I.3.1 Types of Symmetry
There are many types of symmetry, which include: reflective, rotational, translational and scaling, or any mixture of these transformations. When the object or system undergoes symmetrical procedures, its final state remain equal to the initial state, and it is described as invariant under these procedures. A butterfly, for example, exhibits reflective symmetry, so its shape is invariant under reflection. Some flowers and sea creatures exhibit rotational symmetry, so they are invariant under certain rotations. There are also many natural or synthetic patterns that are invariant under translation, such as most shapes found on rugs and wallpaper. A lattice is also a repeating pattern of points in space, such as the arrangement of atoms in many crystalline materials.
Scaling is also a very important type of symmetry where the system remains invariant at different scales. As a simple example, the shape of a circle does not change when we increase or decrease its radius. Scaling invariance is one of the most fundamental characteristics of fractals, which also often exhibit a combination of other types of symmetries.
Moreover, symmetry does not only apply to the apparent shape of objects, but also can be a key feature in many other properties of the system. For example, the speed of light has the same value in all frames of reference, so this forms a symmetry that is described by Poincar group, which is the symmetry group of Special Relativity.
Symmetry can also be observed with respect to the passage of time, like all types of melodies in music, or even in many historical or political events that may be repeated under some similar circumstances, usually mixed with other types of invariance. In social behavior also, there are many interactions and peer relationships that maintain some symmetrical strategies.
Therefore, symmetry is not restricted to the geometrical shape, but it can be generalized in many ways as invariance under some mathematical operation that preserve some property of the system. The set of such operations that preserve a given property form a mathematical group. Groups are studied in mathematics under a particular field called group theory, as we shall discuss further in section I.3.3.
The study of symmetry is essential in the development of physics theories, because it describes the invariance under any kind of transformation between different coordinate systems. It is becoming more evident that all laws of nature originate in symmetries, as we shall discuss further in section I.1. As Anderson stated in his famous 1972 article More is Different : it is only slightly overstating the case to say that physics is the study of symmetry. [[2]]. We shall also see in section I.1.1 that Noether s theorem states that for every continuous mathematical symmetry, there is a corresponding conserved quantity such as energy or momentum.