The question is "Is an n-disk a well-established mathematical term?".
A disc originally had only meaning in 3 dimensions, being " a circular flat object", e.g. a cd or compact disc; http://dictionary.cambridge.org/dictionary/english/disc. Also: a disk is normally "a flat, circular device that is used for storing information"; http://dictionary.cambridge.org/dictionary/english/disk. You reference to mathworld is not supported by other references on that page, so is probably a private generalization of mr. Weisstein. In any case not convincing. I respond with https://www.encyclopediaofmath.org/index.php/Disc where a disc is defined as "The part of the plane bounded by a circle and containing its centre." So they stick to the 2-dimensional case. Also a topological disc is 2-dimensional https://www.encyclopediaofmath.org/index.php/Disc,_topological. Another reference is https://www.wikiwand.com/en/Disk_(mathematics) which states "In geometry, a disk (also spelled disc)[1] is the region in a plane bounded by a circle. " Your reference to math.stackexchange is also an instance of one person using this term once, not convincing at all. John Lee in his book explicitly states on the pages indicated: "In the case n=2, we sometimes call B^2 the (open) unit disk. (...) We sometimes call B^2 the closed unit disk." (where this B has a superbar, which is not copied) So he also restricts a disk to 2 dimensions. Finally, your reference concerning CW-complexes. Perhaps in that, rather restricted area of mathematics it seems disk is used as synonym for ball, see also https://www.wikiwand.com/en/CW_complex. My final conclusion is that an n-disk is NOT a well-established mathematical term, apart from n=2 and, which I cannot decide, perhaps in the area of CW-complexes. R.E. Boss > -----Original Message----- > From: Programming [mailto:[email protected]] > On Behalf Of Murray Eisenberg > Sent: maandag 21 augustus 2017 17:56 > To: [email protected] > Subject: Re: [Jprogramming] "n-volume" of an "n-sphere" > > To the contrary, “n-disk” is a well-established mathematical term, for > arbitrary dimension n = 1, 2, 3, 4, .... See, for example, > http://mathworld.wolfram.com/Disk.html > <http://mathworld.wolfram.com/Disk.html> > (which, alas, mangles the distinction between “disk” and “ball”). > > [Often, for emphasis or disambiguation, the terms “closed n-disk” or “closed > n-ball” are used for the set of points in Euclidean n-space of distance at > most > r from a given point; and then “open n-ball” for the set of points of distance > strictly less than r. (And, in fact, the terms “disk” and “ball” are well- > established even, more generally, for arbitrary metric spaces.)] > > For a typical recent instance of the usage, see: > > https://math.stackexchange.com/questions/24785/the-n-disk-dn- > quotiented-by-its-boundary-sn-1-gives-sn > <https://math.stackexchange.com/questions/24785/the-n-disk-dn- > quotiented-by-its-boundary-sn-1-gives-sn> > > The usage is defined in many sources. See, e.g.: > > John Lee, Introduction to Topological Manifolds, 2nd ed., pages 21-22. > > Soren Hansen, "CW complexes”, page 1 > (https://www.math.ksu.edu/~hansen/CWcomplexes.pdf > <https://www.math.ksu.edu/~hansen/CWcomplexes.pdf>). > > > > On21 Aug 2017 11:15:20 +0000,"R.E. Boss" <[email protected] > <mailto:[email protected]>> wrote: > > > > AFAIR disk is not a mathematically defined term for high dimensions. > > Ball is as you defined it, where one can argue whether the distance is > "strictly less" or " at most" r. > > Sphere is the definition where the distance is equal to r. > > > > > > R.E. Boss > > > > > >> -----Original Message----- > >> From: Programming [mailto:programming- > [email protected] > >> <mailto:[email protected]>] > >> On Behalf Of Murray Eisenberg > >> Sent: zondag 20 augustus 2017 16:33 > >> To: [email protected] <mailto:[email protected]> > >> Subject: Re: [Jprogramming] "n-volume" of an "n-sphere" > >> > >> Re Don Kelly’s comment: > >> > >> NO the ball of radius r at a point p in n-space is the set of all > >> points of distance strictly less than than r from p; > >> the disk of radius r at a point p in n-space is the set of > >> all points of distance at most r from p. > >> > >> “Infinitesimal” has utterly nothing to do with it, nor does > >> transfinite math (although the set of points in such a ball, or such > >> a disk, is definitely infinite and, in fact, uncountable. > >> > >> Re Jimmy Gauvin’s comment: > >> > >> It is utterly trivial to prove that a ball (or disk) in euclidean > >> n-space is convex. It requires nothing more than what is commonly > >> taught in standard courses in sophomore linear algebra today. > >> Specifically, basic properties of the euclidean norm, including the > >> triangle > inequality. > >> > >>> On 2Sat, 19 Aug 2017 18:03:49 -0700,Don Kelly <[email protected] > >>> <mailto:[email protected]> > >> <mailto:[email protected] <mailto:[email protected]>>> wrote: > >>> > >>> If one considers a point as infinitesimal -as usually considered-, > >>> then we have an infinite number of points at an infinitesimal > >>> distance from the origin and at a larger distance from the origin > >>> there are still an infinite number of points on the surface and an > >>> infinite number of points enclosed . Isn't this getting into transfinite > math? > >>> > >>> What's the point? > >>> > >>> Don Kelly > >>> > >>> > >>> On 2017-08-15 8:23 PM, Jimmy Gauvin wrote: > >>>> The construction of the sphere implies it cannot be convex but you > >>>> will have to find a topologist to prove it to you. > >>>> > >>>> The sphere is the collection of points whose distance to the origin > >>>> is equal to the radius of the sphere. > >>>> > >>>> The ball or volume is comprised of the points whose distance to the > > —— > Murray Eisenberg [email protected] > Mathematics & Statistics Dept. > Lederle Graduate Research Tower phone 240 246-7240 (H) > University of Massachusetts > 710 North Pleasant Street > Amherst, MA 01003-9305 > > > > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
