Beyond Three-Dimensional Geometry

Date: 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/