Dimensions

Date: 02/15/97 at 11:35:50
From: Suryadie Gemilang
Subject: One Dimension

I have a question: Does 1-D space exist? How about 5-D space?

Thanks.

Date: 02/16/97 at 02:28:50
From: Doctor Mike
Subject: Re: One Dimension

Hello Suryadie,
  
Yes, such spaces exist.  I will give examples having to do with
temperature.  You know that water freezes at 0 degrees Celsius and 
boils at 100 degrees Celsius. All other temperatures are somewhere 
between these two, above them, or below them.  Temperatures are often 
shown on a thermometer that is a straight line, with colder at one end
and hotter at the other end. This is a one-dimensional way to measure 
the world.  Only one thing is measured. This value goes up or goes 
down. Nothing else is considered.
  
A more complicated way of viewing the world is to consider physical 
position in space, temperature at that point and time. You could 
represent this as (X,Y,Z,C,T) where (X,Y,Z) are the usual space 
coordinates, C is Celsius temperature and T is time. Not all 
quintuples are possible. For instance, at a particular place and 
time, there is going to be only one true temperature.  In three-
dimensional space, it is common to describe a smooth surface by giving 
Z = F(X,Y) as a function of X and Y. Similarly, in five-dimensional 
space you could try to describe temperature by a function 
C = F(X,Y,Z,T) as a function of position and time.  Weather 
forecasters try to do this all the time.  People who listen to the 
weather forecasts want to know how cold or warm it will be at various 
times during the day tomorrow.  Of course, the meteorologist can only 
give very approximate information about this.  But such a function 
exists in thought.
  
These are specific suggestions of why one might want to think of and 
use a 1-dimensional space or a 5-dimensional space. Actually,
mathematicians have generalized this quite a bit. Because we know 
that there are these real-world situations where more than three 
quantities need to be measured, we have developed the idea of an 
N-dimensional space for any positive number N. The term Finite 
Dimensional Vector Space is used for this. This terminology comes from 
the fact that a triple such as (1,2,3) can not only be used to measure 
a position, but can also be used to indicate a direction (like 1 mile 
East, 2 miles North, and 3 miles Up for the direction, or vector, from 
an airport control tower to an airplane).  
  
You will encounter these ideas more and more if you continue your 
studies of mathematics.  I hope this helps for now. 

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