The word dimension means measurement in a direction. Perhaps you
already know that a standard sheet of printer or copier paper is 8'
by 11. That means that the paper is 8' inches wide and 11 inches long.
But the paper isn't particularly thick and we don't usually think
about the thickness of a piece of paper. So normally we consider a
piece of paper to be two-dimensional, having just length and width.
What about the package of paper? The whole package is a couple of
inches thick; it also has a height. The package of paper has length,
width and height; it has three dimensions.
The difference between two and three dimensions is certainly a big
deal! Just about everything we deal with in the real world is three-dimensional
and it is a challenge to developers of virtual reality to create the
visual effect of three dimensions on a two-dimensional computer or
TV screen.
The mathematics for two dimensions is different from the mathematics
for three dimensions. In two dimensions the common shapes are circle,
triangle, square, and rectangle. Do you remember that any flat shape
with three or more sides is called a polygon? Can you name any other
polygons besides triangle, square and rectangle? Can you name other
variations of a circle?
In two dimensions you can measure the length of each side of a polygon
or the circumference of a circle. Because a two-dimensional object
is flat or planar, you can measure the area as well. In geometry you
learned how to calculate the area of different geometric shapes.
Can you think of any way to use area in everyday life? One important
application of area is in real estate. For example, if you want to
rent space in a building, rent can be charged per square foot. "Square
feet" is another term for the area, the length of the space by the
width of the space. Or if you want to buy land, it is sold by the
acre; an acre is 43,578.6 square feet. Why is an acre such a strange
number? The measurement of an acre goes back a long time when measurements
weren't as precise as they are today.
What happens when we get to three dimensions? Real objects
have three dimensions and the names of the shapes changes. A three
dimensional circle can become a sphere, the same as a ball. Or a three
dimensional circle can become a cylinder, like a tin can. Things get
more complicated when you have to calculate in three dimensions. For
example, the cylindrical tin can has a volume, which is the amount
it can hold. Does the tin can have an area as well? You kitchen counter
is a planar surface that has two dimensions, right? So when you set
the tin can upright on the counter, you can calculate the amount of
surface area the can occupies. But how do you calculate the area that
the tin can occupies when you lay it on its side?
Just this simple example shows how complicated the world of mathematics
gets when you start working with three dimensions. But since we live
in a three-dimensional world, mathematicians are always facing the
challenges of finding new equations that can be applied to the real
world around us.