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Talk:Angle

Turn is period?
Reddwarf2956 has been making some improvements to this article (which I support), but we seem to disagree on the definition of a turn. I don't understand how it can be a period. Am I missing something?  D b f i r s   07:29, 15 September 2013 (UTC)
 * A circular movement is periodic, and the turn is its period. Nevertheless this concept seems too technical for this elementary article, and it is better to avoid to mention "period" here. However, I agree that this paragraph is not satisfactory, as giving a circular :-) definition. Therefore, I suggest "the turn is the angle for which the above circular arc is the complete circle". D.Lazard (talk) 09:53, 15 September 2013 (UTC)
 * I agree that there might be some loose association of a turn with a period in some usage, but I still don't see how a turn can be a unit of time. I agree with your suggested improvement.    D b f i r s   06:25, 18 September 2013 (UTC)

Transversal and polygons both have interior angles and exterior angles
Comments at the article edit: "I don't see what the association is for polygons. No mention at the linked article." See Angle. Let's add to the current statements and move them to in a different section of angle. But, we still need some references to both interior angles and exterior angles of Transversal_(geometry) with the Transversal section in angle and with polygons. Maybe combine the transversal and polygons sections together and define the terms?

John W. Nicholson (talk) 02:51, 23 September 2013 (UTC)

Small article merger
I am trying to combine as many of the small articles on different angles into angle (per talk), but I realize that some may not fit very well. Those like transversal, polygon, and both interior angles and exterior angles don't seem to fit currently, and wiki currently has separate articles. Others like Adjacent, Complementary, Supplementary, and Vertical angles seem to easily fit. Please take look at these and decide if any more parts should be moved to this article, angle, as to keep the information or start a AfD for each. Feel free to start a AfD if you feel that it's good. — Preceding unsigned comment added by Reddwarf2956 (talk • contribs) 21:25, 24 September 2013 (UTC)
 * RockMagnetist used a template at 05:06, 20 November 2013 to document nicely, within the initial section (consisting primarily of "header templates") on this talk page, their own merging into the accompanying article page of the 4 (small) angles articles that RD had alluded to as "seem[ing] to easily fit". (I would infer, from RD's (mistaken) ref to AfD that RD did 4 merges, so perhaps RM re-merged, believing there had earlier been a problem with each.)   Of course RD deserves at least our gratitude for the valuable work of calling attention to the opportunity, while (by the way) knowing, for use in this talk section, neither
 * the pipe-trick markup that lets an editor avoid, e.g., a link to the Dab'n of the term "transversal", by linking instead to the actual article "Transversal (geometry)", yet displaying "Transversal", and
 * what you might call "the suffix trick" -- eliminating the last blank in the markup for "Internal and external angle s" to get, voilà, "Internal and external angles", thus avoiding the misleading impression that "interior angles and exterior angles" invites: that WP regards external angles as "the mere bastard siblings" of internal angles.
 * --Jerzy•t 07:16, 21 July 2014 (UTC)
 * Note the date stamps: I did the merges a few months after the comments by Reddwarf2956. RockMagnetist (talk) 15:41, 21 July 2014 (UTC)
 * That made no sense to me on my first several readings, and i responded at User_talk:RockMagnetist, bcz i could not see why RM thot (either or both) that i would care when the merges were made, and/or that i didn't already know. RD's confusing attempts to use wiki-markup to talk about articles, creating the three irrelevant and misleading links "Adjacent", "Complementary", and "Supplementary", may have been a stumbling block for either RM or myself; in any case, i've been at this too long, and not fruitfully enuf, to continue without a generous break. --Jerzy•t 10:14, 25 February 2017 (UTC)

Definition?
The definition of angle on the main page seems rather vague. Perhaps a better definition would be: the fraction of the arc of a circle with a center at the origin of the angle.

That way degrees can be clearly defined as:

(s/c)×360

and radians as:

(s/c)×2π = s/r

where s = arc length, c = circumphrence, and r = radius. '' Evidently unsigned, but from about late 2001; see Jerzy's contrib below which begins "The initial contrib ..." . ''
 * I agree that the definition in the first paragraph of the main page is vague, and in fact inaccurate because an angle is not necessarily bounded and so cannot be defined as a figure; defining them in terms of measure as the second paragraph does is a better approach. However, I don't agree with this comment in defining an angle in terms of a circle because while angles exist in a context that is essential to the specification of their measure definition, angles are not necessarily part of a circle. Skylarkmichelle (talk) 19:47, 19 December 2013 (UTC)
 * "Angle" has two different meanings. It is first a geometrical figure (or shape). As such, the first paragraph is not vague (the figure is precisely defined). It is not only accurate, but also correct, which is important. Note that this definition dates to Euclides, more than 2000 ago.
 * "Angle" is also a measure. As a measure, there are many equivalent ways to define it, which involve either circles or trigonometry. The definition given here seems to be the simplest. It says implicitly that, to measure an angle (figure) in which there is no circle, one has to draw a circle centered at the vertex of the angle and to measure the ratio of the length of the arc delimited by the figure and the radius of the circle. D.Lazard (talk) 20:37, 19 December 2013 (UTC)
 * I want to endorse the thrust of DL's comment, while saying that IMO it's not precise enuf for the article -- a choice that i not only respect, but probably agree with. --Jerzy•t 16:30, 20 July 2014 (UTC)
 * The initial contrib to this section joined the page between 17:54, 30 August 2001 and 15:43, 25 February 2002, i.e. approximately late in the first year of WP; this section's heading, at least, was retrofitted later. (The "user" shown in the edit history, user:Conversion script, was a special piece of code that i've never bothered to acquire any substantive knowledge of, but clearly was used in the very early days -- on many, perhaps all, of  the pages that then existed -- probably in order to add or improve an important, but presumably relatively obscure, design feature in the early wiki engine. It may be that no record of even the IP address of the contributor was ever made. In short, and for certain, i cannot determine anything beyond its being contributed to WP between day 1 -- very early in 2001 -- and the 2002 date.) --Jerzy•t 16:30, 20 July 2014 (UTC)

talk page destruction!
Who eviscerated this talk page? Yesterday it had 36 sections and today it has 4?!! Skylarkmichelle (talk) 22:43, 20 December 2013 (UTC)
 * The discussions have been Archived but are still available for reading. At the header of this discussion page, click Show in the Archive bar. Then click on the numeral 1 that appears at the left. You can refer to issues that have been raised, but bring your discussion to this current Talk. It is common practice to Archive old discussions from lengthy talk pages.Rgdboer (talk) 03:14, 21 December 2013 (UTC)

Proposed merge
I had been contemplating merging the article Internal and external angle into the section Polygon related angles here since this section was better written and there did not appear to be much chance of expanding the current article. I would have done this myself, but a recent spate of activity suggests that I had better ask for some opinions on this merge. Bill Cherowitzo (talk) 18:43, 4 October 2014 (UTC)

 * Support, with sources added. Otherwise delete. - DVdm (talk) 19:21, 4 October 2014 (UTC)
 * weak Oppose - looking at this article leaves with a somewhat dissatisfied feeling as it mixes imho to many things which might be clearer to various readers if they are structured differently and distributed over several articles. That is in particular separate different levels of mathematical approaches from notational, engineering or unit related material. In addition the current mathematical part somewhat switches unstructured between rather abstract notions (without any motivation) and more elementary ones. We would need an angle treatment on highschool level (comprising angle between lines, 2d and 3d vectors, lines and planes, between planes), then the generalisation to curves and surves, more than 3 dimensions and non euclidean spaces. This could be handled in one large well structured article but imho for accessibilty/readabilty reasons 2-3 articles might be better. In any case those should be separated from the other (non-math) content in the current article. Since I favour the splitting and restructuring (as well as extending the splitted parts) of this article, I'm somewhat opposed to merging additional stuff into this article.--Kmhkmh (talk) 18:34, 7 April 2015 (UTC)
 * weak Oppose and conditional strong oppose: This topic is important in the history of mathematics. In addition, I can see this article being expanded in the future to describing how various cultures throughout history have measured angles (this is Ethnomathematics), which would help in countering Systemic bias. I think that articles on English Wikipedia that deal with ancient topics in mathematics are consistently written with a greater focus on mathematics in the western world than elsewhere so if this article to expanded to include more mention of how throughout history other cultures have measured angles then I would say strong oppose. Mgkrupa (talk) 16:23, 3 September 2020 (UTC)

Supplementary & Complementary
These terms are usually used to refer to pairs of angles rather than to sets of three or more. A South African editor believes that the terms are applicable to larger sets of angles (and I asked the same question here several years ago). The OED specifies "pair of angles", but is there a reference for more general usage? (If there is, then I apologise for my reversion of the recent edit.)   D b f i r s   07:05, 14 July 2016 (UTC)

It's all in the reflexes
What about reflex angles? — Preceding unsigned comment added by 75.139.254.117 (talk) 17:58, 24 November 2016 (UTC)
 * Reflex angles are defined in the article, and explained with a figure. The phrase "reflex angle" appears 6 times. What do you want more? D.Lazard (talk) 18:05, 24 November 2016 (UTC)

Merger of Angular unit into Angle
they overlap widely. fgnievinski (talk) 05:09, 7 May 2021 (UTC)
 * Oppose: they are in appropriate Summary style . Klbrain (talk) 20:00, 22 May 2021 (UTC)
 * Support: Angular unit has a lot of material about angles and their measurement that does not belong there, and little detail about units that is not already at Angle. It does not merit a separate page in its current form.  —Quondum 20:15, 22 May 2021 (UTC)
 * Support per nom. In reply to, there is not enough unique and encyclopedic material in Angular unit to merit invoking WP:SUMMARY. — Cheers, Steelpillow (Talk) 17:40, 9 July 2021 (UTC)
 * All fair points, which I'm happy to concede. No objections to a merge from me. Klbrain (talk) 22:57, 9 July 2021 (UTC)
 * ✅ Klbrain (talk) 15:40, 1 August 2021 (UTC)

"measure"?
The statement
 * Angle is also used to designate the measure of an angle or of a rotation.

is meaningless as it stands. A measure is only defined over a set, which is not given: an angle is simply defined here as "the figure formed by two rays [...] sharing a common endpoint". To use a concept from measure theory, even if it were properly done, is totally inappropriate for an article such as this, especially in the lead where every word should be easily understood (and not easily misinterpreted) by a high-schooler. The subject matter of this article is not too complicated to explain in everyday words. I suspect that the average reader will misinterpret the word measure to mean measurement due to the similarity of the words. Note also that angle is naturally interpretable as a signed quantity, especially in relation to a continuous rotation, which makes measure an especially poor fit. Nowhere in the article is a measure used to define the angle as a quantity.

