis properly by giving angles a dimension then turns and revolutions both have a dimension of angle and a magnitude of 2 pi radians, hence are equal. That's the normal meaning of equality of dimensional quantities  it's not an abuse of notation at all. Also "equals" is the phrasing used in the sources. It's like you're pedantically insisting on saying "1+2 corresponds to 3", even though nobody else says that. Mathnerd314159 (talk) 23:57, 9 May 2023 (UTC)
 Dear Quondum, not all unitless quantities are equivalent. For example, 1mg/kg is not equivalent to 1mm/km, even tough both simplify to 1ppm. Similarly, radians are not equivalent to turns, despite both also being unitless. The radian is specified in the SI as a derived unit in terms of the base unit of metre as a ratio of lengths, rad:=1m/m=1, which is remarked in ISO 800003 as follows: "The radian is the angle between two radii of a circle that cuts off on the circumference an arc equal in length to the radius of the circle". ISO 800003 defines "rotations" as a ratio of angles, N=ϕ/2π, where ϕ is plane angle and the radian unit is omitted in the denominator; it also has the following remark: "N is equal to the number (not necessarily integer) of turns [emphasis added], e.g. of a rotating body or in a coil." ISO 800003 also mentions "cycles" as part of the definition of (wave) "period" (inverse of ordinary frequency), where it says: "duration of one cycle [emphasis added]". Finally, ISO 800003 remarks with regard to unit name "one" (symbol 1): "The special name revolution, symbol r, for this unit is widely used in specifications on rotating machines." However, as stated in the beginning, not all unitless or dimensionless quantities are equivalent. So, an arbitrary number of rotations, turns, cycles, or revolutions can be related to the corresponding angle measure by means of a constant of proportionality or scaling factor, equal to the measure of a fullcircle angle (de:Vollwinkel); it is given by ISO 800003 in N=ϕ/2π. It's also restated elsewhere in ISO 800003, such as in regards to rotational frequency dN/dt=ω/2π, where ω is angular speed. If that's not enough, we have other authoritative sources stating that 1 rev corresponds to 2π rad, such as the IET and the European units of measurement directives. Admittedly, sometimes folks trip over notation and oversimplify the correspondence as an equation "1 rev = 2π rad", but that's just abuse of notation, acceptable for most physicists and for all engineers. There are also countless textbooks teaching one revolution equals 2 pi radians (or one revolution equals 360), where "equals" could have been stated more correctly as "corresponds to". Can we settle this issue once and for all, please? fgnievinski (talk) 07:58, 8 May 2023 (UTC)
Dear Mathnerd314159, I do hope the SI and ISQ gets further improved and clarified in the future. But for the time being, we have to recognize their current versions as the prevailing consensus. Any purportedly better formulations, such as Frink's, should be described as fringe theories, especially if they're not published in the peerreviewed literature. More specifically, we should not dispute the formula N=ϕ/2π given explicitly in ISO 800003 as "unphysical", as this is the most authoritative international standard in effect.
That being said, it does seem to exist a significant dissidence which proposes that the quantity "angle" (or angle measure) can be expressed in units of cycles or revolutions, defined as 2π rad. However, this dissident view runs into trouble when they deal with the units of ordinary frequency: they argue it cannot be just reciprocal seconds or hertz  they claim frequency should be, instead, in cycles per second or revolutions per second. However, this difficulty, which Frink so eloquently abhors, is just a consequence of their chosen definition, however misguided. Worse still, sometimes folks like the the IET and the European units of measurement directives state naively 1rev=2πrad unsuspecting of the consequences.
More concretely, applying the ISO 800001 notation in which any quantity Q={Q}[Q] has magnitude {Q} and unit [Q] to Frink's model (for the lack of a better name) based on 1rev=2πrad:
 angle: ϕ={ϕ}[ϕ]={ϕ}rad={ϕ}'rev, where {ϕ}'={ϕ}/2π
 ordinary frequency: f={f}[f]={f}rev/s
 angular frequency: ω={ω}[ω]={ω}rad/s
 angle rate: dϕ/dt={dϕ/dt}[dϕ/dt]={dϕ/dt}rad/s={ω}rad/s={dϕ/dt}'rev/s={f}rev/s, where {dϕ/dt}'={f}={dϕ/dt}/2π={ω}/2π
The above is internally consistent, it just doesn't admit hertz as the unit of frequency, which is a big drawback.
