Wikipedia:Reference desk/Archives/Mathematics/2016 April 13

= April 13 =

What is the strict/technical definition for geometric shapes? Does it include/exclude the area inside the shape?
This question apples to most (all?) geometric shapes. So, I will just use the circle as an example. Strictly speaking, when you define a circle, does that simply refer to the points that make up the circle shape? Or does it include/exclude the area "inside" the circle? Take a look at the picture on the top right corner of this page: circle. Is the "circle" (in the strict definition of the word), just the series of black "points" that encircle the center point? Or does the white space in the middle get included in the definition of circle? This same question would apply to other geometric shapes (square, triangle, rectangle, etc.). At least, the ones that "close" and form a "boundary", I assume. Thanks. Joseph A. Spadaro (talk) 22:19, 13 April 2016 (UTC)
 * Usually when people want to be precise they distinguish between a "circle" (just the boundary) and a "disk" (with the interior). Of course there are finer possibilities, e.g., both the "closed disk" (containing the interior and the boundary circle) and the "open disk" (containing the points on the interior but not the boundary circle itself).  In higher dimensions, one might distinguish between a "ball" (solid) and a "sphere" (just the boundary).
 * In general the right way to handle things like this is not to worry about definitions and instead to think about how I can express myself clearly. --JBL (talk) 22:23, 13 April 2016 (UTC)


 * Think about the quadratic equation for the circle: $$x^2+y^2=r^2.$$ Only the boundary points obey this equation, so the circle is only the boundary points. Likewise for the ellipse: $$x^2/a+y^2/b=1.$$ On the other hand, the convention for polygons is that they include the interior if the boundary is not self-intersecting. Loraof (talk) 23:08, 13 April 2016 (UTC)


 * People do say that x&sup2;+y&sup2;=r&sup2; defines a circle, but on the other hand they also say that this circle has an area of &pi;r&sup2;. In other words, the word "circle" is used both ways.  It should generally be clear from context which one is meant. --69.159.61.172 (talk) 09:53, 14 April 2016 (UTC)


 * In fact we have an article on the area of a circle when technically it should be the area of a disk. As long as we use a natural language such as English to talk about math, such ambiguities and hidden assumptions will crop up. Mathematical language is more precise than everyday language to avoid such ambiguities, but carrying this too far is a hindrance to communication. Natural language computer interfaces are programmed to resolve these ambiguities the way a human would, e.g. if you type "area of circle with radius 4" into Wolfram Alpha you get 16π, but interpreting the phrase literally the answer would be 0. --RDBury (talk) 10:15, 14 April 2016 (UTC)
 * right, changing the "area of a circle" article to "area of a disk" would be utterly insane and would never be approved even if one could make a coherent argument that it's technically proper...68.48.241.158 (talk) 13:04, 14 April 2016 (UTC)
 * I see they say "area of a disk" in the first sentence...I'm not sure that's even proper as a disk is a vaguely defined roundish object, doesn't have to be a circle.......?????68.48.241.158 (talk) 13:09, 14 April 2016 (UTC)
 * it should be "area of a circle's interior" if anything...68.48.241.158 (talk) 13:21, 14 April 2016 (UTC)
 * See Disk (mathematics). "In geometry, a disk (also spelled disc) is the region in a plane bounded by a circle. A disk is said to be closed or open according to whether it contains the circle that constitutes its boundary." -- ToE 14:19, 14 April 2016 (UTC)