I suggest that any reference to measure theory should be omitted. It is neither needed nor used in the article. —Quondum 13:03, 24 May 2021 (UTC)
 * It is only a wrong link, probably added by someone that searched in sections of Measure (disambiguation) without remarking that this is the primary meaning that applies here. I'll fix it. D.Lazard (talk) 17:07, 24 May 2021 (UTC)
 * That was a definite improvement. Wouldn't a more common and natural term be "size of an angle", as opposed to "measure of an angle"?  —Quondum 17:19, 24 May 2021 (UTC)
 * This is a question for a native English speaker. My opinion is that "size" is not convenient here. For example, an angle may have a negative measure, and a size is generally supposed to be positive. Also, "size" is generally qualitative, and "measure" is quantitative. For example, the area and the diameter are two different ways to measure the size of a plane figure. D.Lazard (talk) 17:55, 24 May 2021 (UTC)
 * Fair points. I can't think of a word that works better while addressing what you mention, and it works adequately as is, so I'll leave it be at this stage.  —Quondum 18:16, 24 May 2021 (UTC)

"Ángulo" listed at Redirects for discussion
An editor has identified a potential problem with the redirect Ángulo and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 February 14 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. User:1234qwer1234qwer4 (talk) 06:11, 14 February 2022 (UTC)

"Ângulo" listed at Redirects for discussion
An editor has identified a potential problem with the redirect Ângulo and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 February 14 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. User:1234qwer1234qwer4 (talk) 06:11, 14 February 2022 (UTC)

Proposed merge of Angle of rotation into Angle
Joyous! | Talk 22:55, 19 February 2023 (UTC)

"Angle of rotation" is already covered in Angle. fgnievinski (talk) 03:43, 29 March 2022 (UTC)
 * Support Angle of rotation is just an unsourced stub, there is no benefit to having it separate. --Mathnerd314159 (talk) 18:07, 20 May 2022 (UTC)
 * Support – though it might make sense to delete it and consider leaving a redirect to Rotation instead of to Angle, since an angle might or might not parameterize a rotation. —Quondum 13:10, 19 October 2022 (UTC)
 * Support the alternative target, Rotation. It would be well placed there, perhaps before Rotation which explicitly uses the angle of rotation without describing it in one dimension. Klbrain (talk) 10:08, 2 January 2023 (UTC)
 * This is a bad idea. If someone wiki-links to angle of rotation they are looking for a definition of an angle measure, not a definition of a rotation. –jacobolus (t) 19:45, 2 January 2023 (UTC)
 * Good point; I had overlooked that common use. So, switching to support to the original target, Angle. Klbrain (talk) 08:43, 7 January 2023 (UTC)

 * Support it's a bit silly for this to be a separate article. It's a simple topic, one of the most basic expressions of an angle.

Looks like everyone is in solid agreement. Courtesy pinging. I guess I could do the merge, but I guarantee you'd want someone else, just to avoid the furrowed brows as everyone tries to puzzle out "Now why did she put it there???" Joyous! | Talk 22:55, 19 February 2023 (UTC)

 * ✅ —Quondum 01:41, 23 February 2023 (UTC)
 * Thanks for the merger done sometime ago. I've recently sourced a definition for angular displacement from an authoritative source (ISO 80000-3) which states explicitly it may be a signed quantity. Therefore, I've retargeted the redirect Angle of rotation. fgnievinski (talk) 08:52, 22 July 2023 (UTC)

Math
Like what is explanation of angles 185.76.177.40 (talk) 08:41, 6 May 2022 (UTC)

Dimensional subtlety
In "the measure of the angle in radians", I interpret this to mean an angle, expressed in the unit radians. For comparison, if I ask for someone's height in metres, I expect to get an answer like "1.83 m". That is, I am not asking for their height divided by one metre, which would be "1.83", but that the unit in which the answer is to be expressed be the metre. I could say that "Their height (in feet) is 6 ft", but to say "Their height in feet is 6" sounds wrong. Thus, the "the measure of the angle in radians" is an angle quantity and includes the unit, and is not a bare number. If one assumes angle being dimensionless as defined in the SI, this might not be an issue, but then technically one could then just say "the measure of the angle" (without needing any qualification about units, since it is then (by definition) dimensionless, though one might want a footnote as many would find this confusing).

The SI is not the final word on angle being dimensionless: those who draft the SI are internally conflicted on the dimensionality of angle, some of whom have for many decades been proposing that angle be considered as being an independent dimension. This means that in an article such as this, we would do well to choose wording that is agnostic on the matter. Does anyone have a suggestion for such wording? 172.82.47.242 (talk) 22:30, 13 September 2022 (UTC)

 * Your quotation comes from section . Apparently, you forget that "angle" refers to both a geometrical figure and its measure. So, your comparison with heights is useless as a height is a measure and nothing else. Also your "interpretation" in the first sentence is wrong: an angle (figure) cannot be "expressed" with a number, but it can measured with a number.
 * I have edited the sentence containing your quotation into "For very small values (less than 5%), the slope of a line is approximately the measure in radians of its angle with the horizontal direction." I do not see any way to use your comments to improve this sentence. D.Lazard (talk) 09:04, 14 September 2022 (UTC)

 * The word "angle" originally means a geometrical figure made of two rays emanating from a common vertex, but is also often used to describe the relative orientation between two geometrical objects or the specification/quantification of a rotation or reflection; it often also gets used as shorthand for the "angle measure" (in radians, degrees, or the like). The measure of an angle in radians can be thought of as the natural logarithm of a rotation with the orientation stripped away; scaling between degrees and radians is like scaling between octaves and cents or between nepers and decibels. Even if you don't take the logarithm, rotations and relative orientations don’t really have a “unit” per se, but they do have a magnitude and orientation: they are best thought of as ratios of vectors (line-oriented magnitudes), bivectors (plane-oriented magnitudes), or the like. Anyhow, saying either "the measure of the angle in radians is π/2" or "the measure of the angle is π/2 radians" both seem clear enough.–jacobolus (t) 15:23, 14 September 2022 (UTC)
 * This last ("Saying either ...") tells be that my use of English in my first post is contrary to yours, and that "His height in metres is 1.83" makes sense to you, and not "His height in metres is 1.83 m". Fair enough – I'll not enter into a debate about the semantics of English phrasing, despite my reservations about choosing potentially ambiguous phrasing.
 * Indeed, bivectors provide an excellent way to encode rotation, but as with speed and velocity where one is referring respectively to a scalar and a vector quantity, and one should be clear which. In this article, it is clearly a scalar quantity that is meant, and there are useful nonsimple bivectors of rotation such as double rotations that do not translate to an angle per se, so I regard bivectors as clearly out of scope.  It seems to me that your use may seem so obvious to you that you cannot see how others might differently interpret the wording, and also that you have not been exposed to ongoing discussions around this in standardization bodies that could impact this article.  Anyhow, I'll leave this be.  172.82.47.242 (talk) 17:48, 14 September 2022 (UTC)
 * I think you are missing my point; as you suggest I was probably not clear enough in my description. Let me try again. You can represent a planar rotation or relative orientation between two coplanar vectors $$u$$ and $$v$$ of the same magnitude as the quotient $$R = v/u,$$ an object which multiplies by one vector to yield the other: $$Ru = (v/u)u = v.$$ This can be written as a "complex number" $$a + bI$$ where $$a^2 + b^2 = 1$$ and $$I$$ is a unit-magnitude bivector in the plane of the vectors. The logarithm of this rotation is the (bivector-valued) quantity $$\log R,$$ and if you strip the orientation out (picking arbitrarily between $$\pm I$$) you get an "angle measure" $$\theta = I^{-1}\log R$$ with units of radians. All of these quantities (except the vectors $$u$$ and $$v$$) are dimensionless. –jacobolus (t) 01:26, 15 September 2022 (UTC)
 * The prose sentence “His height in meters is 2.” is not idiomatic English, but if e.g. you asked “What is your height in meters?” then “2.” would be an entirely reasonable reply. Beyond that, the way the unit of “radians” is talked about/used doesn’t too closely match the usage of “meters” as a unit. –jacobolus (t) 01:35, 15 September 2022 (UTC)
 * I am familiar with geometric algebra and all the concepts that you refer to. Your argument that angle is dimensionless is no different from saying that the argument of the $sin$ function is dimensionless, and it is an angle, hence angles are to be considered dimensionless.  Which is a flawed argument, but this is not relevant here; my point is that it would be best to write this article in a way that is agnostic to this point since it is controversial even in the mainstream of metrology.
 * Any quantity can be considered to to have a dimensionless (numeric) part and a dimensional (unit) part, the latter itself being dimensionless for dimensionless quantities. Angles are no different from lengths or any other quantities in this respect.  We can consistently define angles to be dimensional, and to facilitate dimensional analysis, some people prefer to do so.  Mathematica does so, for example.  To be understandable to people working from either perspective, it would be good to use language that will be unambiguously understood about when we are referring to the quantity (including unit) and the dimensionless numeric part after removing (dividing by) the chosen unit.  I seem to have failed to communicate this above.  172.82.47.242 (talk) 13:45, 15 September 2022 (UTC)
 * These quantities are not dimensionless because they are the “numerical part” of a dimensional quantity. They are dimensionless because they are fundamentally ratios. The quantity “a right-angle rotation” is similar to the quantity “double” or the quantity “30%” in that respect. If you want you can put units on both sides of these ratios. Instead of “scale 1:100,000” on your map you can write “scale 1 cm:1 km” or whatever; in just the same way you can talk about a dimensionless angle measure multiplied by the radius (an arc length). You can similarly talk about the sine, secant, etc. of an angle as line segments or their lengths in whatever unit (as was done historically), or as an abstracted (dimensionless) ratio. –jacobolus (t) 15:18, 15 September 2022 (UTC)
 * Once you take the logarithm of a dimensionless quantity, you need to specify “units” again, in the form of the base of the logarithm. So for example you can say “10 doublings is approximately 3 orders of magnitude”. But these quantities are still “dimensionless” in the physical sense. You can apply 10 doublings to quantities measured in meters, seconds, bytes, or grams. –jacobolus (t) 15:23, 15 September 2022 (UTC)
 * Hoo-boy. We're clearly missing each other, and it is not because I am short on any conceptual or theoretical background here.  172.82.47.242 (talk) 15:32, 15 September 2022 (UTC)
 * If you read Molyneux's paper he presents a system where when you compute a logarithmic quantity it creates a logarithmic unit rather than dividing by a unit quantity. So for example one could take the (base 10) log of 10 and 100 meters, get 1 log-meter and 2 log-meters, and then subtract them to get 1 order of magnitude increase (i.e. 1 log-1). The log-meter is an "additive" unit and disappears on subtraction. So there are both dimensionless and dimension-ful logarithmic quantities.
 * Applying Molyneux's concept of logarithmic units to the definition of angle measure as the logarithm of a rotation is left as an exercise for the reader. Mathnerd314159 (talk) 18:28, 4 October 2022 (UTC)
 * Fair enough I guess (though I haven’t seen log-meters or the like before, and I’m not sure I buy that they have significant practical advantage). But an angle measure per se is still the logarithm of a quotient. If you wanted you could maybe take the logarithm of some position in a coordinate grid measured in meters, and end up with a (complex-valued) quantity measured in ln-meters. But the original units for the grid make no difference to the resulting imaginary component: the imaginary component of a particular point in the grid measured in ln-feet, ln-miles, and ln-meters is going to have the same numerical value. Taking the logarithm conveniently separates scale from rotation. –jacobolus (t) 18:51, 8 October 2022 (UTC)
 * FWIW, the log of a dimensional quantity is often used, wherever the decibel is used, with immense practical value. Treating it as "just a quotient" becomes somewhat strained in this context, since the "reference value" is arbitrary (just like a unit) and has indeed become auxiliary information that is attached to the unit.  —Quondum 12:36, 19 October 2022 (UTC)