Now, let's follow the international standard view and introduce a new quantity called "rotation":
 angle: ϕ={ϕ}[ϕ]={ϕ}rad={ϕ}1
 rotation: N=ϕ/2π={N}[N]={N}rad/rad={N}1
 (combined): ϕ=2πN={2πN}[2πN]={2πN}rad*1={2πN}1*1={2πN}
 ordinary frequency: f={f}[f]={f}1/s={f}Hz
 angular frequency: ω={ω}[ω]={ω}rad/s
 angle rate: dϕ/dt={dϕ/dt}[dϕ/dt]={dϕ/dt}rad/s={ω}rad/s={dϕ/dt}'1/s={f}Hz, where {dϕ/dt}'={f}={dϕ/dt}/2π={ω}/2π
Again, the above is internally consistent, and it still admits hertz as the unit of frequency, which is a big advantage.
So my proposal is to describe the two views above, which I believe can be reasonably well sourced, without infringing on WP:OR or WP:ORIGINALSYN. (This is not unlike what happens in the article about Accuracy, in which the ISO definition differs from the more common definition of the term.) We just have to be careful with terms "cycle" and "revolution", which under Frink's model are units of measurement for the quantity of kind angle, and might be taken as synonyms to the quantity of kind rotation in ISO's view. The term "turn" may also arguably fit in either view, although I'd like to think it fits better with ISO's view.
Going back to the original issue in the present talk page section: the interpretation "rev = 1 = rad" doesn't seem compatible with either of the two formalisms above. The edit war mentioned above needs a ceasefire. fgnievinski (talk) 02:16, 12 May 2023 (UTC)
 The system of quantities would probably need further formalization, without which intuition leads one astray (the world's leading metrologists have difficulty for the same reason). I think that the position may be summarized as follows: SI and ISO treat the hertz as being a count rate (which is consistent with arbitrary periodic phenomena, to which no angle can reasonably be attributed in general), and has informally (i.e. in its nonnormative discussion) related (equated?) the cycle per second to the hertz. ISO further treats the revolution in the same way. Many people contrarily equate the cycle to the angle 360°, which is fine as a definition (it just happens to use the same name for an essentially different concept); because no standards body seems to use "cycle" or "revolution" with this meaning, we could call this use "informal". For me, the rub is this: few people properly comprehend this, and there is massive confusion as a result. I'd suggest that if we do include this informal use, we should clearly indicate that it is "nonstandard" or "informal" use being given, and that it is distinct from the SI/ISO use.
 If you really want to get into discussions like those you have been having above, I suggest you start by properly building the underlying system of quantities, complete with equations and definitions, to keep the discussion coherent. Comparing the system of quantities underlying the Gaussian unit system with that underlying SI is handy for training intuition for this. In the end, I don't expect that you will reach a conclusion that differs significantly from my summary here. (As an aside, the turn has no proper sourcing either, but does not have the rather severe problem of confusion arising from indiscriminate dual meanings.) —Quondum 03:00, 12 May 2023 (UTC)
 While the bulk of Fgnievinski's discussion makes sense, the statement the interpretation "rev = 1 = rad" doesn't seem compatible with either of the two formalisms above is problematic: it draws a conclusion that is based on intuition, not logic. SI is internally consistent, and it inherently makes this equivalence. If one accepts the meaning of 'cyc' to be consistent with Hz = cyc/s, SI is based on a system of quantities in which ω = 2πf for a rotational system. If f = 1 Hz, we have that ω = 2πf = 2π × 1 Hz = 2π cyc/s. But ω = 2π rad/s, which unavoidably leaves us with cyc = rad. You cannot argue that there is a "hidden" conversion factor between cycles and radians other than 1 in the SI. SI is consistent, but not intuitive in relation to phase and angle. But hey, even the 8th SI Brochure makes this type of mistake in Table 9 when it says "1 G = 10^{−4} T"; the correct symbol is '≘', for correspondence rather than equality. —Quondum 15:42, 12 May 2023 (UTC)
 "SI is internally consistent, and it inherently makes this equivalence". No. This is *your* intuition. And it is wrong. Actually SI has several units with the same "inherent" dimensions but completely different meanings. For example the becquerel is also s^1 just like Hz. But SI specifically says it is not interchangeable. Similarly torque (N m/s^2) and energy (J) are dimensionally equivalent, but SI considers them separate units. More generally, dimensional analysis is only compatible with SI in one direction  you can determine the dimensions of the quantities of an equation from the units, but the dimensions are not sufficient to determine the correct SI units. The transitive property does not hold  one cannot conclude Bq = Hz from Bq = s^1 = Hz. Similarly, when one has cyc = 1 = rad, one cannot conclude cyc = rad. They have the same (trivial) dimensions, but they are not the same unit. Mathnerd314159 (talk) 16:39, 12 May 2023 (UTC)
