The Fourth Dimension

Date: 8/24/95 at 17:38:37
Date: Thu, 24 Aug 1995 17:38:35 -0400
From: Anonymous
Subject: Fourth dimension

Question:
What is the fourth dimension mathematically?

Date: 8/25/95 at 13:49:57
From: Doctor Ken
Subject: Re: Fourth dimension

Hello!

Well, there are several ways one might interpret things in four dimensions.
Sometimes it's convenient to think of four dimensional space as three of 
ordinary space and one of time.  That's the way that most non-mathematicians
know about.  What a lot of mathematicians like to think of them about though,
is to make them all spatial dimensions.  You can build them up from the
bottom:

If you have one dimensional space, you can only move in one direction, 
i.e. along a straight line:

<--------------------------------------------->


When we add another dimension, we usually think about it as perpendicular
to the ones we already have.  So if we add one here to make 2-d space, we
have a plane:
                        ^
                        |
                        |
                        |
                        |
                        |
                        |
                        |
                        |
                        |
<-----------------------|---------------------->
                        |
                        |
                        |
                        |
                        |
                        |
                        |
                        |
                        v                      

To make three dimensional space, you add one perpendicular to both of these,
that would stick straight out of the page (here's a perspective version).

                        ^
                        |        /
                        |       /
                        |      /
                        |     /
                        |    /
                        |   / 
                        |  /
                        | /
                        |/
<-----------------------|---------------------->
                       /|
                      / |
                     /  |
                    /   |
                   /    |
                  /     |
                 /      |
                /       |
               /        v                      
               
So where do you go from there?  Well, you'd add another dimension 
perpendicular to these three.  We can't visualize it, because all we know
is three dimensions, but that's what you would do.  You then represent
points in this space with four coordinates:  three vectors in four-space
that are all perpendicular to each other would be
(1,0,0,0)
(0,1,0,0)
(0,0,1,0)
(0,0,0,1)

Don't blow your brain up trying to think too hard about this!

-Doctor Ken,  The Geometry Forum