clarifying basic definitions and scope
I think this article could use some significant rework to make definitions and scope more precise, and help especially non-technical readers. The word "angle" is used widely in many related but distinct senses, and it would be good to have a clear and well organized discussion of the relationships between these concepts. Angle can mean (at least):

 * a geometric figure formed by two line segments, rays, directed lines, circles, general curves, or the like (I would guess this is actually a significant minority use of all inbound wiki-links pointing at angle)
 * the relative orientation between two vectors, directed lines, or the like
 * at the most basic here "angle" means the (unsigned and unoriented) relation, a notion of the "distance" between an unordered pair of lines or orientations
 * it can mean the precisely specified rotation taking one object to another, oriented in a particular plane
 * it can sometimes allow for angles to be up to two full turns before returning to the identity state, as with unit quaternions; cf. plate trick
 * it can sometimes allow for a distinction between rotations which are any arbitrary number of full turns apart; this is especially relevant when an angle is used to measure e.g. the total rotation of a spinning physical object, the number of twists of a twist knot, or the like.
 * the angle of incidence between a line (or ray) and a surface; the dihedral angle between two planes or surfaces; in spherical geometry "lengths" are central angles and "angles" are dihedral angles.
 * a spatial coordinate in a circular coordinate system, representing a point on the circle; more generally one of the coordinates in a polar, spherical, etc. coordinate system, partially representing a point on a sphere, in space, or the like
 * the measure of an angle or other quantified representation of an angle such as a unit vector, unit-magnitude complex number, or slope.
 * the phase of a periodic function or the argument of a complex number, among similar concepts
 * an element ("number" or "point") in the periodic interval or circle group, abstracted from representing a specific relative orientation
 * the hyperbolic angle: analog of any of the above oriented in a pseudo-Euclidean plane (Minkowski or Lorentz plane); less commonly the "Galilean angle"(?) analog in a Galilean plane, perhaps usually just called slope. (In the hyperbolic plane, "angles" are (circular) dihedral angles while "lengths" are (hyperbolic) central angles.)
 * solid angle or other higher-dimensional analogs

These concepts are tricky and subtle, and I don't think we currently do an adequate job of organizing or representing them, especially for a non-technical or semi-technical audience. Does anyone know of any sources which explain these clearly? What do others think about the relative importance of these various concepts or ways of relating and describing them? Does anyone have any other recommended sources about the history/etc. of these concepts? –jacobolus (t) 05:26, 5 May 2023 (UTC)

 * 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 transformations, 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 S1 (an angle). The simple Lie groups 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 $$2\pi$$.
 * 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. 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.
 * 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. –jacobolus (t) 17:39, 5 May 2023 (UTC)
 * I think any basic math article should discuss the most general formulation(s) of that concept. For example Derivative has a section devoted to generalizations of derivative. In the case of derivative, there are multiple generalizations, but here there is only one obvious concept (element of the circle group) that makes sense, so it is pretty much canonical, hence can take a more prominent role in the article.
 * I'm not saying the concept of unsigned and unoriented "angle" doesn't have a place, but I looked and there aren't any obvious references that argue against the signed definition. So the relationship I infer from the sources is signed as canonical and unsigned as a simplification, rather than the unsigned being canonical. I guess we could call the small angles "reference angles" (0-90 degrees) or "proper angles" (0-180).
 * I'm not sure what your point is with spherical trigonometry, the angles are still rotations and the sets of rotations around a point forms a circle group. The fact that the rotations form SO(3) and can be labelled with dihedral angles or 3D bivectors is interesting but not really relevant to defining angles.
 * As far as writing for a non-technical audience, WP:TECHNICAL is of course the guide. But notably there is WP:OVERSIMPLIFY: "Encyclopedia articles should not 'tell lies to children'". So dumbing down the definition of angle is not the right approach. I would say the lead paragraph should give a good definition in terms of the circle group and define general concepts, and then a second paragraph can then go on to describe common examples. Lie groups are relevant to defining angles in higher dimensions, but the basic notion of angle is defined in a plane, so the Lie groups can go in a section. Mathnerd314159 (talk) 18:33, 5 May 2023 (UTC)
 * Element of the circle group is too reductive (as you say, we should not WP:OVERSIMPLIFY). First, circle group explicitly describes such elements as complex numbers (not just as elements of an abstract group defined structurally with multiple possible representations), when the most popular representation of angles is as angle measures in a periodic interval. Second, this only covers angles in a particular plane, without allowing the possibility for angles to include orientation. Third, this does not allow for contexts where angle measure per se does not make sense, for example in referring to an angle in an affine space, or an angle in a space where the coordinates are elements of a finite field. Fourth, this does not allow for angles that can spin around many times, as seen in e.g. angular velocity. While we're at it it would be good (not necessarily in the lead but somewhere in the article) to describe the difference between absolute orientations vs. relative "angles" (quotients of orientations or if you like rotors transforming one rotation into another), cf. Baez (2009) "Torsors Made Easy". –jacobolus (t) 18:43, 5 May 2023 (UTC)
 * Where I will agree with you though is that I think we should refocus the lead definition here away from talking about a reference to particular parts of a geometric figure (though this sense of "angle" should be described early in the article) and toward a general concept of relative orientation. –jacobolus (t) 18:59, 5 May 2023 (UTC)
 * Yeah, the lead is the place to start. Maybe something like this:
 * An angle is formed when an object is rotated around a point. In Euclidean geometry, a ray is rotated around its initial point, called the vertex of the angle, and the starting and ending rays are called the sides of the angle. The measure of an angle denotes its magnitude and is a real number with some angular unit such as degrees or radians. The measure is in general any real number, but it may be bounded to [0,180°) or [0,360°) depending on context. Angles are often drawn as circular arcs, and the measure of a counterclockwise angle in radians is the ratio of the length of the circular arc to its radius.
 * Mathnerd314159 (talk) 20:28, 5 May 2023 (UTC)
 * I would instead make the fundamental definition something like "An angle is the relationship between two orientations or directed lines." But maybe we can find some sources (historical surveys, mathematical dictionaries, ...) with a clear definition. –jacobolus (t) 22:04, 5 May 2023 (UTC)
 * Unless you specify it as a rotation, you don't get angles beyond 360 degrees. Hence "relationship" is just too vague. Mathnerd314159 (talk) 14:43, 7 May 2023 (UTC)

"Angles" listed at Redirects for discussion
The redirect [//en.wikipedia.org/w/index.php?title=Angles&redirect=no Angles] has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at  until a consensus is reached. —Lights and freedom (talk ~ contribs) 00:50, 23 May 2023 (UTC)

Euclidean Geometry only?
The lead sentence specifies euclidean geometry, but angles are also a thing in spherical geometry and hyperbolic geometry as well. Is there a common definition to all three that we can use in the lead sentence, and then briefly define the specifics for each geometry? It's a disservice to specify euclidean and then not discuss the other geometries as well, as doing so implies that only euclidean geometry has angles-- which is not true. Fieari (talk) 07:05, 23 May 2023 (UTC)
 * I went ahead and was WP:BOLD and did it. I welcome anyone to review my additions... I believe everything I said is accurate and lead-worthy, but I'd love a 2nd eye on it. Fieari (talk) 07:21, 25 May 2023 (UTC)
 * Your addition would make sense in a specific section named for example "In other geomeries". It is definitively too WP:TECHNICAL for the lead: most people interested in triangles do not know anything on non-Euclidean geometries. So, I'll revert your edit. D.Lazard (talk) 10:14, 25 May 2023 (UTC)

what about grad angle system? And forgot about circular angle?
I think we should add the following topics Yuthfghds (talk) 12:05, 4 June 2023 (UTC)

