Two Dimensions versus Three Dimensions: What's the Big Deal?

Article Summary: "Have you ever seen a 3-D movie? At a 3-D movie you wear some special glasses that change the image so it looks like the action of the movie is going out of the screen right at you! Well, 3-D is an abbreviation for three dimensions and a 3-D movie is different from a regular movie that you see in two dimensions."

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.