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The SI Brochure says that each angle is the ratio of two lengths - i.e. a number - and therefore a derived and dimensionless quantity. On the other hand, the papers [1] and [2] suggest that angles are inherently neither length ratios nor dimensionless, but rather have a dimension which is linearly independent from the base dimensions of the ISQ.

That being said, I believe that you may consider the convention from the SI Brochure less natural for a very simple reason: On the one hand, I think that most of you would intuitively distinguish between angles (e.g. one radiant or one degree) and numbers. On the other hand, I think that you can not consider angles dimensionless and at the same time distinguish between angles and numbers without being inconsistent:

Since each dimensionless quantity is a number, the convention that angles are dimensionless is equivalent to the convention that each angle is a number. Note that this claim is consistent with the fact that the SI Brochure literally says that 1 rad = 1 (see page 151 of the english edition).

Is there a mistake in my reasoning/am I missing something? My knowledge of dimensionless quantities is very restricted, so I would like to hear your opinions.

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    $\begingroup$ Conventions are not right or wrong (if there is no inconsistency). They may be useful or not. But in general, and for this specific case, I would consider it a matter of opinion. $\endgroup$ Commented Jul 30, 2022 at 21:07
  • $\begingroup$ @GiorgioP I understand, but I was not asking if the convention from the SI Brochure is preferable over the convention proposed by the papers. I thought that my question (the title of the question) is not just a matter of opinion. $\endgroup$
    – Filippo
    Commented Jul 30, 2022 at 21:16

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Angles are a peculiar case, and don't really fit the pattern of other physical quantities. But the SI brochure isn't giving a complete picture of what's really going on here. It's quite complicated.

Physical quantities are characterised by their symmetries. The physical quantities exist in their own space, and there are symmetry operations that map them on to one another while preserving, in some sense, the physics. We can build a coordinate system for these physical quantities by picking a 'standard' example, and then specifying any other instance of the physical quantity by listing a set of symmetry transformations needed to get to it from the 'standard' value. The symmetry transformations can be parameterised by various sorts of 'numbers' (that might or might not be the Reals).

This is called a 'homogeneous space' by the mathematicians. If there is a unique transformation to get to any other instance (so the coordinates are unambiguous and well-defined), then it is called a 'principal homogeneous space' or 'torsor'.

Many physical quantities have a scaling symmetry. So we can specify any other quantity by scaling the standard unit by a scale factor. This turns the symmetry group action on some abstract space of physical quantities into a vector space over the Reals, which is much easier to handle.

Angles don't have a scaling symmetry, they have a rotational symmetry, which is periodic. A rotation of $2\pi$ radians is the same as the identity, so there is a natural 'scale' already defined for them. The symmetry is already broken.

However, we can still find a mapping between the angles and the Reals, by wrapping the Real line around the unit circle infinitely often. And then we can apply scaling transformations to the Reals anyway, despite the fact that it breaks the symmetry. That's how we get 'degrees' and 'radians' as alternative angular 'units'.

This mapping uses the Lie algebra for the rotation group. It works a bit like a generalised logarithm. The rotation $\mathrm{exp}(i\theta)$ has logarithm $i\theta$ where we can treat $\theta$ as a Real number and $i$ acts like a 'unit' for rotation. In fact we can generalise this to more than two dimensions by setting $i$ to be a bivector, a quantity that specifies an oriented plane element with magnitude, and has the property that its square is -1. This is actually constructed as a ratio of two vectors pointing in different directions, and rather than just setting a size it also specifies the orientation of the plane in which the angles lie.

As a historical note, this was how Hamilton invented the Quaternions - as the ratio of a pair of vectors in different directions. He had hoped to be able to use them to create an algebra for 3D vectors in the same way Complex numbers worked for 2D vectors, but he was never able to get it to work properly because bivectors don't work the same way as vectors. (Complex numbers aren't vectors, either, for the same reason.) The two can be unified in Clifford algebras, but that's another story.

So, there is an analogy here between scaling transformations and rotations, in that one can be implemented by multiplying by $\mathrm{exp}(\lambda)$ and the other is implemented by multiplying by $\mathrm{exp}(i\theta)$. Multiplication is turned into addition, and a change of unit becomes an additive offset. The choice of unit in these coordinates is effectively a choice of origin.

