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Planes and Lines

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


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 
more about these, you should read about "finite fields".

-Doctor Ceeks,  The Math Forum
 Check out our web site!   

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.
             (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)
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 
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 
was published by Birkhaeuser in 1995.

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!   
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Number Theory

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