Beyond Three-Dimensional GeometryDate: 06/17/2001 at 21:27:44 From: Tanner Wilson Subject: Beyond 3-dimensional geometry If the first dimension is a line and the second dimension is a flat figure, the the third dimension is, say, a cube, then what is the fourth dimension? What is the fifth dimension? How do you figure them out on paper and could you draw them on paper? Date: 06/18/2001 at 14:32:44 From: Doctor Peterson Subject: Re: Beyond 3-dimensional geometry Hi, Tanner. The fourth dimension is something you can't see, so it's impossible to describe. The 4-d equivalent of a cube has been called a tesseract, if that means anything to you. When we draw a cube like this +--------------+ /| /| / | / | / | / | +--------------+ | | | | | | | | | | +----------|---+ | / | / | / | / |/ |/ +--------------+ we are "projecting" the 3-d figure onto a 2-d surface, as if we were looking at the shadow of a wire-frame cube. Notice that it looks as if I drew a square and then dragged it diagonally, with each corner lifting a track as it moved. That corresponds to making a cube in space by making a square vertically in the third dimension. A cube doesn't really look like this, of course; and as we move around it, its shape will seem to change dramatically, though it is not changing at all. We can do the same sort of thing to "project" a tesseract onto a 3-d space, dragging a cube diagonally to represent motion into a fourth dimension. If we project the result onto a piece of paper, it looks something like this: +--------------+ /|\ /|\ / | \ / | \ / | \ / | \ +-------\------+ | \ |\ | +------\-------+ | \ | /| | \ | /| | \+--/-|-----|--\+ / | | /\\/ | | /\\/ | | / +--------------+\ | |/ | \ | |/ | \ | +----|--\|-----+ | \| \ | +------\---|---+ \ | / \ | / \ | / \ | / \|/ \|/ +--------------+ You could continue this into other dimensions, but it gets very crowded, since you are squeezing more and more dimensions into only two, and the edges get tangled. You can find some interesting information by searching our archives for the words "tesseract" or "four dimensions"; here's one that lists sites with pictures: Tesseracts and Hypercubes http://mathforum.org/dr.math/problems/smith5.22.97.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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