My fellow AP-C students and I are working on the Work-Energy unit right now, and in the Webassign there are some questions involving the dot product of vectors. The maximum amount of elements these vectors have is 3, though: <x,y,z> or <i,j,k>. Well, this makes sense, since we live in a 3-dimensional world, of length, width, and depth. Or do we?
Obviously, the concept of 3 dimensions has been around as long as mathematics (even in its most rudimentary of forms) has been around. It's obvious because it's what we see, and touch, and live. Trying to imagine a 4th dimension is like trying to imagine a color we've never seen before - it's impossible for our brains to comprehend.
Mystics used to describe the 4th dimension as a sort of "spirit realm" - a place free of Earth's bounds and restrictions. With the advancement of science and mathematics, many began to view time as the 4th dimension. It makes some sense - just like we humans are constrained to length, width, and depth, we're constrained to constant increments of time (well, some believe it's not constant...but I'll get to that some other time).
(May or may not be inspired by the theory of relativity)
In the theory of relativity, Einstein describes time as a fourth dimension. Space being inseparable from time, spacetime became known as its own continuum, but time is mathematically treated differently from the other spatial dimensions. After all, we can move in all directions 3-dimensionally, but only forward in time. For now.
However, some more modern scientists such as H. S. M. Coxeter state that using time as a 4th dimension is a cop-out. (Paraphrasing) In this regard, there is a spatial/Euclidean description of the fourth dimension: the vector <x,y,z,w>. Try to imagine this: you know that x, y, and z are all perpendicular to each other. Well, w, the fourth dimension, is perpendicular to all of these. It's pretty much impossible to comprehend, since we live in a 3-D world, but its properties can be inferred. Using dimensional analogies, we can see how 3 dimensions relates to 2, or 2 to 1, and infer how 4 would relate to 3. This can be used for even more dimensions - string theory, for example, relies on the existence of 10.
There are mathematical limitations to these inferences. Here's one example:
Circumference of a circle = 2Ï€r
Surface area of a sphere = 4Ï€r^2
So what's the surface volume of a hypersphere, a fourth dimensional figure? Using dimensional analogies, we might be tempted to say 8Ï€r^3, assuming it's multiplied by 2r each time. However, the real answer is apparently 2Ï€^2*r^3. Don't ask me why, I didn't derive the equation.
I can't do this subject justice in a concise blog post, or probably with any. It's incredibly complicated and I don't wish to oversimplify - so research further if you're interested!