 Mathnerd314159, effectively asserting that the many experts who devote their careers to the topic (and define the SI) are wrong is not going to encourage anybody to consider your views. My understanding is not even relevant to my statement here. —Quondum 21:32, 12 May 2023 (UTC)
 Well, I'm sure that many experts have spent countless hours debating these topics. But if you read the CCU report it is clear that writing a standard is a political process, not driven purely by technical merit. And the quality of the outcome of such a process is variable. In this case I think it is clear that SI has compromised between a purely dimensional units system and common usage. Now as far as ISO 80000, ISO has a problem of charging high prices for lowquality standards. In this case the standard is free and seems to have been developed by a reasonably large committee, but I would still say that Wikipedia should not blindly adopt the nomenclature of standards just because they are published by a standards body. The relevant policy is WP:COMMONNAME which dictates that whatever usage is most common is the correct terminology for Wikipedia; whether it is the officially standardized usage is not relevant. Mathnerd314159 (talk) 16:30, 13 May 2023 (UTC)
 Mathnerd314159, effectively asserting that the many experts who devote their careers to the topic (and define the SI) are wrong is not going to encourage anybody to consider your views. My understanding is not even relevant to my statement here. —Quondum 21:32, 12 May 2023 (UTC)
 The statement the meaning of 'cyc' to be consistent with Hz = cyc/s doesn't need to be accepted. It seems possible to use the angular unit 1cyc=1rev=2πrad in the international standard view (ISQ/SI) if one is careful with the interpretation of results:
 N=ϕ/2πrad
 ϕ={ϕ}rad={ϕ}'rev
 N={ϕ}rad/2πrad={ϕ}/2π
 N={ϕ}'rev/2πrad={ϕ}'2πrad/2πrad={ϕ}'*1
 > So, dimensionless quantity "rotation" (or number of rotations) can be taken as the magnitude of quantity "angle" expressed in units of cycles or revolutions.
 Now for rates, it seems possible to use cyc/s=rev/s as a unit of angular frequency only:
 angular frequency: ω={ω}[ω]={ω}rad/s={ω}'rev/s
 Ordinary frequency, defined as f=1/T based on the period T, would remain in units of s^1=Hz:
 ordinary frequency: f={f}[f]={f}s^1={f}Hz
 ISO 800003 takes extra care to distinguish quantity "rotational frequency" (or rotations per unit time), defined as n=dN/dt, with units of s^1.
 rotational frequency: n={n}[n]={n}s^1
 > So, angular frequency ω in units of revolutions per second [ω]'=rev/s has magnitude {ω}'=ω/[ω]' equal to rotational frequency n in units of inverse second [n]=s^1.
 The challenge is how to explain this all across multiple articles in Wikipedia.
 My interpretation of the international standards is that they're internally consistent, just more difficult to explain. The more common views such as "Hz≟cyc/s" may seem simpler but I suspect they are inconsistent; plus, notice 1Hz=s^−1 is embedded in the definition of the second. There are areas for improvement in the ISQ/SI, sure, such as units for countable quantities [5], or the nonequivalence of dimensionless units (mg/kg<>mm/km or 1m/m<>1s/s). But ISQ/SI does represent the prevailing consensus.
 PS: the latest ISO 800003:2019 is available online: [6].
 fgnievinski (talk) 16:45, 12 May 2023 (UTC)
 Fgnievinski, we have a problem that the term "cycle" is used to refer to distinct concepts. I could, for example, swap the names "second" and "metre" and remain consistent, but it would be confusing. I suggest staying away (for this discussion) from the idea that "cycle" must mean some specific thing, and distinguish by, for example, referring to "cycle_{SI}" and "cycle_{informal}", or whatever you choose to use for the purpose.
 On It seems possible to use the angular unit 1cyc=1rev=2πrad in the international standard view (ISQ/SI) if one is careful with the interpretation of results, I disagree. This is incompatible with ISO 800003:2019 on simple mathematical grounds.
 Notice that a name can be applied to anything that is not inconsistently defined. For example, I can define "cycle_{SI}" as the real number 1, and this would be consistent with the SI, so I don't know what you mean by ... "Hz≟cyc/s" may seem simpler but I suspect they are inconsistent. You can consistently define "cycle" either way, just not both ways simultaneously.
