Introduction to geometry

​ ​In the Stone Age, math wasn't needed that much. But around 5,000 years ago when the first civilizations grew, life became much more complex, and more situations arose where people had to be able to calculate things. ​Three children of this farmer inherit his fields and they need to know how big they are. How are they going to calculate that? ​ ​These two fishermen are getting a new mast for their boat and they have to calculate if the tree is tall enough. ​ ​This is a well. How much water can it hold? ​ ​To answer questions like these, people needed to find methods of measuring and calculating lengths, surfaces, angles, and volumes. And they did.

Most of the geometry taught in the elementary school was actually developed during this period between 5,000 and 2,000 years ago. On many occasions, people in different places of the world made the same mathematical discoveries without knowing that someone else had already figured out a solution to that particular problem. ​ ​Some of the places where people solved geometric problems early on included the Indus Valley in modern Pakistan, Babylonia where Iraq is nowadays, and Egypt. 2,300 years ago, a man called Euclid lived in Alexandria. He was a skilled mathematician, but not only that, he also wrote 13 books where he summarized all the knowledge of geometry that people of Egypt had at that time, and they knew quite a lot. ​ ​Let's start with a square. You recognize it.

It has length and width, but no depth. It's completely flat. You can twist it a bit so it becomes easier to see. Length and width, two dimensions. The square is a two-dimensional figure. ​ ​Now, let's remove three of the square's sides.

What do we get? A straight line. We can call it that, but in geometry, it's called a line segment. And now there's only one dimension - length. No width, no area, no thickness, just length.

One dimension. ​ ​So let's take away the length as well and just keep a dot. The dot you see here is perhaps one millimeter in diameter, but we're talking about geometry now, and therefore, it's not a dot but a point. And in geometry, points have no size at all. A point has zero dimensions - no length, no width, and no height. It's just a position - an exact place - but it can exist in a three dimensional space - a room, for instance. ​ ​The point is in a place that can be described using width, length, and height.

We sometimes call them x, y and z. This particular way of describing positions is not in Euclid's Elements books. It was not until the 17th century that the Frenchman, Rene Descartes, came up with this system. But Euclid would probably have liked it if he could have seen it. ​ ​Anyway, here's another point. If we have two points, we can draw a line through them. A line is infinity long, has no width or thickness, and is completely straight.

If a line starts from a point and goes into infinity, it's called a ray. If we cut off the ray at some point, we have a line segment again. There they are, three one-dimensional geometric objects - the line, the ray, and line segment. ​ ​Shall we let the second dimension in? Here's our square again and a triangle and a circle, a rectangle and a rhombus. There are many two-dimensional geometric shapes.

They are flat, completely without thickness. You can measure their area or surface in square meters, for example. But they have no volume. ​ ​If we include the third dimension as well, we can make shapes that have volume - a cube, a rectangular prism, a cylinder, a tetrahedron, and a sphere, for example. They have length, height, and width. They have a measurable surface but also a volume - a body, a content that takes up space.

Three-dimensional figures can be filled with air or water or measured in liters or cubic meters. These figures have three dimensions, and in these three dimensions, they act like you could expect. A triangle cannot have more than one right angle, and two parallel lines never meet. There are other geometries where other rules apply, but you'll learn about them another time. For now, let us settle with the normal, school geometry - or Euclidean Geometry, as we usually call it, as a tribute to good old Euclid.