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### Dimensions

Date: 02/15/97 at 11:35:50
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

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
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/

Associated Topics:
High School Higher-Dimensional Geometry

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