What is the difference 2-Sphere and a 3-Sphere?

Question

What makes a 3-Sphere 3D? Why is a 3rd number required to measure a point on it?

Answer 1

A 2-Sphere is your ordinary sphere, like the cardboard world globe map.  It has a 2-D surface, meaning that it only requires 2 dimensions, for example latitude and longitude, to locate any point on it.  A 3-Sphere, on the other hand, is the next higher dimensional version of the 2-Sphere, and really exists in 4D space.  In order to locate any point "on its surface", 3 dimensions would be required.  It is not the same as the solid spherical ball in 3D space, even though it also takes 3 dimensions to locate any point inside the solid spherical ball.  You might wnat to have a look at the Wiki article on n-spheres.  

Answer 2

Here is one way to think of it.

The standard 2-sphere in standard coordinates (the sphere of radius 1 centered at the origin) is the set of points in R^3 (three dimensional space) that satisfy the equation:

x^2 + y^2 + z^2 = 1

Since the general point in R^3 requires 3 coordinates to specify the point, and we have one equation, it follows that points on the 2-sphere require 2 coordinates to specify uniquely.  This is made precise using the definition of an n-dimensional manifold.  One can find consistent coordinate systems for pieces of the sphere, so that the overlapping coordinate systems behave reasonably.  See: http://en.wikipedia.org/wiki/Manifold

The standard 3-sphere would be the set of points on the hypersphere of radius 1 in R^4:

x^2+y^2+z^2+w^2 = 1

Again since the general point in R^4 requires 4 coordinates to specify the point, and we have one equation, it follows that points on the 3-sphere require 3 coordinates to specify uniquely.