 * What do you mean by "circular angle"? Grads are discussed under . –jacobolus (t) 20:35, 4 June 2023 (UTC)
 * @Jacobolus Sorry my fault I didn't noticed the grad angle system.I am talking about degree measure in which 1°= 60' and 1'= 60''. Here we are only talking about linear angles not about circular degree measure.Thanks for pointing out my mistake, I am a new editor please avoid my faults.I still need feedback from your kind of editors.
 * Thank you
 * Yuthfghds Yuthfghds (talk) 05:49, 5 June 2023 (UTC)
 * I still do not understand what you are trying to say. What is the difference between "linear angles" vs. "circular degree measure"? This article does discuss degrees, minutes, and seconds. –jacobolus (t) 06:03, 5 June 2023 (UTC)
 * @Jacobolus
 * Linear Angles:
 * Linear angles are measured in degrees or radians and represent the amount of rotation or deviation between two lines or line segments.
 * They are used to measure angles in two-dimensional space, such as in geometry or trigonometry.
 * Linear angles have a fixed starting point (usually a reference line) and are measured in a straight line.
 * Linear angles range from 0° to 180° (or 0 to π radians) in a half-circle and extend up to 360° (or 2π radians) in a full circle.
 * Circular Angles:
 * Circular angles are measured in terms of arc length or fraction of a circle and are used to describe rotational positions or displacements.
 * They are commonly used in applications that involve circular or periodic motion, such as physics, engineering, or navigation.
 * Circular angles are based on the concept of a full circle, where one complete revolution is equal to 360 degrees or 2π radians.
 * Circular angles can exceed 360° or 2π radians, representing multiple revolutions or fractional parts of a revolution. Yuthfghds (talk) 06:17, 5 June 2023 (UTC)
 * "Linear angles" is not a term used in technical literature or curriculum that I have seen. Most of the results in a search for "linear angle" seem to be using it as a synonym for "straight angle" (i.e. a half turn, 180°).  Where did you get these names? Do you have a source? I would call this instead something like "angular distance" or "angular separation" (or just "angle"). I agree that this article should do a better job about talking about different concepts of angle, and I have started gathering sources for hopefully an eventual rewrite/rework. –jacobolus (t) 06:45, 5 June 2023 (UTC)
 * @Jacobolus Sir, here are some references for linear and circular angles:- help you.
 * Linear Angle:
 * Elementary Geometry for College Students by Daniel C. Alexander and Geralyn M. Koeberlein.
 * Chapter 1: Basic Geometry Concepts.
 * Chapter 2: Reasoning and Proof.
 * Geometry: A Comprehensive Course by Dan Pedoe.
 * Chapter 2: Angles.
 * Circular Angle:
 * Trigonometry by James Stewart, Lothar Redlin, and Saleem Watson.
 * Chapter 1: Trigonometric Functions.
 * Trigonometry by I.M. Gelfand and Mark Saul.
 * Chapter 1: Angles.
 * Elementary Trigonometry by H. S. Hall and S. R. Knight.
 * Chapter 1: Trigonometrical Ratios of Acute Angles.
 * sir if you still have some queries please question me.Thanks for feedback.
 * Yuthfghds Yuthfghds (talk) 07:25, 5 June 2023 (UTC)

Unless I am doing something wrong with my text searches at the internet archive, the term "linear angle" does not appear in either of the books you listed by Alexander or Pedoe. –jacobolus (t) 11:13, 5 June 2023 (UTC)

 * @Jacobolus Please recheck,if you still don't find it I can't do anything.Because I don't have those books that I mentioned you.Because I red those books few years ago.And I think linear angle is informal term mentioned by the author.And I argued here so it's my fault but still we can add these topics in this article.If I tell you the definition of linear angle and circular angle it would be like this:-
 * linear angle- Angle measured between two rays in degrees.
 * circular angle- Angle measured between two rays also in degrees but including minutes and hours. Technically angles that are placed inside a unit circle with 3 divisions namely degrees, minutes and hours.
 * I am glad that you understand my perspective and debate over this topic. Yuthfghds (talk) 13:03, 5 June 2023 (UTC)

 * The angle between two rays does not depend on whether or not a circle is drawn around their vertex! — When you say "including minutes and hours", are you referring to right ascension (part of the coordinates of a star in the sky) which is at least sometimes given in hours? — The subdivisions of a degree are called minutes and seconds. As far as I know, hours and degrees are never used together, partly because of the confusion between two kinds of "minutes". —Tamfang (talk) 21:01, 7 July 2023 (UTC)
 * @Tamfang You are absolutely right.These are just "informal" terms to distinguish between trigonometric angles and geometric angles. Yuthfghds (talk) 05:25, 9 July 2023 (UTC)
 * Why do they need distinguishing? Is there ever a circumstance in which mistaking a trigonometric angle for a geometric angle (or vice versa) or a linear angle for a circular angle (or vice versa) (are these the same distinction?) would lead to wrong results? —Tamfang (talk) 20:08, 5 August 2023 (UTC)

 * As I mentioned above linear angle is an informal term referred to angles that are measured in only degrees like in protector when we measure angle we say 12.5° or 45.2° just because we can't expand it but in circular angles we expand this system.linear angles are helpful in geometry but circular angles are helpful in trigo Yuthfghds (talk) 13:12, 5 June 2023 (UTC)

 * well the references that I mentioned you are generated by Chatgpt.But I red the book by padeo don't know linear angle is in that book or not but I red somewhere about linear angle in a random book but when Ai generated that book I thought it cover that topic sorry for the inconvenience that you suffered but still together we can make this article near excellence. Yuthfghds (talk) 14:14, 5 June 2023 (UTC)

 * Please don't ever ask ChatGPT for help writing Wikipedia, and be careful using it for any other purpose. It will confidently make up total nonsense. ChatGPT is a statistical language model, not a reference librarian. –jacobolus (t) 15:36, 5 June 2023 (UTC)

 * @Jacobolus You are absolutely right ,my bad. Yuthfghds (talk) 15:47, 5 June 2023 (UTC)

Negative angle definition
The following was removed: "Although the definition of the measurement of an angle does not support the concept of a negative angle," as unhelpful. Of course a definition can be contrived. Essentially one refers to area of a circular sector, using signed area according to the area being above or below the horizontal line of symmetry. This definition is for angles fixed in a standard position. Reference to area for definition is the historic basis of hyperbolic angle, so angle reference to area is one of the unifying theories in mathematics. Rgdboer (talk) 23:03, 9 November 2023 (UTC)

 * This article needs to be clarified and expanded to more clearly describe and discuss the several different concepts of "angle" which are in common use. This is something I plan to do eventually; I've been occasionally gathering sources at User:Jacobolus/Angle but am not yet ready to write a solid survey. –jacobolus (t) 00:45, 10 November 2023 (UTC)

Thank you for those sources on directed angle. The issue of angle definition came to a head in 1893 in Chicago when Felix Klein shot down Alexander Macfarlane’s paper on the topic which was presented to the mathematical congress held in connection with the Columbia Exposition. The paper was included in Papers in Space Analysis (1894) as the Proceedings of the congress only noted the title. The notion of an area-based definition referring to the sector of a circle was included in The Elements of Plane Trigonometry (1892) by R. Levitt & C. Davison (page 158). Robert Baldwin Hayward noted the complete analogy between the circular and hyperbolic cases when area of sectors are used to define the circular and hyperbolic angles. — Rgdboer (talk) 01:19, 12 November 2023 (UTC)

 * Of course, both circular and hyperbolic angle measure can be defined either via area or via arc length, with the latter using a Lorentzian (pseudo-Euclidean) concept of distance in the plane. –jacobolus (t) 02:25, 12 November 2023 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Angle


text:
Acre

The acre is a unit of land area used in the British imperial and the United States customary systems. It is traditionally defined as the area of one chain by one furlong (66 by 660 feet), which is exactly equal to 10 square chains, $1/640$ of a square mile, 4,840 square yards, or 43,560 square feet, and approximately 4,047 m2, or about 40% of a hectare. Based upon the international yard and pound agreement of 1959, an acre may be declared as exactly 4,046.8564224 square metres. The acre is sometimes abbreviated ac but is usually spelled out as the word "acre".

Traditionally, in the Middle Ages, an acre was conceived of as the area of land that could be ploughed by one man using a team of eight oxen in one day.

The acre is still a statutory measure in the United States. Both the international acre and the US survey acre are in use, but they differ by only four parts per million (see below). The most common use of the acre is to measure tracts of land.

The acre is used in many established and former Commonwealth of Nations countries by custom. In a few, it continues as a statute measure, although not since 2010 in the UK, and not for decades in Australia, New Zealand, and South Africa. In many places where it is not a statute measure, it is still lawful to "use for trade" if given as supplementary information and is not used for land registration.

Description
One acre equals $1/640$ (0.0015625) square mile, 4,840 square yards, 43,560 square feet, or about 4047 m2 (see below). While all modern variants of the acre contain 4,840 square yards, there are alternative definitions of a yard, so the exact size of an acre depends upon the particular yard on which it is based. Originally, an acre was understood as a strip of land sized at forty perches (660 ft, or 1 furlong) long and four perches (66 ft) wide; this may have also been understood as an approximation of the amount of land a yoke of oxen could plough in one day (a furlong being "a furrow long"). A square enclosing one acre is approximately 69.57 yards, or 208 feet 9 inches (208.71 ft), on a side. As a unit of measure, an acre has no prescribed shape; any area of 43,560 square feet is an acre.

US survey acres
In the international yard and pound agreement of 1959, the United States and five countries of the Commonwealth of Nations defined the international yard to be exactly 0.9144 metre. The US authorities decided that, while the refined definition would apply nationally in all other respects, the US survey foot (and thus the survey acre) would continue 'until such a time as it becomes desirable and expedient to readjust [it]'. By inference, an "international acre" may be calculated as exactly 4,046.856 422  4 square metres but it does not have a basis in any international agreement.

Both the international acre and the US survey acre contain $1/640$ of a square mile or 4,840 square yards, but alternative definitions of a yard are used (see survey foot and survey yard), so the exact size of an acre depends upon the yard upon which it is based. The US survey acre is about 4,046.872 square metres; its exact value ($1/undefined$ m2) is based on an inch defined by 1 metre = 39.37 inches exactly, as established by the Mendenhall Order of 1893. Surveyors in the United States use both international and survey feet, and consequently, both varieties of acre.

Since the difference between the US survey acre and international acre (0.016 square metres, 160 square centimetres or 24.8 square inches), is only about a quarter of the size of an A4 sheet or US letter, it is usually not important which one is being discussed. Areas are seldom measured with sufficient accuracy for the different definitions to be detectable.

In October 2019, the US National Geodetic Survey and the National Institute of Standards and Technology announced their joint intent to end the "temporary" continuance of the US survey foot, mile, and acre units (as permitted by their 1959 decision, above), with effect from the end of 2022.

