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Planes and Lines

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Date: 10/26/96 at 21:50:23
From: Yuan Tao
Subject: Planes and lines

Do planes and lines have the same number of points? I have noticed
that planes and lines both extend infinitely in multiple directions.
However, a plane is two-dimensional, and a line is one-dimensional.

To compare it with something from actual life, I thought of a plane as
an extremely wide truck, a line as an extremely narrow truck, and
points as apples being loaded onto the trucks. Assuming that both of
the trucks (planes and lines) are the same length, it thus follows
that the wider truck (plane) must hold more apples (points) than the
narrow truck.

I have tried to solve it by graphing the number of points in a plane
as y = x^2, and the number of points in a line as y = x. I used the
two equations stated because a plane has dimensions of infinity on
both axes, and a line has a dimension of infinity on its axis.
When I graphed them, y = x^2 exhibited an upward trend while y = x
remained linear and fairly constant. From the behavior of the two
graphs as they approached infinity, I deduced that the number of
points in a line would never approach the number of points in a plane.
However, my math teachers seem to find fault with this reasoning.
```

```
Date: 10/27/96 at 14:57:13
From: Doctor Ceeks
Subject: Re: planes and lines

Hi,

Whenever you ask a question such as yours (which is rather nice!), you
always have to make sure you have defined the terms in your question.

"Do planes and lines have the same number of points?"

Here, you must settle on a definition for plane, line, and point
before trying to answer.

In your model using trucks and apples, you have adopted a kind of
definition that gives a heuristic understanding of something, but
which differs from the definition of plane, line, and point as used by
Euclid. Of course, in your model, a wide truck will hold more apples
than a narrow truck, so if Euclidean geometry were correctly modelled
by your model, your conclusions would be sound.

In your model using y = x^2 and y = x, the mathematical situation you
are modelling is more like trying to compare how many square feet
exist inside an x by x square versus how many rulers can be put end to
end along the square's side.  This is also close to your model using
apples and trucks, but differs from the notion of point, line, and
plane as used in analytic geometry.

Your teachers may find fault with your argument because they are
thinking of the plane, line, and point in a "Euclidean" sense.
The Euclidean sense is the sense in which these terms are used to
carry out Euclidean geometry and analytic geometry.

In this sense, a point has no dimension (no size).  In your apple and
truck model, an apple does in fact have size, and so does not suitably
model the mathematical definition of point.  In your "y = x^2, y = x"
model, you are using unit squares to model the point, but a unit
square also has size.

As it turns out, a plane and a line contain the same number of points!
That is, you can actually find a function which maps the points in the
line to the points in the plane in such a way that every point in the
plane gets mapped to, and no two points on the line get mapped to the
same point in the plane!  Some of these maps are known as "space-
filling curves".

But your analysis is very intelligent and your "discrete" models are
very interesting. In fact, mathematicians have also found geometries
that are very much like your apples and trucks model.  These finite
geometries involve the construction of finite number systems, and
there, the plane does have more points than the line.  Indeed, your
"y = x^2, y = x" argument is highly applicable! If you'd like to learn

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

```
Date: 10/28/96 at 19:41:51
From: tao
Subject: Thanks for the message, but...

Dr. Math,

Thanks for the message! I forgot to tell you that I was doing this for
a science project. I would also like to know if there are books and
resources in Tennessee that are near Nashville (that's where I live)
that you would recommend for science project research.

Thanks Again!
```

```
Date: 10/29/96 at 12:46:25
From: Doctor Mike
Subject: Re: Thanks for the message, but...

Hello Yuan Tao,

I saw the fine answer you got from Dr. Ceeks. We who are
mathematicians often have differing opinions about what is most
interesting about a question.  Maybe something from my point of view
will help.  By the way, I think it is fantastic to choose a math topic
for a science project.

Perhaps your teachers have trouble imagining your trucks. Perhaps they
have read about conclusions different from yours. Whatever the
reasons, let's get on to your ideas. The concept of infinity is
difficult, so it is good that you are getting an early start. I didn't
seriously begin thinking about infinity until I was almost voting age.

The most important and amazing idea about infinity can be illustrated
well by some of your infinite apple trucks. What I'll try to draw
is 2 trucks that are both pretty narrow. One is wide enough to hold
3 apples side-by-side in each row, and the other can only hold one
apple in each row. The pictures show them with the engine at the left,
the infinitely long tail going off to the right, and (N) as apple
number N.

( 1) ( 4) ( 7) (10) (13) (16) (19) ...
1[FRONT]: ( 2) ( 5) ( 8) (11) (14) (17) (20) ...
( 3) ( 6) ( 9) (12) (15) (18) (21) ...

2[FRONT]: ( 1) ( 2) ( 3) ( 4) ( 5) ( 6) ( 7) ...

The trucks are loaded starting with apple (1), then (2) and so on.
If the trucks were the same finite length, then truck #1 would hold
3 times as many apples. Because these trucks hold the same infinite
number of apples, the apple load in [1] can be matched up one-to-one
with the load in [2]. Each truck has an apple (123); each truck has
apple (57 billion); and on and on. Setting up a "one-to-one
correspondence" like this is the way to show that two infinite sets
are the same size. Another way to think about this is to say that
"3 times infinity = infinity".

I can also use a truck-loading example to show you that "infinity
times infinity = infinity" or "infinity squared = infinity"!  Look at
the following truck picture and try to see the order in which the
apples are being loaded.
.
.
(25)
(16) (24)
( 9) (15) (23)
( 4) ( 8) (14) (22)
3[FRONT]: ( 1) ( 3) ( 7) (13) (21) ...
( 2) ( 6) (12) (20)
( 5) (11) (19)
(10) (18)
(17)
.
.
This truck is not only infinitely long, but infinitely wide as well.
It is loaded in bigger and bigger sideways "V" patterns.  To see this
better, draw lines from (5) to (6) to (7) to (8) to (9). Every spot in
every row of the truck EVENTUALLY has an apple loaded into it, but we
are counting just as before.  Strange but true, this truck has the
same number of apples as the other 2 trucks. So, planes and lines do
have the same number of points.

This is just an introduction to all the awe-inspiring things we can
learn about infinity. Your apple truck examples did show that what
appears to be smaller or larger loads are really the same size of
infinity. On the other hand, we know about different infinite numbers
that REALLY are larger than the infinite number of counting numbers.
In fact, the set of all real numbers (including 5/7, the square root
of 33, pi, etc.) is so large that it is not possible to list them all.
There cannot be a one-to-one correspondence between the reals and the
counting numbers. We can't "count" them like we can apples on the
trucks.

I wish I could give you a good reference to read more about this kind
of thing, but the books I have actually seen are written at the
college level or above. There is a new 150-page book _In Search of
Infinity_ by Vilenkin which got a very favorable review in the October
1996 issue of _American Mathematical Monthly_ magazine. I have not
seen the book myself, but I'm guessing it might be difficult in places
but generally okay for you. The book is translated from Russian and

Your public library will have a math section with books you will find
interesting, even if they don't say much about infinity. Read them.
Your interest in graphing is also important for you to continue. Have
fun with it. Since you are in a metropolitan area, perhaps there are
both a city and county library system to try. Some libraries let you
log in to their catalog from your PC or Mac by modem. Also, since you
are in Nashville you could try Vanderbilt to see if their library is
available to you.

So....good question.  Good luck.  I hope this helps.

-Doctor Mike,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Number Theory

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