I can from experience (>20 years) confirm n-(dimensional)-disk and n-(dimensional)-ball are used interchangeably in mathematics, at least here in Bonn, Germany among these people here:
http://www.hcm.uni-bonn.de/people/faculty/ https://www.mpim-bonn.mpg.de/taxonomy/term/4 https://www.mpim-bonn.mpg.de/peoplelist?pltype=0&plgroup=visitors http://www.math.uni-bonn.de/members?mode=struc Sometimes, we use the terms n-ball vs (n-1)-disk to talk about an n-dimensional ball vs the intersection of an (n-1) dimensional hyperplane with an n-ball. Am 23.08.2017 um 21:13 schrieb Roger Shepherd: > Is there something specifically wrong with common sources of definitions such > as > http://dictionary.sensagent.com/N-sphere/en-en/ > > Just checking > > Sent from Mail for Windows 10 > > From: Jimmy Gauvin > Sent: Wednesday, August 23, 2017 9:20 AM > To: [email protected] > Subject: Re: [Jprogramming] "n-volume" of an "n-sphere" > > 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 > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
