Even in Mathworld it is not clear that disk is in common usage. from http://mathworld.wolfram.com/Ball.html we have :
The [image: n]-ball, denoted [image: B^n], is the interior of a sphere <http://mathworld.wolfram.com/Sphere.html> [image: S^(n-1)], and sometimes also called the [image: n]-disk <http://mathworld.wolfram.com/Disk.html>. (Although physicists often use the term "sphere <http://mathworld.wolfram.com/Sphere.html>" to mean the solid ball, mathematicians definitely do not!) and from http://mathworld.wolfram.com/Disk.html we have : The [image: n]-disk for [image: n>=3] is called a ball <http://mathworld.wolfram.com/Ball.html>, and ... On Wed, Aug 23, 2017 at 6:50 AM, R.E. Boss <[email protected]> wrote: > 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 > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