 Note that the SI (8th Brochure) says "The SI unit of frequency is given as the hertz, implying the unit cycles per second". The 9th SI Brochure says "The SI unit of frequency is hertz, the SI unit of angular velocity and angular frequency is radian per second, and the SI unit of activity is becquerel, implying counts per second. Although it is formally correct to write all three of these units as the reciprocal second, the use of the different names emphasizes the different nature of the quantities concerned. It is especially important to carefully distinguish frequencies from angular frequencies, because by definition their numerical values differ by a factor 1 of 2π. Ignoring this fact may cause an error of 2π. Note that in some countries, frequency values are conventionally expressed using “cycle/s” or “cps” instead of the SI unit Hz, although “cycle” and “cps” are not units in the SI. Note also that it is common, although not recommended, to use the term frequency for quantities expressed in rad/s." [emphasis added] This will not be new to you, but it does seem to suggest that the SI perspective remains that the nonSI unit cycle/s is to be interpreted as a unit of frequency, and not of angular frequency. —Quondum 21:32, 12 May 2023 (UTC)
 Thanks for the quotation, it does shed some light on the intentions of the standardizing committee. Let's check the consequences of defining 1cyc=1 instead of 1cyc=2πrad. The tentative definition is ambiguous, as the unitless unit could imply either a unit of angle, 1cyc'=1rad=1, or a unit of rotation (in ISO 800003 parlance), 1cyc"=2πrad/2πrad=1. The derived unit of temporal rate, cycle per second (cps), could correspond, respectively, to a unit of angular frequency, 1cyc'/s=1rad/s, or a unit of rotational frequency, 1cyc"/s=(2πrad/2πrad)/s=s^1. Only the latter would seem compatible with the following particular statement in the SI brochure (8th or 9th ed.): "Note that in some countries, frequency values are conventionally expressed using “cycle/s” or “cps” instead of the SI unit Hz." I also noticed the SI brochure doesn't mention rotational frequency, refering to ISO 800003 for details; so, when the SI says just frequency, I take it as either ordinary frequency or rotational frequency, as both have dimension of s^1, in contrast to angular frequency, in rad/s. ISO 800003:2019 also describes quantity rotation as "number of revolutions", further suggesting 1cyc=1rev=2πrad/2πrad. Therefore, I reiterate the recommendation to dismis the possibility of defining 1cyc=1rad, as only the definition 1cyc=2πrad/2πrad=1 seems credible. And the third option, 1cyc=2πrad, however more popular, seems incompatible with SI/ISQ, as it doesn't admit hertz as the unit of frequency (demonstrated previously). I end restating my proposal for Wikipedia: to describe the two views above, SI/ISQ (as decyphered above) and the popular alternative 1cyc=1rev=2πrad (with its bad implications with regard to hertz). fgnievinski (talk) 02:02, 13 May 2023 (UTC)
 You are getting much of the reasoning right, but you still seem to be inferring the quantity from the unit, which is not valid. This is why I keep mentioning systems of quantities, and you do not seem to be incorporating that into your thinking yet. To use an example: Consider a cylindrical shaft that is rotating about its axis (in Newtonian, as opposed to relativistic formalism). The shaft has a radius and a circumference. We can define the tangential speed of the shaft as the speed of any point on the cylindrical surface: the derivative with respect to time of its distance travelled along the curve of motion (which will naturally be circular). We can now define two quantities: a, the ratio of the tangential speed to the circumference, and b, the ratio of the tangential speed to the radius. Both of these quantities are welldefined, without any reference to units; we can describe these quantities using whatever suitable unit we chose, from any of many systems of units. These are clearly different quantities (they differ by a factor 2π), and the unit (e.g. m/s) gives no hint of which of the quantities is being quantified. We can even equate a_{1} = b_{2} of two shafts (shafts 1 and 2) of suitably chosen (but obviously different) radii that drive each other through friction. Notice that I have made no reference to angle, cycle, frequency, angular frequency, hertz, radian, second, metre, etc., until this sentence, but a and b clearly refer to the same quantities as the SI names "frequency" and "angular frequency", as these would be associated with a rotating shaft. We tend to throw in the "marker" units rad and cyc to help us remember what quantity we are referring to, but this has no logical significance. From SI: "The symbols rad and sr are written explicitly where appropriate, in order to emphasize that, for radians or steradians, the quantity being considered is, or involves the plane angle or solid angle respectively." Note that these units are stated as being for emphasis, not for correctness.