Spanish acre
The Puerto Rican cuerda (1 cda) is sometimes called the "Spanish acre" in the continental United States.

Use
The acre is commonly used in many current and former Commonwealth countries by custom, and in a few it continues as a statute measure. These include Antigua and Barbuda, American Samoa, The Bahamas, Belize, the British Virgin Islands, Canada, the Cayman Islands, Dominica, the Falkland Islands, Grenada, Ghana, Guam, the Northern Mariana Islands, Jamaica, Montserrat, Samoa, Saint Lucia, St. Helena, St. Kitts and Nevis, St. Vincent and the Grenadines, Turks and Caicos, the United Kingdom, the United States and the US Virgin Islands.

Republic of Ireland
In the Republic of Ireland, the hectare is legally used under European units of measurement directives; however, the acre is still widely used, especially in agriculture. (This is the standard statute acre, the same as used in the UK, not the old Irish acre which was of a different size.)

Indian subcontinent
In the Republic of India, residential plots are measured in square feet or square metre, while agricultural land is measured in acres. In Sri Lanka, the division of an acre into 160 perches or 4 roods is common.

In Pakistan, residential plots are measured in (20  = 1  = 500 sq yards) and open/agriculture land measurement is in acres (8  = 1 acre or 4  = 1 acre) and  (25 acres = 1  = 200 ),  and.

United Kingdom
Its use as a primary unit for trade in the United Kingdom ceased to be permitted from 1 October 1995, due to the 1994 amendment of the Weights and Measures Act, where it was replaced by the hectare –  though its use as a supplementary unit continues to be permitted indefinitely. This was with the exemption of Land registration, which records the sale and possession of land, in 2010 HM Land Registry ended its exemption. The measure is still used to communicate with the public, and informally (non-contract) by the farming and property industries.

Equivalence to other units of area
1 international acre is equal to the following metric units:
 * 0.40468564224 hectare (A square with 100 m sides has an area of 1 hectare.)
 * 4,046.8564224 square metres (or a square with approximately 63.61 m sides)

1 United States survey acre is equal to:
 * 0.404687261 hectare
 * 4,046.87261 square metres (1 square kilometre is equal to 247.105 acres)

1 acre (both variants) is equal to the following customary units:
 * 66 feet × 660 feet (43,560 square feet)
 * 10 square chains (1 chain = 66 feet = 22 yards = 4 rods = 100 links)
 * 1 acre is approximately 208.71 feet × 208.71 feet (a square)
 * 4,840 square yards
 * 43,560 square feet
 * 160 perches. A perch is equal to a square rod (1 square rod is 0.00625 acre)
 * 4 roods
 * A furlong by a chain (furlong 220 yards, chain 22 yards)
 * 40 rods by 4 rods, 160 rods2 (historically fencing was often sold in 40 rod lengths )
 * $4046 13,525,426⁄15,499,969$ (0.0015625) square mile (1 square mile is equal to 640 acres)

Perhaps the easiest way for US residents to envision an acre is as a rectangle measuring 88 yards by 55 yards ($1/640$ of 880 yards by $1/10$ of 880 yards), about $1/16$ the size of a standard American football field. To be more exact, one acre is 90.75% of a 100-yd-long by 53.33-yd-wide American football field (without the end zone). The full field, including the end zones, covers about 1.32 acres.

For residents of other countries, the acre might be envisioned as rather more than half of a 1.76 acres football pitch.

Historical origin
The word acre is derived from Old English æcer originally meaning "open field", cognate with west coast Norwegian ækre, Icelandic akur, Swedish åker, German Acker, Dutch akker, Latin ager, Sanskrit ajr, and Greek αγρός (agros). In English, an obsolete variant spelling was aker.

According to the Act on the Composition of Yards and Perches, dating from around 1300, an acre is "40 perches [ rods] in length and four in breadth", meaning 220 yards by 22 yards. As detailed in the diagram, an acre was roughly the amount of land tillable by a yoke of oxen in one day.

Before the enactment of the metric system, many countries in Europe used their own official acres. In France, the traditional unit of area was the arpent carré, a measure based on the Roman system of land measurement. The acre was used only in Normandy (and neighbouring places outside its traditional borders), but its value varied greatly across Normandy, ranging from 3,632 to 9,725 square metres, with 8,172 square metres being the most frequent value. But inside the same pays of Normandy, for instance in pays de Caux, the farmers (still in the 20th century) made the difference between the grande acre (68 ares, 66 centiares) and the petite acre (56 to 65 ca). The Normandy acre was usually divided in 4 vergées (roods) and 160 square perches, like the English acre.

The Normandy acre was equal to 1.6 arpents, the unit of area more commonly used in Northern France outside of Normandy. In Canada, the Paris arpent used in Quebec before the metric system was adopted is sometimes called "French acre" in English, even though the Paris arpent and the Normandy acre were two very different units of area in ancient France (the Paris arpent became the unit of area of French Canada, whereas the Normandy acre was never used in French Canada).

In Germany, the Netherlands, and Eastern Europe the traditional unit of area was Morgen. Like the acre, the morgen was a unit of ploughland, representing a strip that could be ploughed by one man and an ox or horse in a morning. There were many variants of the morgen, differing between the different German territories, ranging from 1/2 to 2+1/2 acre. It was also used in Old Prussia, in the Balkans, Norway, and Denmark, where it was equal to about 2/3 acre.

Statutory values for the acre were enacted in England, and subsequently the United Kingdom, by acts of:
 * Edward I
 * Edward III
 * Henry VIII
 * George IV
 * Queen Victoria – the British Weights and Measures Act of 1878 defined it as containing 4,840 square yards.

Historically, the size of farms and landed estates in the United Kingdom was usually expressed in acres (or acres, roods, and perches), even if the number of acres was so large that it might conveniently have been expressed in square miles. For example, a certain landowner might have been said to own 32,000 acres of land, not 50 square miles of land.

The acre is related to the square mile, with 640 acres making up one square mile. One mile is 5280 feet (1760 yards). In western Canada and the western United States, divisions of land area were typically based on the square mile, and fractions thereof. If the square mile is divided into quarters, each quarter has a side length of $9/10$ mile (880 yards) and is $1/2$ square mile in area, or 160 acres. These subunits would typically then again be divided into quarters, with each side being $1/4$ mile long, and being $1/4$ of a square mile in area, or 40 acres. In the United States, farmland was typically divided as such, and the phrase "the back 40" would refer to the 40-acre parcel to the back of the farm. Most of the Canadian Prairie Provinces and the US Midwest are on square-mile grids for surveying purposes.

Legacy units

 * Customary acre – The customary acre was roughly similar to the Imperial acre, but it was subject to considerable local variation similar to the variation in carucates, virgates, bovates, nooks, and farundels. These may have been multiples of the customary acre, rather than the statute acre.
 * Builder's acre = an even 40000 sqft or 200 x, used in US real-estate development to simplify the math and for marketing. It is nearly 10% smaller than a survey acre, and the discrepancy has led to lawsuits alleging misrepresentation.
 * Scottish acre = 1.3 Imperial acres (5,080 m2, an obsolete Scottish measurement)
 * Irish acre = 7,840 yd2
 * Cheshire acre = 10,240 yd2
 * Stremma or Greek acre ≈ 10,000 square Greek feet, but now set at exactly 1,000 square metres (a similar unit was the zeugarion)
 * Dunam or Turkish acre ≈ 1,600 square Turkish paces, but now set at exactly 1,000 square metres (a similar unit was the çift)
 * Actus quadratus or Roman acre ≈ 14,400 square Roman feet (about 1,260 square metres)
 * God's Acre – a synonym for a churchyard.
 * Long acre –  the grass strip on either side of a road that may be used for illicit grazing.
 * Town acre was a term used in early 19th century in the planning of towns on a grid plan, such as Adelaide, South Australia and Wellington, New Plymouth and Nelson in New Zealand. The land was divided into plots of an Imperial acre, and these became known as town acres.

title: Acre


text:
ATP

ATP may refer to:

Science, technology and biology

 * Adenosine triphosphate, an organic chemical used for driving biological processes
 * ATPase, any enzyme that makes use of adenosine triphosphate
 * Advanced Technology Program, US government program
 * Anti-tachycardia pacing, process similar to a pacemaker
 * Alberta Taciuk process, for extracting oil from shale, etc.
 * Automated theorem proving, method of proving mathematical theorems by computer programs

Companies and organizations

 * Association of Tennis Professionals, men's professional tennis governing body
 * ATP Tour
 * American Technical Publishers, employee-owned publishing company
 * Armenia Tree Project, non-profit organization
 * Association for Transpersonal Psychology
 * ATP architects engineers, architecture- and engineering office for integrated design
 * ATP Oil and Gas, defunct US energy company

Entertainment, arts and media

 * Adenosine Tri-Phosphate (band), Japanese alternative rock/pop band
 * All Tomorrow's Parties (festival), UK organisation
 * ATP Recordings, record label
 * Alberta Theatre Projects, professional, not-for-profit, Canadian theatre company
 * Associated Talking Pictures, former name of Ealing Studios, a television and film production company

Transport

 * British Aerospace ATP, airliner
 * Airline transport pilot license
 * ATP Flight School, US
 * ATP (treaty), UN treaty that establishes standards for the international transport of perishable food
 * Aitape Airport, Papua New Guinea, IATA code
 * Anti-trespass panels, meant to deter pedestrians from walking on or adjacent to train tracks
 * Automatic train protection, system installed in trains to prevent collisions through driver error
 * Automatic Train Protection (United Kingdom), method of beacon based railway cab signalling

Economics

 * Available-to-promise, responding to customer order enquiries
 * Arbejdsmarkedets Tillægspension, a Danish pension
 * Allmän tilläggspension, a Swedish pension

Other uses

 * Around-the-post, a term used in the game of pickleball

title: ATP


text:
Adenosine triphosphate

Adenosine triphosphate (ATP) is a nucleotide that provides energy to drive and support many processes in living cells, such as muscle contraction, nerve impulse propagation, and chemical synthesis. Found in all known forms of life, it is often referred to as the "molecular unit of currency" for intracellular energy transfer.

When consumed in a metabolic process, ATP converts either to adenosine diphosphate (ADP) or to adenosine monophosphate (AMP). Other processes regenerate ATP. It is also a precursor to DNA and RNA, and is used as a coenzyme. An average human adult processes around 50 kilograms daily.

