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### How Many Dimensions Are There?

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Date: 03/29/2001 at 19:24:09
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

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

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.

>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.

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/
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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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