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:I mean, there are a lot of bad/vague definitions of angle, but that doesn't mean WP has to be vague. And the way I see it your list of examples all points to the same definition. Considering general [[rigid transformation]]s, they can be decomposed into rotations, translations, and reflections. Considering only rotations, they form a [[Lie group]]. A compact Lie group can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones. A connected [[abelian Lie group]] is isomorphic to a product of copies of '''R''' (a vector) and the [[circle group]] ''S''<sup>1</sup> (an angle). The [[simple Lie group]]s are more complex and depend on the structure of the space. But focusing on the [[circle group]], which is simply the complex numbers with magnitude 1, there is a natural labelling of its elements with angle measures, given by the argument or phase, and a natural generalization to the real line to avoid wrapping at <math>2\pi</math>.
:As I see it, only a naive fool would forget the direction of a rotation! Counterclockwise and clockwise rotations are not the same at all. It is simply that angles have a "nice" structure where the inverse of a rotation has a negative angle. One can often be lazy and not worry about the sign, but this doesn't mean the concept of "unsigned" or "unoriented" angle has any real significance. Classical physics is [[time-reversible]] but this does not mean the past is the same as the future. [[User:Mathnerd314159|Mathnerd314159]] ([[User talk:Mathnerd314159|talk]]) 16:32, 5 May 2023 (UTC)
::People use the concept of unsigned and unoriented "angle" ''all over the place'' (with or without quantifying it), and there is nothing wrong with that. (As a simple example, consider the half-angle specifying the shape of a [[cone]] or the [[angular diameter]] of a ball as viewed from some particular point; these kinds of angles do not naturally come with a particular orientation, and giving them a sign is not really meaningful.) In a similar way mathematicians define whole disciplines in terms of a basic concept of unsigned and unoriented "distance" instead of only talking about oriented displacement vectors or the like. This doesn't make anyone defining such a concept a "lazy" "naïve fool", let's please stop with the exaggerated/insulting language, even when discussing people in the abstract. Wikipedia needs to describe the way these concepts are used in practice, not invent/canonize a single editor-preferred definition.
::{{tq|i=yes| list of examples all points to the same definition}} – this is at best overly reductive. For example, in spherical trigonometry it is conventional to use a concept of "angles" at the separate vertices of a spherical triangle each of which is only given a 1-dimensional orientation intrinsic to the surface, but the dihedral angles those come from in the ambient 3-space are all oriented in different rotation planes; and a concept of "side lengths", each of which is given as a 1-dimensional quantity, even though the central angles those come from in the ambient 3-space are also all oriented in different rotation planes. We can make a list of identities relating these quantities without explicit reference to the 3-space orientations, but in a different context it would make sense to consider them as 3-dimensional bivector-valued (or unit-quaternion-valued) quantities, which would lead to a different set of relationships. (If you prefer we might consider the intrinsic 1-dimensional angles to also be bivector- or complex-valued, with bivectors of unspecified orientation.)
::But perhaps more to the point, we need the definition here to be as legible as possible for a non-technical audience. Leading with anything about Lie groups is a non starter. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 17:39, 5 May 2023 (UTC)
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