 With reference to your proposal for WP: including the "popular" definition of the cycle as a unit of angle is problematic. Are you prepared to assert that the popular use of the cycle to denote an angle of 360° means that either (a) the cycle per second is then a unit angular frequency, and not of frequency, or (b) if the cycle per second is still to be said to be a unit of frequency, "frequency" refers the quantity that we now call angular frequency, and not what the SI calls frequency? If you see what I'm saying, the "popular" version cannot be made consistent without redefining almost the entire system. Keeping these two incompatible systems separate in shared articles is just a nightmare. And it is hardly justifible: I have not seen a single coherent treatment of such a topic, notwithstanding that we know that many people think in terms of such a system (usually without understanding that they are completely departing from standard definitions and often with inconsistency); while we can find many uses, we cannot find reliable discussions of this use. —Quondum 15:38, 13 May 2023 (UTC)
 I think you offer a false dichotomy. The "cycle" in 1 cycle = 2 pi radians does not need to be the same unit as the cycle in cycles per second. In fact at fgnievinski's version of Cycle (unit) they were clearly listed as two different usages. 1 cycle is a unit of angle, while cycles per second is a unit of frequency, and only the attempt to assign coherent dimensions is incorrect. It is like English; there are many grammarians who have notions of "incorrect" grammar, but nobody cares about these theories and they just use whatever grammar is natural, without regards for consistency. I guess it is a "nightmare" for encyclopedia writers, but what was wrong with a disambiguation page? I really don't understand your logic that whatever is colloquially popular must be wrong. Mathnerd314159 (talk) 16:53, 13 May 2023 (UTC)
 Wikipedia does not document what is colloquially popular. Do you understand the criteria for inclusion in WP? —Quondum 20:00, 13 May 2023 (UTC)
 We aren't discussing inclusion. We're discussing content. And there the guiding policy is WP:WEIGHT: "represent all significant viewpoints that have been published by reliable sources, in proportion to the prominence of each viewpoint in the published, reliable sources.". WP:SOURCE specifically calls out "universitylevel textbooks" as reliable sources. So the viewpoint that 1 turn = 1 cycle = 1 revolution = 2π rad, being widely printed in textbooks, must be represented, and must have a prominent position. It seems barely anyone has read the ISO standard besides the committee; looking at the Google Scholar / Google Search results for "iso 80000 revolution" there doesn't seem to be any significant adoption of that standard's definition besides a mention by NIST. So it's not prominent and should be in a footnote or a section, rather than the lead. Mathnerd314159 (talk) 05:18, 14 May 2023 (UTC)
 Dimensional analysis with dimensionless quantities is admitedly difficult. You bring a good example: tangential speed s=C/T, in m/s, the circumference (C=R2πrad, in meters) divided by period (T, in seconds). If one asks, "what is the meaning of dividing s by a given length, L", the answer depends for which geometric curve or line that length refers to. Normalizing s by the perimeter length or circumference yields rotational frequency, s/C=n=dN/dt (in ISO 800003 notation and nomenclature). Normalizing s by the assumed circle radius yields angular frequency, s/R=2πn=ω. Normalizing s by the assumed circle diameter yields half the angular frequency, s/2R=πn=ω/2. In general, if the trajectory is not circular or the length is unrelated to the circle, the normalized speed is not any tipe of frequency.