Thus we can see in several ways that the algebra of linear units is not the same as the algebra of angles. We can't use the same "scale a standard unit" trick to represent them. However, this does not mean that angles are Real numbers, either. We can represent them as unit Complex numbers in 2D, or as unit Quaternions in 3D, but using the multiplication operation to combine them instead of addition. (Or we can take the logarithm to get an additive operation, at the cost of making the mapping periodic and thus ambiguous.) But whatever we do, we're not getting the same sort of object as the dimensionless ratio of two scalar lengths - a Real number.

This isn't quite the end of the story. If we consider an angular velocity, we find that the original $0=2\pi$ identification no longer applies. $2\pi$ radians per second is physically different from 0 radians per second. We have to work now in the Lie algebra, which can now be thought of as the tangent space to the rotation group at the identity. Angular velocity is thus properly a bivector, although in 3D we can use the dual to it and treat it as a vector.

Now angular velocity does have the sort of scaling symmetry that other physical quantities do, and thus the angular units in a compound unit like radians per second can more easily be considered to be a 'dimensionful' unit.

As a final twist on the entire topic, the translation, rotation, and dilation transformations can all be considered as special cases of the same group. A translation is 'a rotation about a point at infinity', pointing off perpendicular to the direction of movement. Consider the limit of rotations about a point as that point is moved further away and the angle shrunk proportionately. It is also 'a dilation centred on a point at infinity', this time the infinite point is off in the same direction as the translation. We have a single transformation group covering all cases, and so there is the potential to unify angles and lengths as essentially the same thing.

This is the point of view of the Klein geometries. Euclidean geometry is a special case of a Klein geometry, but its metric is degenerate, which is what stops us directly comparing lengths and angles. However, we can start to see how they fit together if we look at other Klein geometries like the Elliptic geometry. This is the geometry of a sphere with antipodal points identified, or equivalently, the geometry of lines through the origin. Now we discover that lengths on the surface of the sphere are in fact angles, measured from the centre. So treating this angle as a bivector as before, representing the plane of the angle, we find one component spanning the plane is embedded in the geometry, and the other is directed out of the geometry towards the origin. The angle is a ratio of the two lengths.

Now when we switch to the Kleinian picture for Euclidean geometry, we flatten the sphere out into a plane. This is the picture we normally use for projective geometry, where we represent points of the Euclidean plane as a homogeneous triple $(x,y,w)$ and normalise $w$ to get points of the plane $(x/w,y/w)$. This $w$ is measuring the distance from the plane to the origin, and gives the entire geometry a natural scale. And now we can see that what we consider to be 'lengths' are in fact angles measured from this projective origin, where these angles are dimensionless ratios of the length in the geometry and the 'distance' from plane to origin in the $w$ direction perpendicular to the geometry.

Thus, lengths are really just angles measured at right-angles to reality!

We can also interpret the other dimensionful quantities as behaving the way they do because they, too, are essentially lengths. Space and time are of course unified in special relativity. Mass is equivalent to energy, which in quantum mechanics is related to frequency in time. So mass is really just a reciprocal length. And electric charge can be interpreted geometrically as a sort of 'angular momentum' in an internal dimension of space. So fundamentally, all the physical quantities have a geometrical interpretation, and they are all measured by means of angles. The degenerate Euclidean metric causes most of these angles to behave like the Reals, the Lie algebra of the rotations, with a scaling symmetry. But the algebra of non-degenerate angles is quite different, and so the 'unit' construction cannot be applied in the same way.

The SI brochure is simplifying a very complicated issue. Dimensions and units are a lot more complicated than most people think!

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  • $\begingroup$ Thank you for your elaborate answer. You used the word "symmetry" 12 times, but unfortunately it is not clear to me what you mean by a symmetry in this context. Could you please elaborate? $\endgroup$
    – Filippo
    Commented Jul 31, 2022 at 21:40
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Is there a mistake in my reasoning/am I missing something? My knowledge of dimensionless quantities is very restricted

I don’t think the issue lies with your knowledge of dimensionless quantities. I think that the issue you are facing is that you appear to be confounding the concept of commensurable with the concept of dimensionally consistent. Two quantities are commensurable if they represent the same physical quantity. Merely having the same dimensionality is insufficient* for two quantities to be commensurable.

For background, a system of units is a set of standards against which physical quantities can be compared. The choice of standards is completely arbitrary, and different choices constitute different systems of units.

