What is a dimension?
I was wondering if you could give me a solid definition of what a dimension is. I have looked in books but all of their definition are vague if you could help me out I would be most grateful.
Okay, the mathematical definition is something like, "The dimension of a manifold is the minimum integer number of co-ordinates necessary to identify each point in that manifold." A manifold is essentially a generic term for "area, volume, etc." What this means in English is that a surface has, say, two dimensions, if you need two numbers to tell someone where you are on that surface. For instance, the surface of the Earth is two dimensional, because you need two numbers (usually longitude and latitude), to describe your position. A line or the circumference of a circle is one-dimensional, since you only need one number (distance along the line; angle on the circle) to define where you are. The interior area of the circle is two-dimensional, since you need both angle and radius to define your position there. You may have heard about chaos and fractal shapes. Fractal shapes are interesting because they can have non-integer dimension (hence "fractal"!). These non-integer dimensions are defined in a very complicated way somewhat different from what I have said above, but perhaps you can imagine a situation in which one number is not enough, but two is a little more than you need.
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