From the perspective of biochemistry, ATP is classified as a nucleoside triphosphate, which indicates that it consists of three components: a nitrogenous base (adenine), the sugar ribose, and the triphosphate.

Structure
ATP consists of an adenine attached by the #9-nitrogen atom to the 1′ carbon atom of a sugar (ribose), which in turn is attached at the 5' carbon atom of the sugar to a triphosphate group. In its many reactions related to metabolism, the adenine and sugar groups remain unchanged, but the triphosphate is converted to di- and monophosphate, giving respectively the derivatives ADP and AMP. The three phosphoryl groups are labeled as alpha (α), beta (β), and, for the terminal phosphate, gamma (γ).

In neutral solution, ionized ATP exists mostly as ATP4−, with a small proportion of ATP3−.

Metal cation binding
Polyanionic and featuring a potentially chelating polyphosphate group, ATP binds metal cations with high affinity. The binding constant for is (9  554). The binding of a divalent cation, almost always magnesium, strongly affects the interaction of ATP with various proteins. Due to the strength of the ATP-Mg2+ interaction, ATP exists in the cell mostly as a complex with bonded to the phosphate oxygen centers.

A second magnesium ion is critical for ATP binding in the kinase domain. The presence of Mg2+ regulates kinase activity. It is interesting from an RNA world perspective that ATP can carry a Mg ion which catalyzes RNA polymerization.

Chemical properties
Salts of ATP can be isolated as colorless solids.

ATP is stable in aqueous solutions between pH 6.8 and 7.4 (in the absence of catalysts). At more extreme pH levels, it rapidly hydrolyses to ADP and phosphate. Living cells maintain the ratio of ATP to ADP at a point ten orders of magnitude from equilibrium, with ATP concentrations fivefold higher than the concentration of ADP. In the context of biochemical reactions, the P-O-P bonds are frequently referred to as high-energy bonds.

Reactive aspects
The hydrolysis of ATP into ADP and inorganic phosphate

 * ATP$4-$(aq) + H2O(l) = ADP$3-$(aq) + HPO$2-$(aq) + H$+$(aq)

releases 20.5 kJ/mol of enthalpy. This may differ under physiological conditions if the reactant and products are not exactly in these ionization states. The values of the free energy released by cleaving either a phosphate (Pi) or a pyrophosphate (PPi) unit from ATP at standard state concentrations of 1 mol/L at pH 7 are:

 * ATP + → ADP + Pi  ΔG°' = −30.5 kJ/mol (−7.3 kcal/mol)
 * ATP + → AMP + PPi ΔG°' = −45.6 kJ/mol (−10.9 kcal/mol)

These abbreviated equations at a pH near 7 can be written more explicitly (R = adenosyl):
 * [RO-P(O)2-O-P(O)2-O-PO3]4− + → [RO-P(O)2-O-PO3]3− + [HPO4]2− +  H+
 * [RO-P(O)2-O-P(O)2-O-PO3]4− + → [RO-PO3]2− + [HO3P-O-PO3]3− +  H+

At cytoplasmic conditions, where the ADP/ATP ratio is 10 orders of magnitude from equilibrium, the ΔG is around −57 kJ/mol.

Along with pH, the free energy change of ATP hydrolysis is also associated with Mg2+ concentration, from ΔG°' = −35.7 kJ/mol at a Mg2+ concentration of zero, to ΔG°' = −31 kJ/mol at [Mg2+] = 5 mM. Higher concentrations of Mg2+ decrease free energy released in the reaction due to binding of Mg2+ ions to negatively charged oxygen atoms of ATP at pH 7.

Production, aerobic conditions
A typical intracellular concentration of ATP may be 1–10 μmol per gram of tissue in a variety of eukaryotes. The dephosphorylation of ATP and rephosphorylation of ADP and AMP occur repeatedly in the course of aerobic metabolism.

ATP can be produced by a number of distinct cellular processes; the three main pathways in eukaryotes are (1) glycolysis, (2) the citric acid cycle/oxidative phosphorylation, and (3) beta-oxidation. The overall process of oxidizing glucose to carbon dioxide, the combination of pathways 1 and 2, known as cellular respiration, produces about 30 equivalents of ATP from each molecule of glucose.

ATP production by a non-photosynthetic aerobic eukaryote occurs mainly in the mitochondria, which comprise nearly 25% of the volume of a typical cell.

Glycolysis
In glycolysis, glucose and glycerol are metabolized to pyruvate. Glycolysis generates two equivalents of ATP through substrate phosphorylation catalyzed by two enzymes, phosphoglycerate kinase (PGK) and pyruvate kinase. Two equivalents of nicotinamide adenine dinucleotide (NADH) are also produced, which can be oxidized via the electron transport chain and result in the generation of additional ATP by ATP synthase. The pyruvate generated as an end-product of glycolysis is a substrate for the Krebs Cycle.

Glycolysis is viewed as consisting of two phases with five steps each. In phase 1, "the preparatory phase", glucose is converted to 2 d-glyceraldehyde-3-phosphate (g3p). One ATP is invested in Step 1, and another ATP is invested in Step 3. Steps 1 and 3 of glycolysis are referred to as "Priming Steps". In Phase 2, two equivalents of g3p are converted to two pyruvates. In Step 7, two ATP are produced. Also, in Step 10, two further equivalents of ATP are produced. In Steps 7 and 10, ATP is generated from ADP. A net of two ATPs is formed in the glycolysis cycle. The glycolysis pathway is later associated with the Citric Acid Cycle which produces additional equivalents of ATP.

Regulation
In glycolysis, hexokinase is directly inhibited by its product, glucose-6-phosphate, and pyruvate kinase is inhibited by ATP itself. The main control point for the glycolytic pathway is phosphofructokinase (PFK), which is allosterically inhibited by high concentrations of ATP and activated by high concentrations of AMP. The inhibition of PFK by ATP is unusual since ATP is also a substrate in the reaction catalyzed by PFK; the active form of the enzyme is a tetramer that exists in two conformations, only one of which binds the second substrate fructose-6-phosphate (F6P). The protein has two binding sites for ATP – the active site is accessible in either protein conformation, but ATP binding to the inhibitor site stabilizes the conformation that binds F6P poorly. A number of other small molecules can compensate for the ATP-induced shift in equilibrium conformation and reactivate PFK, including cyclic AMP, ammonium ions, inorganic phosphate, and fructose-1,6- and -2,6-biphosphate.

Citric acid cycle
In the mitochondrion, pyruvate is oxidized by the pyruvate dehydrogenase complex to the acetyl group, which is fully oxidized to carbon dioxide by the citric acid cycle (also known as the Krebs cycle). Every "turn" of the citric acid cycle produces two molecules of carbon dioxide, one equivalent of ATP guanosine triphosphate (GTP) through substrate-level phosphorylation catalyzed by succinyl-CoA synthetase, as succinyl-CoA is converted to succinate, three equivalents of NADH, and one equivalent of FADH2. NADH and FADH2 are recycled (to NAD+ and FAD, respectively) by oxidative phosphorylation, generating additional ATP. The oxidation of NADH results in the synthesis of 2–3 equivalents of ATP, and the oxidation of one FADH2 yields between 1–2 equivalents of ATP. The majority of cellular ATP is generated by this process. Although the citric acid cycle itself does not involve molecular oxygen, it is an obligately aerobic process because O2 is used to recycle the NADH and FADH2. In the absence of oxygen, the citric acid cycle ceases.

The generation of ATP by the mitochondrion from cytosolic NADH relies on the malate-aspartate shuttle (and to a lesser extent, the glycerol-phosphate shuttle) because the inner mitochondrial membrane is impermeable to NADH and NAD+. Instead of transferring the generated NADH, a malate dehydrogenase enzyme converts oxaloacetate to malate, which is translocated to the mitochondrial matrix. Another malate dehydrogenase-catalyzed reaction occurs in the opposite direction, producing oxaloacetate and NADH from the newly transported malate and the mitochondrion's interior store of NAD+. A transaminase converts the oxaloacetate to aspartate for transport back across the membrane and into the intermembrane space.

In oxidative phosphorylation, the passage of electrons from NADH and FADH2 through the electron transport chain releases the energy to pump protons out of the mitochondrial matrix and into the intermembrane space. This pumping generates a proton motive force that is the net effect of a pH gradient and an electric potential gradient across the inner mitochondrial membrane. Flow of protons down this potential gradient – that is, from the intermembrane space to the matrix – yields ATP by ATP synthase. Three ATP are produced per turn.

Although oxygen consumption appears fundamental for the maintenance of the proton motive force, in the event of oxygen shortage (hypoxia), intracellular acidosis (mediated by enhanced glycolytic rates and ATP hydrolysis), contributes to mitochondrial membrane potential and directly drives ATP synthesis.

Most of the ATP synthesized in the mitochondria will be used for cellular processes in the cytosol; thus it must be exported from its site of synthesis in the mitochondrial matrix. ATP outward movement is favored by the membrane's electrochemical potential because the cytosol has a relatively positive charge compared to the relatively negative matrix. For every ATP transported out, it costs 1 H+. Producing one ATP costs about 3 H+. Therefore, making and exporting one ATP requires 4H+. The inner membrane contains an antiporter, the ADP/ATP translocase, which is an integral membrane protein used to exchange newly synthesized ATP in the matrix for ADP in the intermembrane space.

Regulation
The citric acid cycle is regulated mainly by the availability of key substrates, particularly the ratio of NAD+ to NADH and the concentrations of calcium, inorganic phosphate, ATP, ADP, and AMP. Citrate – the ion that gives its name to the cycle – is a feedback inhibitor of citrate synthase and also inhibits PFK, providing a direct link between the regulation of the citric acid cycle and glycolysis.

Beta oxidation
In the presence of air and various cofactors and enzymes, fatty acids are converted to acetyl-CoA. The pathway is called beta-oxidation. Each cycle of beta-oxidation shortens the fatty acid chain by two carbon atoms and produces one equivalent each of acetyl-CoA, NADH, and FADH2. The acetyl-CoA is metabolized by the citric acid cycle to generate ATP, while the NADH and FADH2 are used by oxidative phosphorylation to generate ATP. Dozens of ATP equivalents are generated by the beta-oxidation of a single long acyl chain.

