Wikipedia:Reference desk/Archives/Mathematics/2019 December 15

= December 15 =

Euclidean plane?
Please tell me, is the Euclidean plane the concept of two dimensional space only, or is it the concept of 2D space + something else? ~ R.T.G 07:11, 15 December 2019 (UTC)
 * "Plane" specifies that it's two-dimensional. --142.112.159.101 (talk) 07:47, 15 December 2019 (UTC)
 * Yes it's definitely a 2D plane on which to do graphs and plots and stuff, however, if you search for sources "Euclidean plane" specifically... it no longer seems to be just the plane. One catch phrase seems to be, "it satisfies the axioms (Π'A1), (Π'A2), and (Π'A3)", explanations as to what that means are not included... ~ R.T.G 09:38, 15 December 2019 (UTC)
 * Why don’t you try to include enough information in your question so that others have some dim hope of understanding what you’re asking about. Right now answering appears to require being able to read your mind.  —JBL (talk) 13:53, 15 December 2019 (UTC)
 * Pretty sure what the OP is referring to is, the page on sciencedirect.com. The page is cryptic and it's apparent purpose is to sell books rather than educate. I'd suggest trying a different website (like Wikipedia). It's pointless to spend time trying to decipher a confusing or poorly written text when there are so many others to choose from at this level. --RDBury (talk) 14:17, 15 December 2019 (UTC)
 * A Euclidean plane is a two dimensional area for plotting maths in. Is there something else specific about it that makes it a Euclidean plane, rather than just a plane used for geometry?


 * I'm looking for someone who can not only use a Euclidean plane, but describe it before and without, using math terms or figures, to you know, for like, someone who doesn't already know what it is, no really, they don't know what it is, but I want to explain it to them. No, they aren't stupid. It's me. I'm stupid. I can only explain things without math terms and figures. No, it's discriminatory to abuse me. Let's see, what other info was there? One of my favourite quotes from the guides that I am sure is gone now used to say, something like, "Try to write the article as though the reader has a perfect understanding of English, but has never heard of the subject before."


 * Don't worry, if you are just a simple math professor and this is beyond you... that's okay! That just means you are normal! ~ R.T.G 16:49, 15 December 2019 (UTC)


 * If you actually have a question you want answered, I recommend you devote some energy to communicating the question clearly -- performative rambling is not a good substitute. --JBL (talk) 17:10, 15 December 2019 (UTC)


 * The question is obvious and simple. Answer it, improve it, or get out the way? I'm not seeking approval. You are berating me for amusement. I've showed my sense of humour. There is no need to test me for patience. Thanks anyway o/ ~ R.T.G 18:31, 15 December 2019 (UTC)


 * I am telling you (again) that you have not articulated a question, at least not one with sufficient context for anyone else to know what might constitute an answer that would satisfy you. If you want a question answered, you should ask it clearly and concisely, with appropriate context.  If you just want to jerk off, do it in private. --JBL (talk) 21:17, 15 December 2019 (UTC)


 * I am really sorry to inform you you failed. The properties of Euclidean plane appeared very interesting to me some time ago, so now I tried to follow this thread - alas, could not find a sense of humor in it. And not only a sense of humor, but actually little sense at all. What a pity, it could have been an interesting talk... --CiaPan (talk) 21:34, 15 December 2019 (UTC)
 * "A Euclidean plane is a two dimensional area for plotting maths in. Is there something else specific about it that makes it a Euclidean plane, rather than just a plane used for geometry?"
 * "A Euclidean plane is a two dimensional area for plotting maths in. Is there something else specific about it that makes it a Euclidean plane, rather than just a plane used for geometry?"
 * "A Euclidean plane is a two dimensional area for plotting maths in. Is there something else specific about it that makes it a Euclidean plane, rather than just a plane used for geometry?"
 * The original question:
 * "Please tell me, is the Euclidean plane the concept of two dimensional space only, or is it the concept of 2D space + something else?"
 * "Please tell me, is the Euclidean plane the concept of two dimensional space only, or is it the concept of 2D space + something else?"
 * "Please tell me, is the Euclidean plane the concept of two dimensional space only, or is it the concept of 2D space + something else?"
 * The original question consisted of "is it or isn't it". I'm not looking for help reading "science direct". I have found something simple which is not explained very well either on or off the site, across the board. People who cannot see a question and are more interest in discussing contributors than even discussing the content let alone attempting to answer it need not apply. A fool could understand this question. Answer it, or go and look at something else. There is no need for a tirade here. I wouldn't support closing the reference desks, but I might support a witch hunt because the way to stop a major fire is to remove the fuel. There is no need for anyone to respond here who isn't trying to answer this simple question. ~ R.T.G 05:31, 16 December 2019 (UTC)
 * There is no need for a tirade here. Ok then. --JBL (talk) 12:34, 16 December 2019 (UTC)
 * We are just an advanced breed of monkeys on a minor planet of a very average star. But we can understand the Universe. That makes us something very special. ~ R.T.G 13:27, 16 December 2019 (UTC)

