Fourth Dimension

Date: 05/13/97 at 19:19:15
From: alexszeto1
Subject: hi

Hi Dr.Math,

I'm a freshman in high school and I'm doing a research paper on the 
fourth dimension. Will you please help me try to understand what the 
fourth dimension is? If you could, can you also include any 
information you know about the fourth dimension? 

Thank you for your time,
Jody

Date: 06/25/97 at 13:34:41
From: Doctor Barney
Subject: Re: hi 

The fourth dimension is not so much a thing as it is an idea.  This is 
like the fact that the number 3 isn't really a thing in and of itself.  
You can write a symbol to represent 3 on a piece of paper or a chalk 
board, you can use the idea 3 to describe how many apples you have, 
and you can even buy a birthday candle in the shape of a 3, but you 
haven't really bought "three," have you?

Mathematics is a philosophy which we use primarily to describe 
physical phenomena. The most obvious example is the way Euclidean 
geometry (three-dimensional geometry) describes physical space. That 
is because the physical space as we understand it has three 
independent "degrees of freedom," or three directions in which we and 
the other objects in space are able to exist, expand, or move. For 
example: length, width, height; latitude, longitude, altitude; Elm 
street, fourth building down, second floor; x, y, z.  

Now, the idea of the fourth dimension (or the idea of the first four 
dimensions all together) is an idea we use to describe any quality, 
state, object, event, or concept which requires four independent 
degrees of freedom (ways in which it is able to be different) in order 
to describe it completely. For example: length, width, height, weight; 
latitude, longitude, altitude, temperature; Elm street, fourth 
building down, second floor, 9 O'clock on Thursday; x,y,z,w. It's 
really that simple.  

The only reason people get confused about it is because they cannot 
visualize it. If I tell you the length, width and height of an 
object, you can get an idea of what it looks like, perhaps a cube or a 
long slender rod or anything in between. But if I also tell you what 
temperature something is or how much it weighs, what does that look 
like? 

The problem with visualization becomes even more acute when we try to 
graph the data. You can make a drawing of a three-dimensional object 
on a flat piece of paper, and you can even make a model to represent 
three-space, like a relief map that shows the elevation of the ground 
at every point over a given area. But when you try to draw a picture 
of four-dimensional space it is impossible.  

For your paper, you might want to explore some potential applications 
in more detail. Here are some ideas to get you started.  

1) Time is considered to be the fourth dimension in many applications, 
with the first three being the classical representation of physical 
space. For example, to study how an airplane travels through the sky, 
you would need a series of data telling where it was at any given 
time. The individual data points might look like (x,y,z,t) where t is 
the time that the plane was at point (x,y,z). In your paper you might 
discuss what types of functions could describe how the plane travels 
through this four-dimensional space.  To try to visualize the 
function, picture a plane flying through the sky leaving a very long 
contrail. Now picture signs along the contrail depicting constant 
intervals of time. The signs will be farther apart when the airplane 
is going faster. Just as the slope of a line or other function plotted 
on a two-dimensional graph tells you one parameter of the function 
changes relative to another, so the distance between the signs on our 
imaginary contrail would tell you how the airplane's position changes 
relative to time.    

2) You could create or describe a hypothetical representation of the 
temperature in the atmosphere as a function of location. (You could 
also use pressure, or humidity, or any other property of the air.)  
Create several layers, using a transparent media such as vu-graph 
foils. Color the different areas different colors on a spectrum from 
hot to cold to represent the temperature at that location. Be sure 
the temperature is a continuous function (does not jump from the 
hottest color to the coldest) as you go from one layer to another 
directly over the same point.  

In the February '97 National Geographic there is a fascinating article 
on the oceanic science of the Arctic. On page 49 is a really cool 
figure showing one layer from a multi-dimensional model scientists 
have created to describe the temperature as a function of position in 
the Arctic Sea.   

3) You could also create a four-dimensional database that has nothing 
to do with physical space. Collect four different pieces of data on 
20 or so of your classmates, such as age in months, shoe size, number 
of siblings, and amount of money on their person when they answered 
the questions. Then experiment with different tables, charts or graphs 
to present the data and explore any potential correlations.  For the 
purpose of this study, these people would exist in the four-
dimensional space which you define.  

I hope this helps. On a final note, you are welcome to use my 
suggestions and rewrite some of my ideas in your own words, but please 
don't copy large sections of my answer directly into your research 
paper. That would be plagiarism, which is a form of cheating.   

-Doctor Barney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 06/25/97 at 14:34:25
From: Doctor Sydney
Subject: Re: hi

Dear Jody,

Hello!  I want to add just a little to Dr. Barney's response to you.  
I hope you find what I have to say interesting even though it's too 
late for your research paper!

Dr. Barney did a nice job explaining how lots of people understand the 
fourth dimension. However, there is one major way that people think 
about the fourth dimension that was not addressed in his response to 
you. That is what I will talk about with you.  

Let's start with what we know. Let's consider first a plane. We know 
that a plane is 2-dimensional, right?  What does that really mean?  
Well, one way to think of it is that we can put a pair of axes on the 
plane (an x-axis and y-axis, for example), and then we can assign to 
each point on the plane a pair of numbers, right?  You've probably 
worked with graphing things on a plane, so you are probably familiar 
with what I am talking about. If I told you to graph the triangle 
with vertices (1,0), (0,0), and (0,1) you would know what to do, 
right? The space is 2-dimensional because we use PAIRS of numbers to 
define points in space. Each number in the pair tells us how far we 
should move from the origin in a certain direction, right?  So, each 
number in the pair tells us where to go in space. 