 * well, that's fine then...but certainly "area of a circle" means what we all know that it means so saying it's erroneous would be redefining the language...it doesn't mean and it's never meant the area of the connecting line of points that make a circle or whatever....68.48.241.158 (talk) 14:34, 14 April 2016 (UTC)
 * Could you please restrain yourself from making comments that could not possibly be of interest to anyone? Or, at the very least, spend enough time on them to format them in standard sentence format?  Thanks in advance.  --JBL (talk) 15:11, 14 April 2016 (UTC)
 * the only thing technically not grammatical about what I wrote is probably the "or whatever" which should have had a comma before it...otherwise, I'm substantively discussing the notion brought up by RDBury that "area of a circle" if interpreted technically means something other than what we all know it to mean...I'd say this is entirely false and that the distinction he's trying to make is artificial and doesn't in fact exist...all of this relates to the OP..."area of a circle" equals "area of a circle's interior" by definition and it equals absolutely nothing else... (maybe you don't like all my (....) but I don't think it's somehow against the rules to do that....)68.48.241.158 (talk) 15:30, 14 April 2016 (UTC)
 * I think JBL is referring to your use of ellipses. Regarding the discussion though, I think you've missed the point about the difference between natural and technical language. While in natural language "area of a circle" is perfectly correct (to mean area enclosed by a circle), in technical language "area of a circle" is trying to describe the area of a line, which is invalid. Also, could you clarify what you meant when you said I'm not sure that's even proper as a disk is a vaguely defined roundish object, doesn't have to be a circle? A disk appears to me to be well defined as the region in a plane bounded by a circle. — crh 23   &thinsp;(Talk) 17:12, 14 April 2016 (UTC)
 * it appears the area interior to a circle has been defined mathematically as a "disk"....that's fine, it's just an arbitrary convention...the normal meaning of that word just means a roundish type object....but my point is that "area of a circle" has the same exact technical meaning as "area of a disk"...and is the phrase that everyone uses when referring to the concept that both refer to...there's nothing erroneous about it in the technical sense nor otherwise...it couldn't possibly meaning anything else and it's never meant anything else...in any event, I shouldn't particularly have responded to JBL as his comment was out of line and inappropriate...really, his comment should have simply been reverted...68.48.241.158 (talk) 19:42, 14 April 2016 (UTC)
 * No: your comments, which are rambling, poorly thought-out, and rendered without basic concern for things like completeness of thoughts (demonstrated superficially through your refusal to use sensible sentence structure, punctuation, and capitalization) are largely worthless. Someone should tell you this.  I am doing so in a rude way because I am in a bad mood (partly because I have just read a dozen rambling, structureless comments; but I digress) and someone who was more thoughtful than I would do it in a polite way.  But the fact that I am rude should not be allowed to disguise the facts that your comments show a lack of effort and thought on your part, and that you should put more effort into crafting them, both substantively and superficially.  --JBL (talk) 23:14, 14 April 2016 (UTC)
 * Not sure where to reply, but I thought I'd chime in. Arguably "Area of a circle" is wrong, and should be "Area enclosed by a circle".  Most people are not that pedantic about such niceties though.   S ławomir  Biały  19:51, 14 April 2016 (UTC)
 * my point is it's not wrong as the definition of the phrase "area of a circle" is "area enclosed by a circle." the phrase can be thought of as one word with a particular definition....68.48.241.158 (talk) 20:00, 14 April 2016 (UTC)
 * It is wrong for conventional meanings of "area" and "circle". The area (the two-dimensional Lebesgue measure) of a circle (the set of points $$x^2+y^2=r^2$$) is equal to zero for every circle.  If you want to define the phrase "Area of a circle" to mean something other than the meanings of the words appearing there, then that does not facilitate clear communication.   S ławomir  Biały  20:17, 14 April 2016 (UTC)
 * let's put it like this, do you think the name of the article "area of a circle" should be changed? there's no way consensus would be reached to do so...as thisis the phrase used for the concept and therefore means the concept....68.48.241.158 (talk) 20:25, 14 April 2016 (UTC)
 * The words "circle" and "area" have established meanings, and the intended meaning of the phrase "Area of a circle" does not correspond to the established meanings of its constitutive terms, arranged as they are according to the conventional rules of the English language. Just because many people use a phrase does not mean that it is strictly correct formal English.  We do not suspend the meaning of words in the sciences, as we would in a kind of dystopic Quinian gedanke that you would have us inhabit.  Sentences should mean what they say.   S ławomir  Biały  23:04, 14 April 2016 (UTC)
 * I agree with this to an extent. If you are writing a formal mathematics paper in a modern-day context, it is certainly better to say "area of a circular disk", if that's what you mean.
 * However, the notion of the "area of a circle" is a traditional geometric one that well predates modern notions of a circle being a set of points, or of "area" meaning "two-dimensional Lebesgue measure". Within the context of traditional plane geometry, it seems to me, it's perfectly fine phrasing.  It's only when you want to understand traditional plane geometry in the paradigm of modern mathematics that you run into any conflict. --Trovatore (talk) 23:16, 14 April 2016 (UTC)
 * let's put it like this, do you think the name of the article "area of a circle" should be changed? The article was titled Area of a disk until three days ago:
 * 03:44, 12 April 2016 (UTC) GeoffreyT2000 (talk | contribs) moved page Area of a disk to Area of a circle over redirect: Per WP:COMMONNAME.
 * See Talk:Area of a disk. -- ToE 12:40, 15 April 2016 (UTC)