I guess the simplest answer AFAIK, if not a totally precise one, is that what distinguishes the Euclidean plane from a general 2D vector space is that it is over the real numbers plus the notion of length and angle. (Strictly speaking, the Euclidean plane is not a vector space, but an affine space; but you can naturally find its associated vector space by picking a point on the plane and saying "this is my origin". Double sharp (talk) 13:56, 16 December 2019 (UTC))

In more detail: you can think of the Euclidean plane as just R2 (the set of all ordered pairs of real numbers, interpreted as coordinates like (0,0) or (4,3) and so on), equipped with the usual dot product to define lengths and angles. As stated at Euclidean space, every n-dimensional Euclidean space behaves exactly like (i.e. is isomorphic to) Rn with the dot product, and you can make that explicit by picking an origin and drawing axes at right angles marked at unit length (i.e. picking an orthonormal basis). My apologies if I have made some inaccuracies in attempting to simplify this and leave the precise terms to parentheses... Double sharp (talk) 05:56, 16 December 2019 (UTC)
 * (As JBL helpfully remarked on my talk page, this is indeed a bit oversimplified: Euclidean space is not a vector space, but an affine space, as there is no origin or binary addition of points. So really what I should have said is that the associated vector space of Euclidean space has the properties I mentioned, which basically means being an inner product space over the reals. It is naturally easy to fix this just by picking an origin somewhere, of course; I've added a sentence to my answer to remedy this.) Double sharp (talk) 13:54, 16 December 2019 (UTC)


 * To simplify even further, the Euclidean plane is a plane in which the distance between two points is defined to be the Euclidean distance. There are many other ways to define the distance between two points, and Metric (mathematics) and Metric space contain further information.--Wikimedes (talk) 07:25, 16 December 2019 (UTC)


 * Wikimedes, you are the least technical in your attempt to describe. However, you refer to euclidean distance... After an amount of math terms and figures, "euclidean distance" says this: "The Euclidean distance between points... ...is the length of the line... ...connecting them." So, given that Euclidean in lines seems to mean "straight" line? Am I given to assume that Euclidean in planes simply means "flat" plane? If it is a simple as that, even I am going to be surprised... ~ R.T.G 08:20, 16 December 2019 (UTC)
 * Euclidean space can be considered to be "flat" space, though the notion of "flat" relies on the notion of curvature. Unfortunately, "flat" alone isn't enough. Minkowski space is considered to be flat, even though its metric... is not really a metric at all in the strict sense of the term (since it assigns zero magnitude to nonzero vectors).--Jasper Deng (talk) 12:26, 16 December 2019 (UTC)
 * So, although this doesn't solve simplification enough for articles yet, it's like a plane embedded with other dimensions, for instance, like the addition of space to time in as a tool in physics. Could it be, just for the sake of whether I have the concept or not, like a flat plane with a topography? A bumpy sheet, so to speak, representing something tied as a constant, like spacetime? ~ R.T.G 13:23, 16 December 2019 (UTC)
 * It does not have to be embedded in anything: curvature can be an intrinsic notion. A sphere has curvature, but it would have it even if the surface of the sphere was the whole 2D world. A plane lacks curvature. But it is not the only surface that lacks curvature: a cylinder lacks it too, because you can cut it along a meridian and flatten it into a plane without bending it. You cannot do that with a sphere. So while Euclidean space is "flat", this property is not enough to distinguish Euclidean space from other surfaces admitting Euclidean geometry. Double sharp (talk) 13:52, 16 December 2019 (UTC)
 * It's a flat plane, but it is distinguished as Euclidean because it represents more than two dimensions? Is that sufficient? ~ R.T.G 13:57, 16 December 2019 (UTC)
 * No, higher dimensions don't have to come into it. A Euclidean plane is two-dimensional, and it has its properties regardless of how many dimensions the space you embed it into has. The important feature that distinguishes the Euclidean plane is how translations (basically panning) act on its points, and how the dot product determines lengths and angles on it. The translations let you define lines and what it means for them to be parallel, among other things. Double sharp (talk) 13:58, 16 December 2019 (UTC)
 * So it's a 2D space with (bumps, curves, peaks?) representing constants?


