Random Thought 001 - Hypercubes

In contemplating the very overused (and never applied) phrase "think outside the box," I was wondering if one could say, "Think outside the three dimensional hypercube." This of course leads to the thought about how a hypercube is defined. Clearly a definition could include some orthogonality in the lines created by each set of points. Here are some thoughts on the characteristics of three dimensional parallelpipeds and some ponderances on how these can be extrapolated to higher spacial dimensional hypercubes.

1) Each line created by a set of vertices is parallel to three other lines created by vertices. It is also perpendicular to four of the other lines created by vertices. It does not intersect any of the other four lines, nor is it parallel to any of those.

2) In a two dimensional world, the "parallelpiped" is a rectangle and has four sides and four lines making up the edges. This is 2^2, 2*2 or 2+2. But in three dimensions there are six sides and 12 lines. Six sides could be 2*3 indicating the pattern is 2*(number of dimensions). Hmm... might there be some combination/permutation thing going on here?

Lunch time is over. Somebody comment, please, this is driving me nuts.

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