Imagine a
point - a
singularity. That's the
0th dimension. Only
one point exists. Everything within this
universe exists at exactly the
same place.
Now run a
coordinate axis through the
point. Suddenly you've got a
line. Everything within this
coordinate system has a single
coordinate - we'll call it the
x coordinate, and we'll call the
axis the
x axis. That's the
1st dimension. You probably know it as a
number line from
algebra.
Now run a new
axis perpendicular to the
x axis. We'll call it the
y axis. Now
everything addressable has
two coordinates, an
x and a
y coordinate. This is
2-space, a
plane, the
2nd dimension. We can put points anywhere on this
infinite plane.
Now we're going to run an
axis perendicular to the
x and
y axes. We'll call this new axis
z. We now have
3-space, the
3rd dimension. We can address
points in
space - we can describe a
cube or a
sphere.
Now we will
induce a mindfuck. Run a
fourth
axis perpendicular to your
x,
y, and
z axes. We'll call this the
w axis.
Impossible, you say? Can't find a directon that will place this new
line perpendicular to all the others? That's because you probably
cannot conceive of this direction. You can't
move in it, you can't
see it, and you probably can't
think of it in anything but an
abstract way because you have
no experience with it. Depending on which
physicists you listen to, this
dimension is either
too damn small (see
string theory), only
gravity propagates in its
direction (will try to find
reference), or it doesn't
physically exist. This is the
fourth dimension. It
may or may not have
physical relevance, but
mathematically it's
completely valid, and even
necessary for modelling certain things in
3-space such as
temparature in a solid (anything that's a function of
3 variables can describe a
four-space curve).
You can continue adding as many
new dimensions as you please, without compromising
validity. AFAIK the greatest number of
physical dimensions that have been seriously
speculated is
seven (again, they are
too damn small for you to interact with), but
the magic of math lets you deal with
n-space. Mathematically, adding
more dimenstions is sufficiently
trivial that
textbooks may introduce
extra dimensions without
fanfare.
See also:
Flatland,
Flatland: A Romance of Many Dimensions,
vector calculus