User blog:Edwin Shade/Higher-Dimensional Spaces

We begin by defining a space in which there exists an infinitude of points that all lay on one axis. This of course, is one dimensional space, and a point in it can be expressed using one number enclosed in brackets. (i.e. [5] or [-18.1])

Next there is two-dimensional space which consists of the set of points definable relative to two mutual orthogonal axis. An interesting property of two-dimensional space is that is it possible to construct a line bent in such a way that a point moved across that line eventually loops back to where it started. This is a very well known fact, and the shape of the bent line is a circle. We can take this further however, by conceiving of a 3-D sphere, which loops 2-D space on itself, so that an ant on the surface of this sphere would eventually loop around, like a game of Pac-Man. Of course, we do not have to stop here, and we can define the concept of higher dimensions by using this concept: The (n+1)th dimension is the lowest dimensional space in which it is possible to construct a nth dimensional manifold that loops on itself.

This definition is important when we reach $$\omega$$ space, or the space of infinite dimensions. It would seem as if we are at an impasse here, but in fact we are not. We can easily define $$\omega+1$$ space as the lowest dimensional space in which it is possible to construct an $$\omega$$ space manifold that loops back on itself. We've clearly transcended anything that might be visualized or drawn geometrically, but none the less we can analyze mathematically the properties of $$\omega+1$$ space.

For instance, just as one must use n-coordinates to express a point in nth dimensional space, one can use a single coordinate in union with an infinite set of coordinates to express a point in $$\omega+1$$ space. (i.e. $$[a_1]\cup [b_1,b_2,b_3,...,b_\omega]$$)

[I'll finish this up tomorrow.]