[Jprogramming] "n-volume" of an "n-sphere"

Wed Aug 23 10:50:58 UTC 2017

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-bounces at forums.jsoftware.com]
> On Behalf Of Murray Eisenberg
> Sent: maandag 21 augustus 2017 17:56
> To: programming at jsoftware.com
> 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" <r.e.boss at outlook.com
> <mailto:r.e.boss at outlook.com>> 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-
> bounces at forums.jsoftware.com
> >> <mailto:programming-bounces at forums.jsoftware.com>]
> >> On Behalf Of Murray Eisenberg
> >> Sent: zondag 20 augustus 2017 16:33
> >> To: programming at jsoftware.com <mailto:programming at jsoftware.com>
> >> 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 <dhky at shaw.ca
> >>> <mailto:dhky at shaw.ca>
> >> <mailto:dhky at shaw.ca <mailto:dhky at shaw.ca>>> 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                                murray at math.umass.edu
> Mathematics & Statistics Dept.
> Lederle Graduate Research Tower      phone 240 246-7240 (H)
> University of Massachusetts
> 710 North Pleasant Street
> Amherst, MA 01003-9305
> 
> 
> 
> 
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