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Why is a sphere in an $n $-dimensional space called $(n-1) $-sphere? Isn't it natural to call a sphere in 3D a 3-sphere, a sphere in 2D (i.e. a circle) a 2-sphere, etc?

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  • $\begingroup$ It is the dimension viewed as a manifold $\endgroup$ May 18 '14 at 11:52
  • $\begingroup$ It takes only two coordinates to specify a point on a sphere $S \subset \mathbb R^3$, hence $S$ is two dimensional. A plane in $\mathbb R^3$ is certainly two dimensional, and a sphere is also. $\endgroup$
    – littleO
    May 18 '14 at 11:56
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tl;dr: You are describing the sphere itself, not a particular space in which it is represented.

It should first be made clear the distinction between a sphere and a ball: a sphere consists of the surface only, whereas a ball consists of both the surface and the contained volume.

Now that that's out of the way, then there's the question of how you define a sphere. A sphere in n-dimensional space with radius $r$ centered at $(c_1,c_2,\cdots,c_n)$ is very easily represented by $(x_1-c_1)^2+(x_2-c_2)^2+\cdots+(x_n-c_n)^2=r^2$. However, this is an implicit representation, not an explicit representation. An explicit representation entails a mapping of some other space, and in the case of a sphere in $n$-dimensional space, it happens to be the case that the lowest possible dimension of this other space is $n-1$. As such, spheres in $n$-dimensional space are said to be $n-1$-spheres.

The reason for this nomenclature is that this creates a description specific to the sphere itself and not to a specific environment of the sphere in which it is convenient to describe.

Consider a circle, that is, a $1$-sphere, for instance. Certainly, the equation $x_1^2+x_2^2=r^2$ may come to mind, and there are only two variables in this equation, $x$ and $y$, which might inspire you to call this a $2$-sphere. But a circle can also be represented in $\mathbb{R}^3$, $\mathbb{R}^4$, and so on (although of course, you would need to fix $x_3, x_4$ and so on as well). Why, then, should this circle not also be called a $3$-sphere or $4$-sphere? For that matter, is a $3$-sphere a circle or sphere?

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To expand a little - a small piece of a circle is like a line, a small piece of the surface of a sphere is like a plane.

The only way that a line can live in one dimension is if it is straight - a curve can live in many dimensions and still look like a line when you look at a short piece.

Likewise a surface can only live in two dimensions if it is flat. Surfaces exist in many dimensions, but they look like a plane when you take a small enough piece.

If we are creating a coherent useful idea of dimension, it turns out that we want a curve to be one-dimensional and a surface to be two dimensional - but what do we mean by a curve or a surface if we have many-dimensional space?

The idea of a manifold captures this - if our curve is locally like a line, and the local pieces can be coherently glued together to make the whole, it is a one-dimensional manifold. Likewise a surface is a two dimensional manifold if it looks locally like a plane wherever you look and you can glue the local pieces together in a coherent way.

There are some technicalities about precise definitions, to make these intuitive ideas rigorous. But that is why the dimensions come out as they do.

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Because sphere in $n$-dimensional space is $(n-1)$-dimensional object.

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  • $\begingroup$ So, we include the temporal dimension aswell? $\endgroup$
    – user132181
    May 18 '14 at 11:49
  • $\begingroup$ I do not know what is temporal dimension, but sphere in an $n$-dimensional space is a $(n-1)$-dimensional manifold. This is waht I meant in the answer. $\endgroup$
    – Antoine
    May 18 '14 at 11:51
  • $\begingroup$ ball ≠ sphere. The surface of the ball exists in 3D but is is a 2 dimensional object in the sense that you only need 2 real numbers to specify the point on the sphere. $\endgroup$
    – Shady_arc
    May 18 '14 at 12:12
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This is because the "$n-1$" sphere is the surface of the $n$ dimensional solid contained in the $n-1$ sphere, I.e it is one dimension lower.

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