Points and Lines
The fundamental building blocks of Euclidean Geometry are points and straight lines. Intuitively, the idea of a point and a line occurs to the mind easily.
Euclid talked about a point as a geometrical entity which has no part. If you touch a paper with a very sharp-tipped pencil, you will have created an approximation to a point. It will be an approximation because if you were to zoom onto this “point” using a microscope, you would find that it is not a point at all, but a region with some finite, non-zero area! No matter how sharp-tipped your pencil is, you can never make an exact point. In this sense, a point is an abstract entity which exists in our minds – in the real world, we can only draw an approximation to a point. However, when we are studying geometry and we talk about points, it is understood that we are talking about the abstract concept of points – that a point is an entity with exactly 0 length, 0 breadth and 0 area.
Euclid also talked about the (abstract) concept of a straight line. He said that a straight line is a “straight”, width-less length. You must have drawn many lines, using rulers or other such tools. However, whenever you think you are drawing a line (say, with a sharp-tipped pencil and a ruler), you are once again drawing an approximation to a line. This is because if you were to zoom onto this “line” of yours with a microscope, you would find that it has some finite, non-zero width! It is not an exact line. In fact, you can never draw an exact line. Once again, the perfect line exists only in the abstract world – in the real world, we can only draw approximations. However, as in the case of points, whenever we talk about lines, it should be understood that we are talking about “perfect” lines – straight lengths with exactly 0 width (How should we define the adjective straight? Think about this!).