 * Or... It's a 2D plane within which unusual rules apply, for instance, light might be curved or something like that, or gravity might act in a corkscrew manner, that type of quality? ~ R.T.G 14:04, 16 December 2019 (UTC)


 * It is a 2D plane with a corresponding set of transformative equations to represent constant variables not defined in a standard 3 or 2 dimensional space, similar to the attachment of time to space in astrophysics? ~ R.T.G 14:15, 16 December 2019 (UTC)
 * No, absolutely none of all that. An infinite sheet of flat, blank paper, that you could push and turn around without curving it, and that a ruler and a protractor would work on, in exactly the way you would expect, is a perfectly good picture of a Euclidean plane. Double sharp (talk) 14:25, 16 December 2019 (UTC)
 * What I've got now is an image of a flat sheet workspace, with notations representing constants which do not have a dimensional form, similar to the non dimensional constant of time added to space in physics, except possibly with more complexity noted in the additionals..? ~ R.T.G 14:39, 16 December 2019 (UTC)
 * Like... a gravity bump map or something? ~ R.T.G 14:43, 16 December 2019 (UTC)
 * I mean, let's say here on the Earth were huge metallic areas which altered gravity from one area to the next. If you created a topographical map, that would be a distorted plane. You couldn't represent the variations in gravity with more distortions without disturbing the topography. So you'd need another dimension in the graph which did not rely on the standard dimensional model. Then at the same time you might have an extremely volatile environment where it made sense to have a map of hot areas also superimposed, again a further dimension attached to but not fitting a standard dimensional model, and as it seems to appear, these extra dimensions may also have relativity to each other outside their relativity to the topographical part? ~ R.T.G 14:56, 16 December 2019 (UTC)
 * No, none of that at all gets involved into what a Euclidean plane is. As we have been repeating, a Euclidean plane is just 2D space with conventional notions of distance. (A little more precisely; conventional notions of movement, rotation, reflection, distance, and angles.) The reason why we have all those formal terminology and axioms is because just being 2D alone does not necessarily imply all of that works normally in general. So we formalised those nice properties and then defined a Euclidean plane as something satisfying them. The Euclidean plane is flat. No extra dimensions, no distortions, nothing else. Double sharp (talk) 16:07, 16 December 2019 (UTC)
 * So, and I am almost at a loss now but, a Euclidean plane is a flat plane with no distortions of things like distance, angles, size I suppose, but unlike reality, it is particularly undistorted? I understand very well the concept you suggested earlier of curving a flat surface after it has been worked on, but other than the possibility of flattening a curved surface... within the plane itself, all things are flatter than flat... and when books and articles start off with the phrase, satisfying "the axioms (Π'A1), (Π'A2), and (Π'A3)", they are talking about stuff like angle, distance, etc, being uncorrupted, such as we might say about an axis in engineering if we wanted to say, up is definitely up, left is definitely left... this sort of thing? If not any of these so far, I may be at a loss how it can be a flat plane with no special notation of distortion, but saying so is not sufficient to describe it if the further variable cannot be described... ~ R.T.G 17:54, 16 December 2019 (UTC)
 * An abstract plane, but a specifically pure plane..? ~ R.T.G 17:55, 16 December 2019 (UTC)
 * Hi RTG. I think you need an example of a plane without the Euclidean metric to compare it to one with the Euclidean metric.  Take a look at the first picture in Taxicab geometry and read the description.  (This will probably be better than me trying to describe it in words without a picture.)  After you've chewed on that, what I write next might be easier to understand.  The plane doesn't really change, you just measure distances differently on it.  If you use the the Pythagorean Theorem to measure distances, it's called the Euclidean plane.  If you use the taxicab metric, it's called ... "a flat plane with the taxicab metric"? I'm not sure if it has a special name.  (For completeness I'll mention that you can probably curve, distort, or poke holes in a plane so that the Euclidean metric is not a useful metric to work with, but at it's root, the Euclidean plane is not about the shape of the plane, it's about how you decide to measure distances on it.)--Wikimedes (talk) 20:20, 16 December 2019 (UTC)
 * Hi, I read that image and it says, in my words, if a taxi has to go through a grid of square blocks, it can take many routes of the same distance, owing to the right angle corners. Then it says, in Euclidean geometry, the true shortest distance is as the crow flies. In fact, the addition of "Euclidean" added nothing to the statement. The shortest distance between two points is as the crow flies, Euclid or not. ~ R.T.G 12:16, 17 December 2019 (UTC)