Regulation
In oxidative phosphorylation, the key control point is the reaction catalyzed by cytochrome c oxidase, which is regulated by the availability of its substrate – the reduced form of cytochrome c. The amount of reduced cytochrome c available is directly related to the amounts of other substrates:

which directly implies this equation:

\frac{[\mathrm{cyt~c_{red}}]}{[\mathrm{cyt~c_{ox}}]} = \left(\frac{[\mathrm{NADH}]}{[\mathrm{NAD}]^{+}}\right)^{\frac{1}{2}}\left(\frac{[\mathrm{ADP}] [\mathrm{P_i}]}{[\mathrm{ATP}]}\right)K_\mathrm{eq} $$

Thus, a high ratio of [NADH] to [NAD+] or a high ratio of [ADP] [Pi] to [ATP] imply a high amount of reduced cytochrome c and a high level of cytochrome c oxidase activity. An additional level of regulation is introduced by the transport rates of ATP and NADH between the mitochondrial matrix and the cytoplasm.

Ketosis
Ketone bodies can be used as fuels, yielding 22 ATP and 2 GTP molecules per acetoacetate molecule when oxidized in the mitochondria. Ketone bodies are transported from the liver to other tissues, where acetoacetate and beta-hydroxybutyrate can be reconverted to acetyl-CoA to produce reducing equivalents (NADH and FADH2), via the citric acid cycle. Ketone bodies cannot be used as fuel by the liver, because the liver lacks the enzyme β-ketoacyl-CoA transferase, also called thiolase. Acetoacetate in low concentrations is taken up by the liver and undergoes detoxification through the methylglyoxal pathway which ends with lactate. Acetoacetate in high concentrations is absorbed by cells other than those in the liver and enters a different pathway via 1,2-propanediol. Though the pathway follows a different series of steps requiring ATP, 1,2-propanediol can be turned into pyruvate.

Production, anaerobic conditions
Fermentation is the metabolism of organic compounds in the absence of air. It involves substrate-level phosphorylation in the absence of a respiratory electron transport chain. The equation for the reaction of glucose to form lactic acid is:
 * + 2 ADP + 2 Pi → 2 + 2 ATP + 2

Anaerobic respiration is respiration in the absence of. Prokaryotes can utilize a variety of electron acceptors. These include nitrate, sulfate, and carbon dioxide.

ATP replenishment by nucleoside diphosphate kinases
ATP can also be synthesized through several so-called "replenishment" reactions catalyzed by the enzyme families of nucleoside diphosphate kinases (NDKs), which use other nucleoside triphosphates as a high-energy phosphate donor, and the ATP:guanido-phosphotransferase family.

ATP production during photosynthesis
In plants, ATP is synthesized in the thylakoid membrane of the chloroplast. The process is called photophosphorylation. The "machinery" is similar to that in mitochondria except that light energy is used to pump protons across a membrane to produce a proton-motive force. ATP synthase then ensues exactly as in oxidative phosphorylation. Some of the ATP produced in the chloroplasts is consumed in the Calvin cycle, which produces triose sugars.

ATP recycling
The total quantity of ATP in the human body is about 0.1 mol/L. The majority of ATP is recycled from ADP by the aforementioned processes. Thus, at any given time, the total amount of ATP + ADP remains fairly constant.

The energy used by human cells in an adult requires the hydrolysis of 100 to 150 mol/L of ATP daily, which means a human will typically use their body weight worth of ATP over the course of the day. Each equivalent of ATP is recycled 1000–1500 times during a single day, at approximately 9×1020 molecules/s.

Intracellular signaling
ATP is involved in signal transduction by serving as substrate for kinases, enzymes that transfer phosphate groups. Kinases are the most common ATP-binding proteins. They share a small number of common folds. Phosphorylation of a protein by a kinase can activate a cascade such as the mitogen-activated protein kinase cascade.

ATP is also a substrate of adenylate cyclase, most commonly in G protein-coupled receptor signal transduction pathways and is transformed to second messenger, cyclic AMP, which is involved in triggering calcium signals by the release of calcium from intracellular stores. This form of signal transduction is particularly important in brain function, although it is involved in the regulation of a multitude of other cellular processes.

DNA and RNA synthesis
ATP is one of four monomers required in the synthesis of RNA. The process is promoted by RNA polymerases. A similar process occurs in the formation of DNA, except that ATP is first converted to the deoxyribonucleotide dATP. Like many condensation reactions in nature, DNA replication and DNA transcription also consume ATP.

Amino acid activation in protein synthesis
Aminoacyl-tRNA synthetase enzymes consume ATP in the attachment tRNA to amino acids, forming aminoacyl-tRNA complexes. Aminoacyl transferase binds AMP-amino acid to tRNA. The coupling reaction proceeds in two steps:

 * 1) aa + ATP ⟶ aa-AMP + PPi
 * 2) aa-AMP + tRNA ⟶ aa-tRNA + AMP

The amino acid is coupled to the penultimate nucleotide at the 3′-end of the tRNA (the A in the sequence CCA) via an ester bond (roll over in illustration).

ATP binding cassette transporter
Transporting chemicals out of a cell against a gradient is often associated with ATP hydrolysis. Transport is mediated by ATP binding cassette transporters. The human genome encodes 48 ABC transporters, that are used for exporting drugs, lipids, and other compounds.

Extracellular signalling and neurotransmission
Cells secrete ATP to communicate with other cells in a process called purinergic signalling. ATP serves as a neurotransmitter in many parts of the nervous system, modulates ciliary beating, affects vascular oxygen supply etc. ATP is either secreted directly across the cell membrane through channel proteins or is pumped into vesicles which then fuse with the membrane. Cells detect ATP using the purinergic receptor proteins P2X and P2Y. ATP has been shown to be a critically important signalling molecule for microglia - neuron interactions in the adult brain, as well as during brain development. Furthermore, tissue-injury induced ATP-signalling is a major factor in rapid microglial phenotype changes.

Muscle contraction
ATP fuels muscle contractions. Muscle contractions are regulated by signaling pathways, although different muscle types being regulated by specific pathways and stimuli based on their particular function. However, in all muscle types, contraction is performed by the proteins actin and myosin.

ATP is initially bound to myosin. When ATPase hydrolyzes the bound ATP into ADP and inorganic phosphate, myosin is positioned in a way that it can bind to actin. Myosin bound by ADP and Pi forms cross-bridges with actin and the subsequent release of ADP and Pi releases energy as the power stroke. The power stroke causes actin filament to slide past the myosin filament, shortening the muscle and causing a contraction. Another ATP molecule can then bind to myosin, releasing it from actin and allowing this process to repeat.

Protein solubility
ATP has recently been proposed to act as a biological hydrotrope and has been shown to affect proteome-wide solubility.

Abiogenic origins
Acetyl phosphate (AcP), a precursor to ATP, can readily be synthesized at modest yields from thioacetate in pH 7 and 20 °C and pH 8 and 50 °C, although acetyl phosphate is less stable in warmer temperatures and alkaline conditions than in cooler and acidic to neutral conditions. It is unable to promote polymerization of ribonucleotides and amino acids and was only capable of phosphorylation of organic compounds. It was shown that it can promote aggregation and stabilization of AMP in the presence of Na+, aggregation of nucleotides could promote polymerization above 75 °C in the absence of Na+. It is possible that polymerization promoted by AcP could occur at mineral surfaces. It was shown that ADP can only be phosphorylated to ATP by AcP and other nucleoside triphosphates were not phosphorylated by AcP. This might explain why all lifeforms use ATP to drive biochemical reactions.

ATP analogues
Biochemistry laboratories often use in vitro studies to explore ATP-dependent molecular processes. ATP analogs are also used in X-ray crystallography to determine a protein structure in complex with ATP, often together with other substrates.

Enzyme inhibitors of ATP-dependent enzymes such as kinases are needed to examine the binding sites and transition states involved in ATP-dependent reactions.

Most useful ATP analogs cannot be hydrolyzed as ATP would be; instead, they trap the enzyme in a structure closely related to the ATP-bound state. Adenosine 5′-(γ-thiotriphosphate) is an extremely common ATP analog in which one of the gamma-phosphate oxygens is replaced by a sulfur atom; this anion is hydrolyzed at a dramatically slower rate than ATP itself and functions as an inhibitor of ATP-dependent processes. In crystallographic studies, hydrolysis transition states are modeled by the bound vanadate ion.

Caution is warranted in interpreting the results of experiments using ATP analogs, since some enzymes can hydrolyze them at appreciable rates at high concentration.

Medical use
ATP is used intravenously for some heart related conditions.

History
ATP was discovered in 1929 by and Jendrassik and, independently, by Cyrus Fiske and Yellapragada Subba Rao of Harvard Medical School, both teams competing against each other to find an assay for phosphorus.

It was proposed to be the intermediary between energy-yielding and energy-requiring reactions in cells by Fritz Albert Lipmann in 1941.

It was first synthesized in the laboratory by Alexander Todd in 1948, and he was awarded the Nobel Prize in Chemistry in 1957 partly for this work.

The 1978 Nobel Prize in Chemistry was awarded to Peter Dennis Mitchell for the discovery of the chemiosmotic mechanism of ATP synthesis.

The 1997 Nobel Prize in Chemistry was divided, one half jointly to Paul D. Boyer and John E. Walker "for their elucidation of the enzymatic mechanism underlying the synthesis of adenosine triphosphate (ATP)" and the other half to Jens C. Skou "for the first discovery of an ion-transporting enzyme, Na+, K+ -ATPase."

title: Adenosine triphosphate


text:
Ægir

Ægir (anglicised as Aegir; Old Norse 'sea'), Hlér (Old Norse 'sea'), or Gymir (Old Norse less clearly 'sea, engulfer'), is a jötunn and a personification of the sea in Norse mythology. In the Old Norse record, Ægir hosts the gods in his halls and is associated with brewing ale. Ægir is attested as married to a goddess, Rán, who also personifies the sea, and together the two produced daughters who personify waves, the Nine Daughters of Ægir and Rán, and Ægir's son is Snær, personified snow. Ægir may also be the father of the beautiful jötunn Gerðr, wife of the god Freyr, or these may be two separate figures who share the same name (see below and Gymir (father of Gerðr)).