Think now about the third dimension. In three dimensions, we assign 
TRIPLETS of numbers to each point in space, right? Again, each of the 
coordinates in the triplets assigned to points in 3-D space tells us 
how far to move in a certain direction from the origin, right?  For 
instance, if we think of a 3-D space with the standard x, y, and z 
axes and origin (0,0,0), then the point (1,2,3) would tell us to move 
1 unit down the x-axis from the origin, 2 units down the y-axis from 
the origin, and 3 units down the z-axis from the origin, right? So, 
all of the coordinates are space coordinates; they tell us how far to 
move in a specific direction.

So, if we wanted to think of the fourth dimension as a space that has 
4 SPACE parameters (just in the way that the plane has 2 space 
paramaters and the world we live in has 3 space parameters), then we 
have to use our imagination a little, right? If what I have said above 
confused you don't worry. Just start following from here.  This is 
confusing stuff, believe me!  

One way to understand what the fourth dimension "looks like" is to 
carefully examine what the 3rd dimension looks like to "creatures" 
living in a 2-dimensional world. If we can understand this, then we 
can understand some of what the fourth dimension looks like to us 
creatures living in the 3-D world by using appropriate analogies.  

Here is what I mean. Suppose you live in a 2-dimensional world. For 
simplicity, let's say that you live in a plane. Let's call this world 
"Flatland." You don't see anything that is outside of this plane. As 
far as you know, that is the only space there is. Let's say that you 
are a solid yellow square. I am a hollow green circle. Now, what 
will you see when you look at me?  Remember that you can see only 
things that are in the plane. It might be fun and helpful for you to 
cut out of construction paper a yellow square and a green circle.  
Then, to figure out what you see when you look at me, put the square 
and circle on a flat surface (like a table), and think about what you 
would see at eyelevel with the table. You might be surprised! Below, 
I will talk about what you see because it will help us to understand 
the fourth dimension; however, it will be more fun for you if you 
don't read what the answer is but figure it out yourself.

Okay, so now you have done the experiment or perhaps thought about 
what you would see, so I can talk about what the answer is. As you 
probably found, a person who lives in this plane sees a green line 
when he or she looks at me. When I look at you, I see a yellow line.  
When I first see you, if we are both sitting still, I can't tell if 
you are a square, a circle, a line, or some other shape, right?  All 
of these things look the same to a person in Flatland. How do you 
think people in Flatland determine what shapes their friends are?  Or, 
do they not care?  (:  These are fun things to think about.  

Okay, so now we understand a little bit about what life is like in 
Flatland. It might be fun for you to think about what life is like in 
Flatland. What do houses look like?  If I have a twin sister who is a 
solid green circle and is the same size as me, can you tell us apart 
(Recall that I was a hollow green circle)?  Drawing pictures will help 
you. 

Let's think about some objects in Flatland. Think about a safe in 
Flatland. What would it look like?  We would want it to be a container 
such that when locked, inhabitants of Flatland cannot get to the 
objects inside of it, right?  What kind of shape would work for this?  
In our 3-D real world, we often use a hollow cube, right?  Well, in 
Flatland, they  use, among other things, hollow squares. Does that 
make sense?  Think about it - an object that is in a hollow square 
cannot be taken by an inhabitant of Flatland; it is completely secure.  

Good. Now, we introduce the third dimension into our 2-D world. Think 
about someone who is an inhabitant of this world that we live in, our 
3-D world. Suppose they walk by Flatland. They see Flatland as a flat 
land (duh!) with lots of flat creatures and objects in it, right?  
Now suppose that they come across the safe used by me. I have put a 
necklace in my safe. Is my necklace safe from someone in the 3-D 
world? No! A 3-D person can easily reach into my safe and take my 
necklace, right?  He or she does not need to break down the barrier of 
the safe since he or she can just reach in from the top. Does that 
make sense?  

So, one way to understand the third dimension from the perspective of 
the second dimension is that enclosed spaces like safes can broken 
into by 3-D creatures without touching any of the barriers that 2-D 
creatures see.  

Let's think about how we could think about the fourth dimension from 
the perspective of the third dimension in an analogous way. Think 
about the safes we use in the 3-D world. They are, as we said before, 
often hollow cubes, right?  Once we put something in them and lock the 
door, we can feel sure that no one will be able to get to them without 
breaking down one of the sides of the safe, right?  Well, in the same 
way that a 3-D creature could break into a 2-D safe without breaking 
down any of the sides, a 4-D creature can break into our 3-D safe 
without breaking down any of the walls. Now, that is pretty wild, eh?  
It just doesn't seem to make sense. However, it is one way to think 
about the fourth dimension.

I could go on for a long time about similar ways to understand the 
fourth dimension, but instead I will refer you to a book that I think 
you might really like if you found what I said above to be 
interesting.  The book is called _Flatland_, and it was written about 
100 years ago by Edwin A. Abbott. It describes in detail the world we 
called "Flatland," and it discusses many more ways to understand the 
fourth dimension from the perspective of the third dimension.  If you 
have access to the WWW, you can find the entire book at the following 
website:

  http://www.alcyone.com/max/lit/flatland/index.html   

It is fun to look through, even if you don't read the whole thing.  I 
hope you enjoy it.  

If you have any more questions about the fourth dimension, please feel 
free to write back.  I think that dimensionality is one of the most 
interesting things that mathematicians study.  Good luck!

-Doctor Sydney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/