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### 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 is? If you could, can you also include any
information you know about the fourth dimension?

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
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

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

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

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/
```
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High School Higher-Dimensional Geometry
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