Monsters from other dimensions are a staple of B-movies. The Millennium Falcon flies through hyperspace to get from planet to planet in Star Wars. The theory of relativity describes our universe as being made up of a four-dimensional spacetime.
Ideas of dimensions and higher dimensions show up frequently in popular culture and are extremely important in science.
But what, exactly, does it mean to have four dimensions?
Dimension is a surprisingly complicated mathematical concept. Here, we will look at the basics of how we go from zero dimensions to the kind of higher dimensional spaces prevalent in science and science fiction.
The Single Point
We start with a lowly, single point. A mathematical point has zero dimension — it has no length, area, or volume. We cannot say too much about a single point on its own. However, points are the fundamental building blocks of geometry — more interesting spaces and structures are made out of uncountably many points related to each other in some way.
Now we look at something more interesting. The basic one dimensional space is a line, extending out infinitely in both directions, as indicated by the arrows on the left and right ends:
Whenever we are working with a space like a line, or the higher dimensional spaces we will be looking at later, we want to have a way to describe where we are in the space. So, we need a system of coordinates describing the positions of all the points on our line.
Here is where the idea of a number line will be useful to us. We assign a fixed point in the middle of the line to be our origin, or zero, point, choose a fixed unit length, and then identify each point on the line with its distance in our fixed units from the zero point, with points to the right of the origin being positive, and points to the left being negative:
By turning our line into a number line like this, we can say where any point is on the line by referring to its number.
This gives us an idea of what it means for a line to be one-dimensional. We only need one number, or coordinate, to describe where we are.
Much of geometry depends on being able to find the distance between two points in a space. On a line, we can describe the distance between any two points using their numbers by subtracting the smaller number from the larger number. The points represented by 2 and 4 are two units away from each other — 4 – 2 = 2.
Number lines are perfect for representing quantities where we are considering only a single variable — time, incomes, temperatures, anything that can be represented by a scaled quantity.
Before we move on to higher dimensions, there are a couple important subsets of the number line to look at.
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