I am the OP. So, can I safely assume that the above discussion also applies to the other geometric shapes (square, rectangle, triangle, etc.)? Not just circles, correct? Joseph A. Spadaro (talk) 17:03, 14 April 2016 (UTC)
 * I don't think the answer to that question is very clear cut. A circle is a curve, and a well studied one, so it has a very clear definition: the locus of points a given distance from a set point. Polygons are less clear: Wolfram MathWorld says that there is some contention over the issue. — crh 23   &thinsp;(Talk) 17:20, 14 April 2016 (UTC)


 * Thanks. But, huh?  Things like square, rectangle, and triangle are not clearly defined?  And they are contentious?  I find that hard to believe.  Really?    Joseph A. Spadaro (talk) 18:17, 14 April 2016 (UTC)
 * The answer to your question is: "In general the right way to handle things like this is not to worry about definitions and instead to think about how I can express myself clearly." If you have some particular idea you are trying to communicate to some particular audience, and you want to know whether you have expressed that idea in a way that is likely to be understood, I suggest you share the details. --JBL (talk) 18:59, 14 April 2016 (UTC)


 * In most contexts it doesn't matter whether you are talking about a square that includes or excludes its interior. It is simple to define them, but there is no consensus on which way to define them. As JBL said, it doesn't matter which way you use, as long as you clearly explain which definition you are using, if it matters (which it probably doesn't, if I'm honest). — crh 23   &thinsp;(Talk) 21:16, 14 April 2016 (UTC)


 * Just a few points here, responding to things I noticed in the above discussion:
 * Nothing is contentious about the definition of square, rectangle, triangle, in the ordinary sense of the word "definition". The only thing at issue here is whether you mean the curve, or the region of the plane enclosed by that curve.  In most cases this will be clear from context.
 * Don't rely on MathWorld for questions about terminology. The actual math in MathWorld is usually fine.  But sometimes they just make up names, or use names pulled from an obscure discussion without warning the reader that the name has no particular currency in the community.
 * Literally speaking, there is nothing wrong with "the area of a circle" when the circle is understood as a curve (1-d manifold). That area is perfectly well-defined.  It just happens to be zero.
 * Mathematics is not immune from using pragmatics as a key to understanding its terminology. If you encounter text that refers to the area of a circle, except in very special cases, it does not mean zero.  Why wouldn't the author just say "zero"?  See also Gricean maxim.
 * --Trovatore (talk) 22:13, 14 April 2016 (UTC)

OP here. I am asking what is the "strict" (precise) definition from a geometry/mathematical point of view. Thanks. Joseph A. Spadaro (talk) 17:41, 15 April 2016 (UTC)
 * From the strict modern perspective, the "circle" is just the curve, not the region of the plane it encloses, and its "area", strictly speaking, is zero. --Trovatore (talk) 17:48, 15 April 2016 (UTC)


 * And I assume the same for squares, triangles, etc.?  Joseph A. Spadaro (talk) 19:21, 15 April 2016 (UTC)
 * The answer to your question is: "In general the right way to handle things like this is not to worry about definitions and instead to think about how I can express myself clearly." There is no "the definition" of the word "square" or "triangle". --JBL (talk) 19:27, 15 April 2016 (UTC)


 * I find it pretty difficult to believe that mathematicians (geometry) do not have a formal definition for those geometric shapes. In fact, I don't believe that at all.  I am sure they are defined.  Probably very specifically and explicitly.  Mathematics is nothing, if not precise.  Joseph A. Spadaro (talk) 01:17, 16 April 2016 (UTC)
 * I think this discussion has run its course..as JBL stated, there's not necessarily a "the definition" (and this holds within any subject matter, really)..BUT mathematicians are entirely capable of very precisely and very technically stating what they are talking about within a given context...68.48.241.158 (talk) 13:56, 16 April 2016 (UTC)
 * Joseph A. Spadaro, what is "the definition" of "normal" in mathematics? Maybe it's at normal (mathematics).  JBL (talk) 14:42, 16 April 2016 (UTC)


