I want to comment on Mitchell Porter's post above, regarding what exactly is meant by an n-sphere.There are a couple of difficulties with using "geometer's convention" (see above).
Firstly, if we decide that the dimension of a sphere (or a manifold in general) is to be the dimension of the ambient space that it is embedded in, we quickly run into problems. A circle, for a example, could be embedded in a plane or 3-space or 4-space or even 18 dimensional space. However, having an 18 dimensinal circle is highly unsatisfying. Thus we better be more precise, and define this extrinsic dimension to be the dimension of the *smallest* euclidean space that supports an embedding of our manifold. It is a fact that for a circle this dimension is 2, for a sphere it's 3, and for an n-sphere it is n+1.
It turns out, however, that this kind of dimension could be quite hard to compute. Even for a simple thing such as a circle, a proof that one cannot embed a circle into a line is required. It surely seems obvious from an intuitive standpoint, but mathematically, it is not all that trivial. Things get even harder when one tries to prove the same for a 2-sphere (i'm using a "topologist's convention" here). Can you prove that a 2-sphere cannot be embedded in a plane? It's not easy. Finally, to convince you of non-triviality of this problem, let me ask if a 7-sphere can be embedded in 7-space? I, for one, cannot visualize either a 7-sphere nor a 7-space, so the fact that a 7 sphere does not embed into 7-space, certainly cannot be visualized by mere mortals such as myself.
Secondly, let me also point out that certain surfaces that are topologically 2-dimensional (that is, locally they look like a plane), cannot be embedded into a 3-dimensional space. A projective plane and a klein bottle are examples of these strange surfaces. They can, however, be embedded in 4 space. Thus, for these surfaces, the extrinsic dimension, defined above, would be 4.
In conclusion, let me mention that a theorem of Whitney states, roughly, that any n-dimensional smooth manifold can be embedded in a 2n-dimensional space. It turns out that this is in fact the best estimate.
For these reasons I urge everyone to stick to the "topologist's convention" and call a sphere - 2-dimensional and a circle - 1-dimensional. After all, we don't want a knotted circle, a sphere and a solid ball to all be 3-dimensional! But then again, I speak as an aspiring topologist :)
Cheers,
Michael
-- (February 11, 2005) on Pi
Very nice webpage, Eve. I thought I'd mention that a 3 dimensional sphere is a boundary of a 4 dimensional ball and lives in 4 space. Thus the volume of it is not the same as the volume of a 3-ball (equivalently the inside of a 2-sphere) which is 4/3*pi*r^3 (i think :)). By the way, the actual volume of a 3-sphere of unit radius is 4pi^2/2!! = 2pi^2Best,
Michael
-- (February 9, 2004) on Pi