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How Many Dimensions Are There?


Date: 03/29/2001 at 19:24:09
From: Adam
Subject: Four dimensions

Doc,

I don't get it. I was playing with some simple shapes and looking at 
some of Escher's work, and I started thinking about four dimensions. 
I asked some people and got some brief references, and I understand 
what everyone's saying, but there's still something I'm uneasy about. 
Let's say we have a picture of a four-dimensional cube. Now, we see 
things in three dimensions, so the picture is a representation of a 
four-dimensional object in a three-dimensional environment, right? So 
the picture is just a representation. 

Now on to the questions:

1. We're just representing four-dimensional objects, so how do we know 
   there are even four dimensions? Is it an assumption? How many 
   dimensions are there?

2. What is the significance of drawing four-dimensional cubes?

3. If I were in a two-dimensional environment/world/realm/universe/
   whatever, how would I be able to represent three dimensions?

Could you recommend any books to me that you think I might find 
interest in reading?

Thanks for your time,
Adam


Date: 03/29/2001 at 23:20:07
From: Doctor Peterson
Subject: Re: Four dimensions

Hi, Adam.

First, I wonder if you have tried searching our archives for the 
phrase "four dimensions" or "fourth dimension"? We get a lot of 
questions in this area, and you can find some good insights there.

Let's look at your questions:

>1. We're just representing four-dimensional objects, how do we know 
>   there are even four dimensions? Is it an assumption? How many 
>   dimensions are there?

Mathematically, it doesn't matter whether there "are" four dimensions. 
We can still talk about the concept. In fact, we can talk about 
infinite-dimensional spaces! Actually, any time we work with something 
that involves four different variables, we are working in a 
four-dimensional space; for example, a problem involving three space 
dimensions plus time involves four dimensions. And if we deal with 
both position and velocity, we are working with six dimensions.

I think you're really asking about extra dimensions of SPACE. That 
takes you out of math and into physics; there are some answers about 
that in our archives, but I can't say much about it.

>2. What is the significance of drawing four-dimensional cubes?

It just stretches your mind. I don't know that it has any special use.

>3. If I were in a two-dimensional enviornment/world/realm/universe/
>   whatever, how would I be able to represent three dimensions?

Well, you are probably already used to representing three dimensions 
in a two-dimensional world; that's what you're doing when you draw on 
paper. This is a two-dimensional representation of a cube:

       +----+
      /|   /|
     +----+ |
     | +--|-+
     |/   |/
     +----+

The usual representations of a four-dimensional "cube" are done very 
similarly. (In fact, if you've seen them in books, you were really 
looking at 2-dimensional versions.)

If we actually lived in two dimensions, I don't think we could draw 
at all, unless everything were somehow transparent so we could see 
through it. Think about it - if you just drew a square, how much of 
it could you see from inside the piece of paper?

Here are a few answers I found in our archives:

   The Fourth Dimension
   http://mathforum.org/dr.math/problems/dixon24.html   

   Beyond the Third Dimension
   http://mathforum.org/dr.math/problems/rob5.16.96.html   

   Fourth Dimension
   http://mathforum.org/dr.math/problems/zeto5.13.97.html   

   Tesseracts and Hypercubes
   http://mathforum.org/dr.math/problems/smith5.22.97.html   

   Visualizing a Klein Bottle
   http://mathforum.org/dr.math/problems/sean.8.19.99.html   

The third is especially relevant to your questions.

You may also want to read _Flatland_ by Edwin A. Abbott, about worlds 
of different dimensions: 

   http://www.math.brown.edu/~banchoff/gc/Flatland/   

- Doctors Peterson and TWE, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Physics/Chemistry
Middle School Geometry
Middle School Higher-Dimensional Geometry

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