There is a maxim in physics (rather in science), a necessary basic axiom and is that under a reproducible experiment it must give the same value wherever (place) and whenever (time).

This axiom (well, these two) is what is called symmetries.

A symmetry gives the same value under a translation in some (or several of its axes), so, that an experiment of the same (or that a formula of the same) in all places is a spatial symmetry in which it moves On the axes. Or that the results are the same at one time and then give exactly the same in another, it is a temporal symmetry.

The symmetries make the systems interchangeable by the mere fact of being worth the same.

There are two groups of symmetries, continuous and discrete. A symmetry is continuous when we can apply a continuous function (in an interval, which can be infinite) in front of what we want to modify. Thus, for example, a translation is a continuous function that we can apply as spatial symmetry, so this symmetry is continuous.

A discreet symmetry is one that happens only under certain values. This is something more mathematical but easily understood if, for example, we consider a triangle not as a set (its three sides), but as three points and their vectors separately. If we want to keep the shape of the triangle (ie, it is a triangle with the same orientation and shape) we will have to move it and rotate it determined angles (2pi / radian sides). It is simple to understand.

The symmetries are three, C, P, T. C of electric charge (one particle has the same charge as its opposite). P of parity (spatial transformation) and T of temporal (time transformation). Depending on what we do, function, symmetries are combinations of these three basic, so a symmetry TP is a translation in space and time or a symmetry CPT is a change of charge, in space and time.

In the end it is just algebraic functions of all life that we apply to algebraic groups, that is, mathematical functions on physical equations (which are also mathematical and so work so well). And as they are algebraic functions (continuous or not) at the end we can move all this to matrices and calculating matrices (the base of algebra so hated but so necessary). All very simple and coherent, after all.

Before some purist jumps (remember that I make introductions to topics), there are “violations” to these symmetries, the symmetry CP (time and space) with the theme of the disintegration of certain elementary particles that do not meet that for each Particle there is an antiparticle and that is tried to solve through the theory of strings thanks to the propagation of energy in other dimensions but that everything is still in phase of study.