 * Let's say that I am a high school student, and I open up a "standard" Geometry textbook, and I look up the definitions of "circle", "square", "triangle", and "rectangle", etc. The textbook is going to say "those words have no definition".  Or "those terms are ambiguous in geometry and, as such, we can't provide a good definition".  Oh, ok. I didn't realize that such simplistic terms like "circle" and "square" were so controversial.  So much so, that they cannot be defined to a math student who is studying them.  Oh, ok.    Joseph A. Spadaro (talk) 15:24, 16 April 2016 (UTC)


 * Sometimes, people take a simple issue/question and overly complicate matters. Not because they are trying to help answer the question.  But, because they like to hear themselves talk and they want to "show off" to "prove" how smart they are.  And when you say "complicated" things, that makes you "appear" to be"smart".  I guess that's the theory.  And, quite frankly, that's pretty common with "math" type of people.  Trying to outshine or outsmart the next guy.  Of course, not on this particular thread.  Of course.   Joseph A. Spadaro (talk) 15:28, 16 April 2016 (UTC)


 * Your belief that school-level textbooks can be expected to provide precise, consistent definitions seems baseless to me. (What is the high school geometry definition of "area", "point" or "plane"?) Your implicit belief that different school-level books will provide the same definition also seems baseless.  Sometimes the right answer to a question is "this is not the right question."  This is one of those times.  --JBL (talk) 15:37, 16 April 2016 (UTC)


 * I am no expert on the subject of "Geometry high school textbooks in the USA". But I am pretty sure that every high school geometry textbook in the USA looks pretty much like every other geometry high school textbook in the USA. Especially on such a simplistic matter (i.e., the definition of the most basic geometric shapes).  You state: Sometimes the right answer to a question is "this is not the right question."  This is one of those times.  That's pretty silly.  When the question itself is: what's the geometry definition of a simple word?   Joseph A. Spadaro (talk) 03:58, 17 April 2016 (UTC)


 * P.S. The concept of "controversy" is totally irrelevant here -- do you understand this? (Just as there is no "controversy" about the meaning of the word "normal.")  If you would like help understanding something I have written, please ask, I am happy to help. --JBL (talk) 15:41, 16 April 2016 (UTC)


 * As explained above, mathematicians define each shape precisely in plane geometry, but allow context to determine whether the definition applies to the boundary only, or to the area bounded by the shape. It's just convenient that way.  Where it's important to distinguish, for example in moment of inertia calculations, they refer to either a circular wire (assumed negligible thickness), or a circular disc.  The same applies to any other shape.  Sometimes the word "lamina" indicates that the area is to be considered.    D b f i r s   15:43, 16 April 2016 (UTC)


 * Sometimes, people ask questions on a volunteer forum, and when they don't like the answers they get, they start being rude, offensive and disrespectful, and engage in ad hominem attacks.
 * You've received great answers here, if you don't see that, you need to think about it and try to figure out what it is that you're missing - and if necessary, ask followup questions politely.
 * For my own part, I'll emphasize one thing - it is very common in mathematics for a term to have a rigorous, consistent definition within a specific textbook/class, but that the definition will be different for different textbooks. Even for things which are relatively clear-cut, there are always generalizations which have a different formal definition. So it is often folly to seek "the one true universal definition" of a given term. It's more important to understand the underlying concepts and communicate clearly. -- Meni Rosenfeld (talk) 17:47, 16 April 2016 (UTC)
 * The area(s) along any abstract line(s) and curve(s) are zero simply because these are one-dimensional. Of course, my calculus textbook, which is quite modern and good, has a table of "Formulas from Geometry" which gives the two-dimensional area enclosed by a circle as pi*r^2. BTW, back when I was taking computer graphics at NCSU, I managed to improve slightly Bresenham's decision algorithm for drawing circles that was being taught at the time (by factoring out the repeated multiplication by two, I worked out the math to show why it could be done, and which was an interesting exercise in modular arithmetic). Here is an article on its current form the older algorithm I think, but I need to relocate and check my work to be sure, and with it you only draw the pixels located nearest to the circle (other pixels such as those in the interior are ignored).  -Modocc (talk) 19:15, 16 April 2016 (UTC)
 * Merriam-Webster's Dictionary has more than one definition for the circle:

a : ring, halo b : a closed plane curve every point of which is equidistant from a fixed point within the curve c : the plane surface bounded by such a curve --Modocc (talk) 22:11, 16 April 2016 (UTC)

Thanks, all. Joseph A. Spadaro (talk) 03:56, 18 April 2016 (UTC)