Talk:Hypersphere

Embedding

A hypersphere is a hypersurface so it is presumed an embedding in a higher dimensional space. On the other hand, the n-sphere may be taken as the global space of a study such as Conformal geometry#The Euclidean sphere, where the term hypersphere is used. Therefore it was thought necessary to distinguish the two with a separate article. There is a contrast with the case hypercube, which receives redirect from n-cube, as this context connotes expanding out, as in hyperspace. For the sake of inversive geometry#Inversion in higher dimensions the parallel embedded status of hyperplanes and hyperspheres is important.Rgdboer (talk) 02:10, 20 February 2015 (UTC)

Relation to n-sphere

I agree with Tamfang's comment on Rgdboer's talk page. As the articles n-sphere and Hypersphere stand now, they define these two things in the same way. From n-sphere:

an n-sphere of radius r is defined as the set of points in (n + 1)-dimensional Euclidean space which are at distance r from a central point.

a 1-sphere is the circle, which is the one-dimensional circumference of a (two-dimensional) disk in the plane,

From Hypersphere:

The surface of the hypersphere is a manifold of one dimension less than the ambient space.

A circle ... is a hypersphere in the plane.

Seems identical to me. Clarification, please? Loraof (talk) 22:45, 14 April 2015 (UTC)

I think the key word is Euclidean. Personally I would expect typing "hypersphere" to take me to the contents of the n-sphere article. --Rumping (talk) 08:00, 7 October 2015 (UTC)

I think user:Rgdboer's point in #Embedding above is valid. The key distinction is that a hypersphere is defined as a subset of the points of a Euclidean space, whereas an n-sphere is a manifold that has properties that allow it to be embedded in a Euclidean space of dimension ≥n+1. Ergo, the properties of the embedding space at other points, including its dimension, are not relevant to the definition of an n-sphere, but are to the definition of a hypersphere. The article n-sphere should be clarified in this regard. —Quondum 21:33, 23 July 2016 (UTC)

Well, the thing to do would be to quote some kind of embedding theorems, for finite-dimensional real/complex manifolds, and give counterexamples e.g. for non-differentiable manifolds, or talk about spheres in infinite-dimensional hilbert spaces or banach spaces or something, because, yes, the knee-jerk reaction is that this is some kind of strange neologism that no one actually uses. 67.198.37.16 (talk) 21:24, 7 May 2019 (UTC)

Illustration

The graph illustrating the article is nonsensical. You cannot compare volumes of different dimensions.

The graph displays the ratio between the volume of a hypersphere and the volume of a hypercube with a side equal to the radius of the hypersphere. This is an arbitrary choice. If you use instead the diameter as side, i.e. compare the volume of the hypersphere to the volume of the surrounding box, then you get a different graph which is starting at volume 1 and decreasing. The maxima at 5 and 7 have no meaning. They depend on an arbitrary choice.

In short the graph is mathematically confusing and it doesn't show any fundamental property of hyperspheres, so it should be removed.

imho Florian Fischer — Preceding unsigned comment added by 178.238.166.202 (talk) 08:00, 25 January 2021 (UTC)