Today I want to talk about something that I tell you a lot, imagining that, you know what I mean: geometry.
Normally I assume that this basic part of mathematics is known by all, but today, I will make a small incision explaining that e is geometry.
Geometry is the part of mathematics that is responsible for the study of figures in different spaces. This helps to know their properties and, more important, their transformations so that we can move from one geometry to another.
There are many geometries and in the sense of the type of space to which they refer. In short, and as a summary, the figures can be expressed and expressed in different ways depending on the number of dimensions to which we refer or the type of geometry that supports it. This is a bit of what is called the topology.
That is, there are rules that help to move from one geometry and topology to another so that an object can be another type of object depending (geometry) depending on the space where it is represented (topology).
These branches of mathematics are very important to a physicist since they help us to calculate simpler characteristics of something that seems very complex. That is, for example, the old joke of the cow with spherical shape that explains you with the theme of conforming transformations, a long theme that if you get bored you can re-read.
In principle, the definition of geometry was made of what we see and how we see it. Let us define the method in which the world is seen and as we see it, in the three dimensions. This was called Euclidean geometry because of the old Euclid. In these are defined concepts that are the base as point, straight, plane, space… and then redefined those concepts for what was seen to be other types of geometries.
And you ask, is there other types of geometry ?. Of course, hundreds, thousands, millions of them, although all are based or have to follow the same principles, 4 to be exact.
These postulates are, that two points are joined by a line, that the lines are infinite or at least expandable, that the circles can have any radius, and that the angles between two lines are congruent, let’s keep the angles.
There is a fifth postulate that… well I skip it because it is not met in other geometries and therefore is not valid.
In fact, which is the most beautiful, the passage from one geometry to another is done through functions. As you know a function is a norm or rule that assigns to other values, to a set of values assigns them other values different or equal, modifying them. These new values can be in the same geometry or in a totally different one and the study of how the properties of the first geometry are maintained in the second is what the topology does.
In physics, what is created are these functions which, what they do, is to indicate the rules or norms of nature. That is, physicists observe nature and try to create rules that explain what happens, happened and will happen. For this we have to rely on mathematics and, therefore, on geometry and topology since, many times, the functions we create open a new mathematical world and we have to know if what we have “seen” really corresponds With reality and works as it tells us.
As I said, these functions have given us the creation of different topologies, which, the topology, I will define and explain the properties later. For now I am satisfied that you understand and understand that geometry (and algebra, therefore) is fundamental but highly hated.
Remember that this is a brief intro about geometry, very little about it. If you like it and you want to learn more it’s nice to go to a library where you will found a lot of books about.