And what exactly does "as the crow flies" mean? A geodesic is the formal concept of that. In Euclidean space, the geodesics are straight lines, but in general relativity they aren't, since the space itself isn't straight. Google Maps' calculation of distance between two points also must use a curved geodesic since Earth is curved. A maze is also an elementary example where the shortest path between any two points can be not a straight line, or in some cases, not even unique. The taxicab metric is another example. To be quite honest, you're not going to receive a better answer than what has already been posted here; please learn some geometry first instead of posing a hand-waving question whose meaning is unclear.--Jasper Deng (talk) 12:35, 17 December 2019 (UTC)


 * And that is one of those intuitions that do not hold generally, which is why in mathematics we specifically refer to "the Euclidean plane", "Euclidean space", and "Euclidean distance" to specify when it does. "The shortest distance between two points" is the straight line between them only in Euclidean geometry. And indeed, since a crow is flying on the Earth's surface, its answer to "what's the shortest distance between London and Tokyo" is assuredly not drawing the straight line that connects them, but approximating the great circle distance: the surface of the Earth approximates a sphere, which is two-dimensional, but not Euclidean. From the perspective of the taxi, the "straight-line" route that seems the shortest to you as an external outsider is not a route at all because it cannot be taken. The particular intuitive properties of the Euclidean distance are not shared by all metrics (distance measures), and that's why we have a word for them.
 * I have to agree with Jasper Deng: you're not going to receive a better answer because you don't understand the concepts needed to make one understandable. You're going to have to learn some geometry first. For now, since anything else seems to be confusing you further, you can just read "Euclidean plane" as "idealised infinite flat plane where all your intuitions about lengths, angles, and all that hold true". Double sharp (talk) 12:42, 17 December 2019 (UTC)


 * you write that as though it's been explained at length that Eulcidean space is where nothing blocks a straight line. In fact, nothing blocks a straight line, Euclid or not. There is no answer going on here which describes the difference between Euclid geometry and just geometry, let alone the plane. I've made a very clear statement as to why the taxicab diagram is unrevealing. Use of the word added nothing but the word itself. Impatience is understandable.


 * You say that the shortest point between London and Tokyo is a straight line unless you follow the curve of the Earth... and the surface of a sphere is not Eulcidean. Again you have used the word Euclidean but you have not described the relevance. Instead you repeated the statement in slightly more technical terms with less example, i.e., not all metrics fit the Euclidean plane... But I can say the shortest point between London and Tokyo, is a straight line in any case, even if it is not possible to traverse. Euclidean_geometry says, Euclidean geometry postulates axioms for plane geometry in terms of constructions. But an axiom is a truth. "To draw a straight line from any point to any point" is not a truth in contemporary discourse. It is a purpose, an excuse, a motivation, etc...
 * There's five axioms. They are not unlike something I've heard before. Please tell me when I am wrong describing them...
 * A straight line is a distance between two points.
 * A single line is only single if it is continuous.
 * Any straight line can be used as a radius.
 * The right angle is a constant.
 * Parallel lines never cross.
 * Is it not the case, that Euclid simply formalised concepts like up is up, down is down, and describing anything unusual or complex relative to that, is the opposite of describing Euclidean geometry..? ~ R.T.G 14:13, 17 December 2019 (UTC)