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:[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
