# Points

Any discussion of geometry should start by examining the basic building blocks from which more complex geometric forms are constructed. These building blocks are *points*, *lines*, *rays*, and *angles*. They have the same significance in three-dimensional space as they have on a two-dimensional plane, but we will initially be looking at the properties of these objects, and how they relate to each other, in a two-dimensional context. You can perhaps best think of a *plane* as a flat sheet of paper that extends indefinitely in every direction. A kind of two-dimensional universe, if you like, completely flat and without depth. All of the points, lines and angles that we are interested in are going to appear somewhere on that sheet of paper. Of course, it would help if we had some point of reference.

The point of reference we are talking about is usually referred to as the *origin*, and is itself by definition a *point*. Imagine yourself standing on the piece of paper, which stretches endlessly to the horizon in every direction. When you move around, the origin (which appears as a tiny dot on the paper) provides your only means of knowing how far you have moved. Even then, you have no idea in which direction you have moved, and only approximately how far you have strayed from the origin. You could, after all, walk round and round in a circle and still be the same distance from the origin. Nevertheless, at least the paper is not completely blank. So what *is* a point?

The first thing to realise is that a point is *dimensionless*. We imagine a point as an infinitely small dot and indeed this is how we represent it on paper. In reality, however, we should not really be able to see a point, since it has no physical existence. It lacks length, breadth, and depth. We can only define it, in fact, by using numbers to specify its position on the plane. We can only do *that*, of course, if we have some *point of reference* (the origin) to start with, some *unit of measurement* that we can use to specify *how far away* our point is from the origin, and some further means of reference to establish *in what direction* the point lies, relative to the origin. The trouble is that from the origin, we have an infinite number of directions to choose from, and they all look the same. How this problem is resolved is dealt with in the page entitled "Lines, rays and the coordinate system".