In a given system of units, the different standards are assigned a dimension. The choice of dimensionality is arbitrary, and depends on the system of units. Dimensionality is used when converting units within a system of units or between systems of units. Quantities with the same dimensions are called dimensionally equivalent.

As mentioned above, dimensional consistency is insufficient for two quantities to be commensurable. Consider torque and energy. These two quantities are dimensionally equivalent in SI units, but they are clearly incommensurable since torque is a vector and energy is a scalar. A scalar quantity cannot be the same physical quantity as a vector quantity, regardless of the units. Just as you can distinguish between torque and energy, despite their dimensional equivalence, similarly you can distinguish between angles and numbers, despite their dimensional equivalence. A right angle is not the same kind of physical quantity as a dozen.

*Having the same dimensionality may not even be necessary for two quantities to be commensurable. For instance, the SI unit of charge is the coulomb and the cgs unit of charge is the statcoulomb. Since both represent electrical charge I would say that they are commensurable quantities, but they have different dimensions. Similarly, speed has dimensions in SI units but is dimensionless in geometrized units, but I would consider those quantities to be commensurable.

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  • $\begingroup$ I'm pretty sure coulombs and statcoulombs do have the same dimensions, just like Newton and Dyne have the same dimensions. (just different units, as one is SI and other is CGS) $\endgroup$
    – AVS
    Commented Jul 31, 2022 at 6:34
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    $\begingroup$ @AVS I disagree, the statcoulomb has the dimension $\mathrm{L}^{3/2}\mathrm{M}^{1/2}\mathrm{T}^{-1}$. $\endgroup$
    – Filippo
    Commented Jul 31, 2022 at 7:10
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    $\begingroup$ My mistake. You are right. I overlooked the fact that the base units are different in SI and CGS. $\endgroup$
    – AVS
    Commented Jul 31, 2022 at 7:37
  • $\begingroup$ Thank you for your answer. AFAIU, your point is that different physical quantitites can have the same dimension. But would you really go as far as to say that a dimensionless quantity is not a number? $\endgroup$
    – Filippo
    Commented Jul 31, 2022 at 8:18
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    $\begingroup$ "I often use “Dale-SI” units wherein angles are dimensionful. They are helpful for doing dimensional analysis in my research" - I think this supports my current point of view that declaring angles dimensionless in the SI Brochure was a bad decision. Fortunately the SI Brochure allows us to write "rad", so it seems like we can use the SI without necessarely having to accept the convention that angles are dimensionless :) Thank you for the discussion (+1). $\endgroup$
    – Filippo
    Commented Jul 31, 2022 at 12:31
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The claim that angles are dimensionless (being the ratio of two lengths) is simply incorrect. You only have to look at Euclid's Elements to see this. Proposition 33 in Book VI states, among other things that, in a circle, any two central angles (say theta1 and theta2) lie in the same proportion as the two corresponding arc-lengths on the circumference (say s1 and s2, respectively). This means: theta1/theta2 = s1/s2. Let s1 be a general arc-length, s, with corresponding central angle, theta; and let s2 be the full circumference, c, for which the corresponding angle is one full revolution, rev. Then we have: s/c = theta/rev [arc-length relative to circumference = central angle relative to one revolution].

In a plane, the circumference is 2 pi times the radius: c = 2 pi r. Now we can write: s/r = theta/rad [arc-length relative to radius = central angle relative to one radian], where the radian is defined as rad = rev/(2 pi). This is a fundamental formula:

s/r = theta/rad

It is generally written as "s/r = theta, where theta must be in radians". But "in radians" here means that the "theta" in the formula (which is the ratio of two lengths) is the numerical value of the angle theta when theta is expressed in radians--i.e "theta" = theta/rad.

In the fundamental formula, s/r = theta/rad, any angle unit may be used for theta--conversion factors take care of themselves!

In summary, plane angle is a base quantity and has its own independent dimension: angle, symbol A. Solid angle is a derived quantity and has dimension A-squared. This fact is not currently recognised by the SI, but will have to be eventually. Ironically, solid angle is treated correctly, using the steradian, sr, which is a square radian:

solid angle = (spherical area)/(radius-squared) sr.

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  • $\begingroup$ You seem to be confusing dimension and unit. If angles had a dimension, then explain a Taylor expansion like $\sin(x)\simeq x$: if $x$ had a dimension, then so would $\sin(x)$. $\endgroup$
    – Miyase
    Commented Sep 18, 2023 at 4:53

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