 Now returning to the "popular" definitions, I agree they are not internally consistent, for the reasons you and I have explored. But its popularity is exactly what makes it notable enough to be covered in Wikipedia  at least within exisiting articles. Our job, then, is to describe those definitions with due weight and balance  including their flaws and all. There are many broadconcept articles covering such "hairy" concepts. ISQ/SI are notable by the very dictionary definition of the term "international standard", however intricate and misunderstood they may be. I remain convinced ISQ/SI would serve well as a framework for explaining the informal/popular definitions and their eventual inconsistencies. ISO 800003 is widely cited in Google Books [7] and in Google Scholar [8].fgnievinski (talk) 05:33, 15 May 2023 (UTC)
 Well, the overall standard is cited by a few people (I get nonsense results after 4 pages in Books and 8 pages in Scholar), but the specific definitions at issue here (turn, revolution, cycle, etc.) are not. So the question is why writers do not bother with writing out "this is the definition of revolution according to ISO 800003". My guess is that it is simply that most writers consider the definitions of these terms "obvious" and do not bother to consult the standard in the first place. The results you give for the ISO standard are referring to it for terms that have a tricky definition, like the second where nobody will remember the magic constant of 9192631770 Hz. Mathnerd314159 (talk) 21:30, 16 May 2023 (UTC)
 I'll prepare a draft of what I have in mind so that we can have something more concrete to discuss. fgnievinski (talk) 02:19, 18 May 2023 (UTC)
 Well, the overall standard is cited by a few people (I get nonsense results after 4 pages in Books and 8 pages in Scholar), but the specific definitions at issue here (turn, revolution, cycle, etc.) are not. So the question is why writers do not bother with writing out "this is the definition of revolution according to ISO 800003". My guess is that it is simply that most writers consider the definitions of these terms "obvious" and do not bother to consult the standard in the first place. The results you give for the ISO standard are referring to it for terms that have a tricky definition, like the second where nobody will remember the magic constant of 9192631770 Hz. Mathnerd314159 (talk) 21:30, 16 May 2023 (UTC)
 We aren't discussing inclusion. We're discussing content. And there the guiding policy is WP:WEIGHT: "represent all significant viewpoints that have been published by reliable sources, in proportion to the prominence of each viewpoint in the published, reliable sources.". WP:SOURCE specifically calls out "universitylevel textbooks" as reliable sources. So the viewpoint that 1 turn = 1 cycle = 1 revolution = 2π rad, being widely printed in textbooks, must be represented, and must have a prominent position. It seems barely anyone has read the ISO standard besides the committee; looking at the Google Scholar / Google Search results for "iso 80000 revolution" there doesn't seem to be any significant adoption of that standard's definition besides a mention by NIST. So it's not prominent and should be in a footnote or a section, rather than the lead. Mathnerd314159 (talk) 05:18, 14 May 2023 (UTC)
 Wikipedia does not document what is colloquially popular. Do you understand the criteria for inclusion in WP? —Quondum 20:00, 13 May 2023 (UTC)
 I think you offer a false dichotomy. The "cycle" in 1 cycle = 2 pi radians does not need to be the same unit as the cycle in cycles per second. In fact at fgnievinski's version of Cycle (unit) they were clearly listed as two different usages. 1 cycle is a unit of angle, while cycles per second is a unit of frequency, and only the attempt to assign coherent dimensions is incorrect. It is like English; there are many grammarians who have notions of "incorrect" grammar, but nobody cares about these theories and they just use whatever grammar is natural, without regards for consistency. I guess it is a "nightmare" for encyclopedia writers, but what was wrong with a disambiguation page? I really don't understand your logic that whatever is colloquially popular must be wrong. Mathnerd314159 (talk) 16:53, 13 May 2023 (UTC)
 Thanks for the quotation, it does shed some light on the intentions of the standardizing committee. Let's check the consequences of defining 1cyc=1 instead of 1cyc=2πrad. The tentative definition is ambiguous, as the unitless unit could imply either a unit of angle, 1cyc'=1rad=1, or a unit of rotation (in ISO 800003 parlance), 1cyc"=2πrad/2πrad=1. The derived unit of temporal rate, cycle per second (cps), could correspond, respectively, to a unit of angular frequency, 1cyc'/s=1rad/s, or a unit of rotational frequency, 1cyc"/s=(2πrad/2πrad)/s=s^1. Only the latter would seem compatible with the following particular statement in the SI brochure (8th or 9th ed.): "Note that in some countries, frequency values are conventionally expressed using “cycle/s” or “cps” instead of the SI unit Hz." I also noticed the SI brochure doesn't mention rotational frequency, refering to ISO 800003 for details; so, when the SI says just frequency, I take it as either ordinary frequency or rotational frequency, as both have dimension of s^1, in contrast to angular frequency, in rad/s. ISO 800003:2019 also describes quantity rotation as "number of revolutions", further suggesting 1cyc=1rev=2πrad/2πrad. Therefore, I reiterate the recommendation to dismis the possibility of defining 1cyc=1rad, as only the definition 1cyc=2πrad/2πrad=1 seems credible. And the third option, 1cyc=2πrad, however more popular, seems incompatible with SI/ISQ, as it doesn't admit hertz as the unit of frequency (demonstrated previously). I end restating my proposal for Wikipedia: to describe the two views above, SI/ISQ (as decyphered above) and the popular alternative 1cyc=1rev=2πrad (with its bad implications with regard to hertz). fgnievinski (talk) 02:02, 13 May 2023 (UTC)
