Vtyjkyuk

According to the definition, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere.[given on wikipedia]

Let us take the case of the sphere.

My question is: When we locate a point we start from the origin. So if we want to locate a point on the spherical surface,first we would have to move the distance=radius and reach on the surface, then only we can use latitudes and longitude or we can use the concept of $theta$ and $phi$. No doubt radius is given for a spherical surface but still we can't reach the surface from the origin without knowing its value. We need minimum three coordinates to locate a point. So, shouldn't the dimensions of a spherical surface be three even though one dimension(radius) is fixed?

Similarly, I have read somewhere that a circle in a plane is one-dimensional because we only need the length along the circumference to reach a point or (say) we need $phi$ to reach a point. But even in this case we start from the origin and need radius so it should also be two-dimensional even though one dimension(radius) is fixed.

All that I have said becomes wrong if we are already on the surface or on the arc of the circle but I can't find a reason for it.

I think the key issue here is the following:

No doubt radius is given for a spherical surface but still we can't reach the surface from the origin without knowing its value.

This is an important point which can be slippery in the "number of coordinates" definition: to what extent are we allowed to "bring information in from outside?"

The right way to think about this is that you know the surface ahead of time. E.g. I tell you "I'm thinking of a point on the unit sphere." Now, you can find my point by asking two questions, namely latitude and longitude, because the additional information was built into the "context" of the problem.

Thinking of the situation as a game can be - in my experience - quite helpful. For example, if you take the surface of a seventeen-holed torus instead of a sphere, there aren't nice words like "longitude" or "latitude" which apply very well, but you can cook up your own rather artificial pair of coordinates to solve the problem for you, and this feels (to me at least) more natural if you think of yourself as "playing against" some imaginary opponent who's hiding a point.

Now that I've hopefully clarified things a bit, let me make the situation worse.

We can cook up a bijection $f$ between $mathbbR$ and the surface of the sphere. Once we've done this, we only need one number to locate a point. Indeed, we can do this for all of three-dimensional space, or all of fifteen-dimensional space, or ... So does every shape have dimension $1$?

In some sense this approach is cheating - especially since $f$ is pretty ugly! - but it's not necessarily clear at first why this is wrong. Ultimately I find the "how many coordinates do you need" approach to be a bit problematic, for this reason (although one's mileage may vary). For first learning about dimension, I prefer the "local character" approach: if you were a very small ant on the shape, what sort of space would you think you were living in?

For example, if I'm on the surface of a sphere and I'm very small (or the sphere is very large), I'll probably think I'm in $mathbbR^2$: the sphere is "locally flat." We see this in everyday life (the earth isn't actually flat). It's easy to see that the same holds of any other "surface" (e.g. torus, Klein bottle, projective plane, ...) - in fact, this is what a surface "really is." Thinking along these lines will lead you to the notion of a manifold - this is just a space which is "locally Euclidean" (and perhaps enjoys other properties if we stick a fancy adjective in front), and figuring out a precise way to define this and its various cousins is a fundamental first step towards "general geometry."

All you need to know to locate a point on the surface of the Earth is the latitude and the longitude. You don't need to know where the center of the Earth happens to be in its orbit around the sun, or the orientation of the north-south axis. The surface of the Earth (mathematically, essentially a sphere) is two dimensional, so two coordinates will do.

The dimension of an object is an inherent property of the object, independent of how it happens to be embedded in some space. A circle in three dimensional space is still just one dimensional, although it would take more numbers to describe just where that circle is in three space.

A one dimensional object has length but not area.

For example a curve on the plane or in the space is not considered two or three dimensional it is still a one dimensional manifold. You can describe it with just one parameter.

A two dimensional object has surface area not volume. The surface of a given sphere is a two dimensional manifold because you only need two parameters to describe it and it does not hold any volume.

The solid sphere or a ball is a three dimensional object since it has volume.