With so many people you can provide a long list of papers which support your memory, I suppose.
R.E. Boss > -----Original Message----- > From: Programming [mailto:[email protected]] > On Behalf Of Jens Pfeiffer > Sent: donderdag 24 augustus 2017 07:56 > To: [email protected] > Subject: Re: [Jprogramming] "n-volume" of an "n-sphere" > > 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:programming- > [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 ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