 "SI is internally consistent, and it inherently makes this equivalence". No. This is *your* intuition. And it is wrong. Actually SI has several units with the same "inherent" dimensions but completely different meanings. For example the becquerel is also s^1 just like Hz. But SI specifically says it is not interchangeable. Similarly torque (N m/s^2) and energy (J) are dimensionally equivalent, but SI considers them separate units. More generally, dimensional analysis is only compatible with SI in one direction  you can determine the dimensions of the quantities of an equation from the units, but the dimensions are not sufficient to determine the correct SI units. The transitive property does not hold  one cannot conclude Bq = Hz from Bq = s^1 = Hz. Similarly, when one has cyc = 1 = rad, one cannot conclude cyc = rad. They have the same (trivial) dimensions, but they are not the same unit. Mathnerd314159 (talk) 16:39, 12 May 2023 (UTC)
Way forward[edit]
@Quondum and Mathnerd314159: I've started Draft:Angles and related quantities in the International System of Quantities. The intention is to make sure we have the same understanding about the ISQ definitions; it carefully distinguishes between angle and rotation (and angular frequency and rotational frequency). Later it's my intention to try to summarize the more common simplistic or naive alternative. But at the end it'll be hard to split the centralized discussion over many separate articles. fgnievinski (talk) 09:00, 14 July 2023 (UTC)
 I have reservations, which will probably largely exclude me: (a) this sounds as though its intention is to be a tutorial or thesis, rather than a specific concept, making an article problematic, and to split it, as you say, will be tough, and (b) I anticipate that I would find it to be a frustrating exercise, since I have repeatedly found myself at odds with both of you on this general topic. I have found that staying away from topics where there are recurrent differences of opinion works best for me, but still find myself being triggered at times.
 A comment on the phrasing used in the draft: "... in radians"; I suggest reading §7.11 In fact, this form is still ambiguous because no clear distinction is made between a quantity and its numerical value. —Quondum 12:25, 14 July 2023 (UTC)
 As a draft, this suffers from what I was pointing out before, namely that there are no secondary sources discussing the ISO 800003 definitions of angle. I've looked and there aren't such sources. If it was in mainspace your draft would be immediate deleted as nonnotable, 0 independent/secondary sources. I don't see much of a purpose for it in draft space either, the standard is freely accessible anyway.
 If you want to mention the ISQ definition in this article (Turn (angle)), by all means go ahead  it does appear to be missing. Just don't attempt to present it as "the" definition or indeed as a "mainstream" definition. Mathnerd314159 (talk) 17:17, 14 July 2023 (UTC)
 The draft was only meant to settle any potential disputes about the interpretation of ISO 800003. For example, I'm still curious how could one derive 1rev=1rad or what flaws exactly have been identified in the SI/ISQ. I've offered it as a courtesy before editing the respective main articles to include parts of the draft.
 Regarding the phrasing "... in radians" and the claim it doesn't satisfy section 7.11 (Quantity equations and numericalvalue equations) of the NIST Guide to SI. I'm afraid that's not applicable, because angle is defined in SI and ISQ (ISO 800003) as a quantity of dimension one, whose special name (radian) "can be used or not depending on the circumstances", as per section 7.10 (Values of quantities expressed simply as numbers: the unit one, symbol 1). In fact, a very similar wording is found in the SI Brochure (9th ed.), section 5.4.8 (Plane angles, solid angles and phase angles): "The plane angle, expressed in radian [emphasis added] ... is the length of circular arc s, swept out between the lines by a radius vector of length r from the common point divided by the length of the radius vector, θ=s/r rad." Unfortunately, arbitrary angular units are not fully supported by the SI or ISO yet; they only allow degrees for historical reasons, as an afterthought. Hopefuly in a future revision they'll rephrase the definition as ""The plane angle's numerical value, expressed in radians, equals ... {θ}_{rad}= θ/rad=s/r." But then they'll also have to introduce a new physical dimension of plane angles (symbol A?) and angles will no longer have dimension one. Perhaps physicists will be happy although mathematicians might be sad. (See also: Radian#Dimensional_analysis.)
