Understanding Surface Dimensions for Beginners

[Red_CCF]

How do we define the dimension of a surface? I know surfaces are 2-D but I don't really get where that comes from.

 

[Office_Shredder]

There are a couple of ways to define dimension. One common one (which explains why surfaces are 2 dimensional) is that around any point if you take a small enough open subset of your surface, there is a homeomorphism (a continuous bijection with a continuous inverse) from that piece of the surface to an open subset of the plane (or R³ or some other power of the reals depending on the dimension of your set). This is the idea of a manifold:

http://en.wikipedia.org/wiki/Manifold

For a surface, you probably have a parametrization which basically describes how to form these homeomorphisms either immediately or with only a little bit of work

 

[HallsofIvy]

Perhaps more importantly, how do you define "surface"?

 

[Red_CCF]

Perhaps more importantly, how do you define "surface"?

I don't really have a formal definition, but I would say something like a plane or a hollow sphere on R3 (ex. x^2+y^2+z^2 = k) is a surface.

 

[blue_raver22]

The dimension of a surface is defined as the number of coordinates needed to specify a point on that surface. In other words, it is the minimum number of variables required to describe the location of a point on the surface.

Surfaces are commonly referred to as 2-dimensional because they can be represented on a 2-dimensional plane, such as a piece of paper. This is because a surface can be defined by two variables, typically referred to as x and y coordinates. These coordinates can be used to locate any point on the surface, thus making it 2-dimensional.

One way to visualize this is to think of a flat surface, such as a tabletop. To locate a point on the tabletop, you would need to provide two coordinates, such as the distance from one edge of the table and the distance from the other edge. This concept can be extended to more complex surfaces, where additional coordinates may be needed to fully describe the location of a point.

In summary, the dimension of a surface is determined by the minimum number of coordinates needed to describe a point on that surface, and surfaces are typically referred to as 2-dimensional because they can be represented on a 2-dimensional plane. I hope this helps clarify the concept for you.