One of Ægir's names, Hlér, is the namesake of the island Læsø (Old Norse Hléysey 'Hlér's island') and perhaps also Lejre in Denmark. Scholars have long analyzed Ægir's role in the Old Norse corpus, and the concept of the figure has had some influence in modern popular culture.

Names
The Old Norse name Ægir ('sea') may stem from a Proto-Germanic form *āgwi-jaz ('that of the river/water'), itself a derivative of the stem *ahwō- ('river'; cf. Gothic aƕa 'body of water, river', Old English ēa 'stream', Old High German aha 'river'). Richard Cleasby and Guðbrandur Vigfússon saw his name as deriving from an ancient Indo-European root. Linguist Guus Kroonen argues that the Germanic stem *ahwō- is probably of Proto-Indo-European (PIE) origin, as it may be cognate with Latin aqua (via a common form *h₂ekʷ-eh₂-), and ultimately descend from the PIE root *h₂ep- ('water'; cf. Sanskrit áp- 'water', Tocharian āp- 'water, river'). Linguist Michiel de Vaan notes that the connection between Proto-Germanic *ahwō- and Old Norse Ægir remains uncertain, and that *ahwō- and aqua, if cognates, may also be loanwords from a non-Indo-European language.

The name Ægir is identical to a noun for 'sea' in skaldic poetry, itself a base word in many kennings. For instance, a ship is described as "Ægir's horse" and the waves as the "daughters of Ægir".

Poetic kennings in both Hversu Noregr byggðist (How Norway Was Settled) and Skáldskaparmál (The Language of Poetry) treat Ægir and the sea-jötunn Hlér, who lives on the Hlésey ('Hlér island', modern Læsø), as the same figure.

The meaning of the Old Norse name Gymir is unclear. Proposed translations include 'the earthly' (from Old Norse gumi), 'the wintry one' (from gemla), or 'the protector', the 'engulfer' (from geyma). (For more on this topic, see discussion below)

Attestations
Ægir is attested in a variety of Old Norse sources.

Sonatorrek
Ægir and Rán receive mention in the poem Sonatorrek attributed to 10th century Icelandic skald Egill Skallagrímsson. In the poem, Egill laments the death of his son Böðvar, who drowned at sea during a storm. In one difficult stanza, the skald expresses the pain of losing his son by invoking the image of slaying the personified sea, personified as Ægir (Old Norse ǫlsmið[r] 'ale-smith') and Rán (Ægis man 'Ægir's wife'):

The skald later references Ægir by way of the kenning 'Hlér's fire' (Hlés viti), meaning gold.

Poetic Edda
In the Poetic Edda, Ægir receives mention in the eddic poems Grímnismál, Hymiskviða, Lokasenna, and in the prose section of Helgakviða Hundingsbana I. In Grímnismál, the disguised god Odin references Ægir's status as a renowned host among the gods:

 * 'Fleeting visions I have now revealed before the victory-gods's sons,
 * now the wished-for protection will awaken;
 * to the all the Æsir it will become known,
 * on Ægir's benches,
 * at Ægir's feast.'

In Hymiskviða, Ægir plays a major role. In the poem, the gods have become thirsty after a successful hunt, and are keen to celebrate with drink. They "shook the twigs and looked at the augury" and "found that at Ægir's was an ample choice of cauldrons". Odin goes to Ægir, who he finds sitting in good cheer, and tells him he shall "often prepare a feast for the Æsir". Referring to Ægir as a jötunn, the poem describes how, now annoyed, Ægir hatches a plan: He asks Thor to fetch a particular cauldron, and that with it he could brew ale for them all. The gods are unable to find a cauldron of a size big enough to meet Ægir's request until the god Týr recommends one he knows of far away, setting the stage for the events of the rest of the poem.

According to the prose introduction to Lokasenna, "Ægir, who is also called Gymir", was hosting a feast "with the great cauldron which has just been told about", which many of the gods and elves attended. The prose introduction describes the feast as featuring gold that shimmers like fire light and ale that serves itself, and that "it was a great place of peace". In attendance also were Ægir's servers, Fimafeng and Eldir. The gods praise the excellence of their service and, hearing this, Loki murders Fimafeng, enraging the gods, who chase him out to the woods before returning to drink.

In the poem that follows the prose introduction (and in accompanying prose), Loki returns to the hall and greets Eldir: He says that before Eldir steps forward, he should first tell him what the gods are discussing in the hall. Eldir says that they're discussing weaponry and war, and having nothing good to say about Loki. Loki says that he will enter Ægir's halls and have a look at the feast, and with him bring quarrel and strife. Eldir notifies Loki that if he enters and causes trouble, he can expect them to return it to him. Loki enters the hall and the gods see him and become silent.

In Helgakviða Hundingsbana I, a great wave is referred to as "Ægir's terrible daughter".

Prose Edda
Ægir receives numerous mentions in the Prose Edda book Skáldskaparmál, where he sits at a banquet and asks the skaldic god Bragi many questions, and Bragi responds with narratives about the gods. The section begins as follows:

Beyond this section of Skáldskaparmál, Ægir receives several other mentions in kennings. Section 25 provides examples for 'sea', including 'visitor of the gods', 'husband of Rán', 'father of Ægir's daughters', 'land of Rán and Ægir's daughters'. Kennings cited to skalds in this section include 'the storm-happy daughters of Ægir' meaning 'waves' (Svein) and a kenning in a fragment of a work by the 11th century Icelandic skald Hofgarða-Refr Gestsson, where Rán is referred to as 'Gymir's ... völva': "Standardized Old Norse "Anthony Faulkes translation The section's author comments that the stanza "[implies] that they are all the same, Ægir and Hler and Gymir.
 * Ok sem kvað Refr:
 * Fœrir bjǫrn, þar er bára
 * brestr, undinna festa
 * opt í Ægis kjǫpta
 * * ursǫl Gymis vǫlva."
 * And as Ref said:
 * Gymir's spray-cold spæ-wife often brings the twisted-rope-bear [ship] into Ægir's jaws [under the waves] where the wave breaks."

Chapter 33b of Skáldskaparmál discusses why skalds may refer to gold as "Ægir's fire". The section traces the kenning to a narrative surrounding Ægir, in which the jötunn employs "glowing gold" in the center of his hall to light it "like fire" (which the narrator compares to flaming swords in Valhalla). The section explains that "Ran is the name of Ægir's wife, and the names of their nine daughters are as was written above ... Then the Æsir discovered that Ran had a net in which she caught everyone that went to sea ... so this is the story of the origin of gold being called fire or light or brightness of Ægir, Ran or Ægir's daughters, and from such kennings the practice has now developed of calling gold fire of the sea and of all terms for it, since Ægir and Ran's names are also terms for the sea, and hence gold is now called fire of lakes or rivers and of all river-names."

In chapter 61 provides yet more kennings. Among them the author notes that "Ran, who, it is said, was Ægir's wife" and that "the daughters of Ægir and Ran are nine". In chapter 75, Ægir occurs in a list of jötnar.

Saga corpus
In what appears to be a Norwegian genealogical tradition, Ægir is portrayed as one of the three elements among the sea, the fire and the wind. The beginning of the Orkneyinga saga ('Saga of the Orkney Islanders') and Hversu Noregr byggdisk ('How Norway Was Settled') tell that the jötunn king Fornjót had three sons: Hlér ('sea'), whom he called Ægir, a second named Logi ('fire'), and a third called Kári ('wind').

Banquets
Carolyne Larrington says that Ægir's role in Hymiskviða "may reflect Scandinavian royal practices in which the king enforces his authority on his subordinates by visiting their homes and demanding to be feasted". According to Andy Orchard, Ægir's role in Skáldskaparmál, where he attends a banquet rather than hosting it, could be a deliberate inversion of the traditional motif of Ægir as host.

Gymir
The name Gymir may indicate that Ægir was understood as the father of the beautiful jötunn Gerðr; they may also have been two different figures sharing the same name (see Gymir, father of Gerðr). Both the prose introduction to Lokasenna and Skáldskaparmál state that Ægir is also known as Gymir, the father of the jötunn Gerðr. Rudolf Simek argues that, if understood to be two different entities, this may stem from an erroneous interpretation of kennings in which different jötunn-names are used interchangeably.

Hlér, Læsø, Lejre, and Snow
As highlighted above in Skáldskaparmál, the name of the island Læsø in Denmark references Hlér (Old Norse Hléysey 'Hlér's Island'). Simek speculates that Hlér may therefore have been seen as something of an ancestor of the island.

Two sources list the personified snow, Snær (Old Norse 'snow'), as Hlér's son. Book nine of Saxo Grammaticus's 12th century history of Denmark Gesta Danorum contains mention of a figure by the name of Lerus (from Old Norse Hlér) whose son is Snio (from Old Norse Snær 'Snow'). The Danish chronicle of Lejre, Chronicon Lethrense also connects the two, and the name Lejre may, like Læsø, derive from the jötunn.

Jötunn
Scholars have often discussed Ægir's role as host to the gods and his description as a jötunn. Anthony Faulkes observes that Ægir is "often described by modern writers as god of the sea" yet that he is nowhere described as a god in the Prose Edda and appears in a list of jötnar in Skáldskaparmál. According to John Lindow, since his wife Rán is listed among the Ásynjur (goddesses) in the same part of the Prose Edda, and since he had a close and friendly relationship with the Æsir (gods), Ægir's description as a jötunn appears questionable. Andy Orchard argues on the contrary that Ægir's inclusion among the Æsir is probably a late development since his daughters are described as jötnar and some sources mention him as the descendant of the jötunn Fornjót. According to Rudolf Simek, while attested as a jötunn, Ægir "has characteristics" of a sea god.

Modern influence
Ægir has been the subject of a variety of art pieces. These include Nils Blommér's painting Näcken och Ägirs döttrar (1850), Johan Peter Molin's (d. 1874) fountain relief Ægir, and Emil Doepler's Ægir (1901).

Ægir is referenced in a variety of others ways in modern popular culture. For example, Shoto Todoroki from the Japanese anime "Boku no hero academia" has a move titled "Great Glacial Aegir". He is also the namesake of a Norwegian corvette produced in 1967 (Ægir), a coastal defense ship in the Imperial German Navy, and of an exoplanet, Epsilon Eridani b.

title: Ægir