 About the secondary sources, ISO 800003 is cited in both the SI Brochure, section 2.3.4 (Derived units), and the NIST Guide to the SI, section 8.1 (Time and rotational frequency). It's also discussed at length in metrology journals [9]. I do agree the ISQ view is not widespread, so it should be covered with due weight in Wikipedia. However, there are articles missing mathematical formulations, as in rotational displacement; or indirect formulations, as in rotational frequency; or messy formulations, as in angular frequency. For those, I propose to follow the international standard (unless someone brings better sources):
 Here are the proposed changes:
 in Turn (angle):
 (...) Alternatively, the International System of Quantities (international standard ISO 800003) introduces the quantity rotation (symbol N) defined as "number of revolutions":
The abovedefined rotation or number of revolutions is a quantity of dimension one, resulting from a ratio of angles.N is the number (not necessarily an integer) of revolutions, for example, of a rotating body about a given axis. Its value is given by: N=φ/2π, where φ denotes the measure of rotational displacement [also called angular displacement].
 Angular displacement (symbol θ), also known as rotational displacement, is defined in the International System of Quantities (international standard ISO 800003) as the ratio between a circular arc length (signed), s, and the radius, r (or distance to the axis of rotation): θ=s/r. In the SI, it has units of radians (rad); it may be negative and greater than 2π in modulus. It is also known as angle of rotation or signed angle.[if agreed about synonyms, then retarget the last two redirects, currently pointing to angle.]
 Rotational frequency (symbol n), also known as rotational speed or rate of rotation, is defined in the International System of Quantities (international standard ISO 800003) as the rate of change of rotation (the number of revolutions, N) with respect to time, t: n=dN/dt. It has dimension of reciprocal time and units of reciprocal seconds s^{1} in the SI; the equivalent unit of hertz (Hz) is recommended to avoid confusion with angular frequency, in rad/s (also equivalent to s^{1}).[cite SI Brochure] Related customary units include cycles per second (cps) and revolutions per minute (rpm).
 Angular frequency (symbol ω), also known as angular speed or angular rate, is defined in the International System of Quantities (international standard ISO 800003) as the rate of change of phase angle, ϕ, with respect to time, t: ω=dϕ/dt. In the SI, it has units of radians per second (rad/s). It is related to (ordinary) frequency, f, by a proportionality factor (2π rad): ω=2πf. It is equivalent to the temporal rate of change of angular displacement: ω=dθ/dt.^{[citation needed]}
 I've amended the draft with two pertinent quotes about units. fgnievinski (talk) 05:10, 16 July 2023 (UTC)
 Yeah, the proposal for turns seems reasonable. I think I would just dump it in a section "ISO 800003" and drop the "alternatively", and mention the synonyms cycles, turns, etc. ISO 800003 defines so that people aren't confused as to why it's in an article on turns. It also looks like the definition of angular displacement you give for ISO 800003 assumes units of radians, so I would say that should be mentioned, either in the text "where φ denotes the measure of rotational displacement in radians", or by changing the formula to N=φ/(2π rad). Personally I prefer changing the formula, because a dimensionallycomplete equation is just more useful all around, but maybe you want to adhere to the standard more closely.
 The other stuff looks suspicious, because its phrasing puts forward ISO as a definitive source rather than as one of many opinions. But I think it's offtopic here and I would say to repropose those changes on their respective talk pages. Mathnerd314159 (talk) 16:01, 16 July 2023 (UTC)
 OK, I've implemented the proposed changes in Turn (angle). I've also updated the lead to summarize the new section.
 Notice the two concepts, angular unit and unitless angular ratio, can be related as in N=φ/tr, for arbitrary similar units of the angle and the full turn. They can also be related as N={φ}_{tr}: the number of turns equals the angle's numerical value when expressed in units of turns.
 I've included a call to template:infobox physical quantity (modified to mention synonyms). I intend to use the same template in the remaining articles: angular displacement, rotational frequency, angular frequency. fgnievinski (talk) 05:30, 17 July 2023 (UTC)
 BTW, wikidata:Q76435127 already fully described quantity rotation count in great detail; now it's interlinked here via redirect Rotation (quantity). fgnievinski (talk) 21:22, 22 July 2023 (UTC)
Related articles[edit]
@Quondum and Mathnerd314159: I've implemented the above proposal in angular displacement and in rotational frequency. Now, only angular frequency needs to be updated; its animation would need a correction: "Rotation (in radians)" should be "Angle (in radians)". I've also moved some long quotes around, including from revolutions per minute. Finally, I've made an addendum in Turn (unit) about the special names for the dimensionless unit (1rev=1 and 1=1rad) as given in old ISO 800003:2006. fgnievinski (talk) 05:22, 31 July 2023 (UTC)
 I've updated the GIF animation (File_talk:AngularFrequency.gif) and also the article on Angular frequency. fgnievinski (talk) 05:50, 16 August 2023 (UTC)