Dimension Explained in 5 Minutes

Dimension Explained in 5 Minutes

There are many mathematical definitions of dimension. It's an important concept. Here's one way to think about dimension.. it allows the description of some infinite sets in terms of a finite number of parameters.

What? You were probably expecting a statement like..

"We live in a 3-dimensional world", or perhaps "Einstein discovered the 4th dimension".

Both these statements are meaningless without some definition of dimension. Here's one definition..

Suppose I have an infinite set of mathematical objects S={a,b,c,...}

What type of objects? Any type, it does not matter.

Now suppose you ask me to describe each object. There are an infinite number, and they are all different, so I cannot do it. At least not within a finite amount of time. I have a set and I can't describe it. Not a good situation!

But after investigating my objects I notice something interesting.. it's possible to produce a representation of each object that uses only a finite number of parameters. It's true that each of the parameters can assume an infinite number of values, but there are only a finite number of parameters. Let's call them p1,p2,p3,...,pn

So, any object in the set S can be completely specified by giving the value of its parameters..

(p1,p2,p3,...,pn)

Let's suppose n is the smallest number of parameters than can be used to describe the objects, and let's suppose that all the n parameters are independent. This simply means no parameter has any relationship with any other.

Now I have dramatically simplified the description of my set. Now, when you ask me to describe any object I simply give you the list of n parameters for that object and I'm done!

Mathematicians say the set S has n dimensions and call the set an "n-dimensional space".

Clearly, this definition of dimension only applies to some sets, not all sets. But when it does apply it's incredibly powerful.

This is the abstract definition. Here's a few real world examples..

Consider the set of all points on a sheet of paper. This is an infinite set. But it's possible to draw a coordinate system so that every point can be represented by just two parameters, x and y. So the set of all points on the sheet of paper is suddenly simplified. It's a 2-dimensional set because all I need are two parameters to describe any object in the set.

The set of all points on the surface of a sphere is 2-dimensional.

The set of all points on the circumference of a circle is 1-dimensional.

The set of all temperature measurements in the atmosphere is 4-dimensional.
That's 3 spacial dimensions to specify the point in the atmosphere and 1 extra to specify the temperature at that point.

The set of all events that Einstein used in Special Relativity is 4-dimensional.
That's 3 spacial dimensions to specify the location of the event and 1 extra to specify the time of the event.

This definition of dimension is very general, but it assumes the number of dimensions n is a positive integer. Needless to say, mathematicians have produced other definitions of dimension that are not so restrictive!

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Content written and posted by Ken Abbott abbottsystems@gmail.com