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.
The science of Time is a noble science, that reveals the secret of Eternity. Only the Elites of Sages may ever come to know this secret. It is called the First Age, or the Age of ages, from which time is emerging.
