# Talk:Turn (geometry)

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## Others than Hoyle

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

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   30° 36° 45° 60° 72° 90° 180° 270° 360°
Radians 0 ${\displaystyle {\tfrac {1}{6}}\pi }$ ${\displaystyle {\tfrac {1}{5}}\pi }$ ${\displaystyle {\tfrac {1}{4}}\pi }$ ${\displaystyle {\tfrac {1}{3}}\pi }$ ${\displaystyle {\tfrac {2}{5}}\pi }$ ${\displaystyle {\tfrac {1}{2}}\pi }$ ${\displaystyle \pi \,}$ ${\displaystyle {\tfrac {3}{2}}\pi }$ ${\displaystyle 2\pi \,}$
Grads 0g 33⅓g 40g 50g 66⅔g 80g 100g 200g 300g 400g

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 "π-radians" for "radians in terms of π", which I'm not sure I like (it certainly isn't self-explanatory). 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. "π-radians" 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 ${\displaystyle {\tfrac {\tau }{12}}}$ ${\displaystyle {\tfrac {\tau }{10}}}$ ${\displaystyle {\tfrac {\tau }{8}}}$ ${\displaystyle {\tfrac {\tau }{6}}}$ ${\displaystyle {\tfrac {\tau }{5}}}$ ${\displaystyle {\tfrac {\tau }{4}}}$ ${\displaystyle {\tfrac {\tau }{2}}}$ ${\displaystyle {\tfrac {3}{4}}\tau }$ ${\displaystyle \tau \,}$
0 ${\displaystyle {\tfrac {1}{6}}\pi }$ ${\displaystyle {\tfrac {1}{5}}\pi }$ ${\displaystyle {\tfrac {1}{4}}\pi }$ ${\displaystyle {\tfrac {1}{3}}\pi }$ ${\displaystyle {\tfrac {2}{5}}\pi }$ ${\displaystyle {\tfrac {1}{2}}\pi }$ ${\displaystyle \pi \,}$ ${\displaystyle {\tfrac {3}{2}}\pi }$ ${\displaystyle 2\pi \,}$
Degrees 30° 36° 45° 60° 72° 90° 180° 270° 360°
Grads 0g 33⅓g 40g 50g 66⅔g 80g 100g 200g 300g 400g
But maybe that's just redundant? - 193.84.186.81 (talk) 11:26, 15 February 2012 (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 ${\displaystyle {\tfrac {\tau }{12}}}$ ${\displaystyle {\tfrac {\tau }{10}}}$ ${\displaystyle {\tfrac {\tau }{8}}}$ ${\displaystyle {\tfrac {\tau }{6}}}$ ${\displaystyle {\tfrac {\tau }{5}}}$ ${\displaystyle {\tfrac {\tau }{4}}}$ ${\displaystyle {\tfrac {\tau }{2}}}$ ${\displaystyle {\tfrac {3}{4}}\tau }$ ${\displaystyle \tau \,}$
Radians 0 ${\displaystyle {\tfrac {1}{6}}\pi }$ ${\displaystyle {\tfrac {1}{5}}\pi }$ ${\displaystyle {\tfrac {1}{4}}\pi }$ ${\displaystyle {\tfrac {1}{3}}\pi }$ ${\displaystyle {\tfrac {2}{5}}\pi }$ ${\displaystyle {\tfrac {1}{2}}\pi }$ ${\displaystyle \pi \,}$ ${\displaystyle {\tfrac {3}{2}}\pi }$ ${\displaystyle 2\pi \,}$
Degrees 30° 36° 45° 60° 72° 90° 180° 270° 360°
Grads 0g 33⅓g 40g 50g 66⅔g 80g 100g 200g 300g 400g

Glas(talk)Nice User skin 03:48, 24 February 2013 (UTC)

But ${\displaystyle \tau \,}$ 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 ${\displaystyle \tau \,}$   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 ${\displaystyle {\tfrac {1}{6}}\pi }$ ${\displaystyle {\tfrac {1}{5}}\pi }$ ${\displaystyle {\tfrac {1}{4}}\pi }$ ${\displaystyle {\tfrac {1}{3}}\pi }$ ${\displaystyle {\tfrac {2}{5}}\pi }$ ${\displaystyle {\tfrac {1}{2}}\pi }$ ${\displaystyle \pi \,}$ ${\displaystyle {\tfrac {3}{2}}\pi }$ ${\displaystyle 2\pi \,}$
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 no-one will be helped if it's added, it's just unnecessary clutter.--JohnBlackburnewordsdeeds 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°=400g. 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°, 400g 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

FWIW I didn't explain it very well, but the form of Euler's identity:

${\displaystyle e^{i\tau }=1(+0)}$

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 tongue-in-cheek"). Kingdon (talk) 01:37, 1 July 2011 (UTC)
It's much more important that ${\displaystyle e^{ik\tau }=1}$ 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 ${\displaystyle e^{i\pi }=-1}$. And anyone who understands Euler's identity should know that it is referring to rotation (as if ${\displaystyle e^{ix}=cos(x)+isin(x)}$ 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

"τ is the radian angle measure for one turn of a circle." (below figure 8 http://tauday.com/tau-manifesto#fig-tau_angles )

-- (unsigned) 2013-09-14T02:31:44‎ Reddwarf2956