What is Symmetry?
This is an interesting question, namely, because when I ask myself this my mind immediately starts thinking in terms of spatial symmetries. For example, putting my hands in front of me, I see mirror symmetry. This isn't entirely correct because my hands have immutable chirality that don't allow me to turn a left hand into a right hand or vica-versa ( I'd have to remove my fingers and reposition them, something I'm interested in doing). The reason that I think in terms of spatial symmetries is because my experience occurs in a 3D spatial world and my brain and eyes have evolved to identify objects with symmetries to characterize and label. For example faces have a high-degree of symmetry and it would make sense if I could remember faces. The truth however, is symmetry is much more ubiquitus in the construct of our universe. That is to say, we see symmetries in the most abstract of concepts.
So how does one understand symmetries in an abstract sense? The most fundamental place to start is of course mathematics, specifically within group theory. This branch of mathematics is primarily concerned with the grouping (hence the name) of structures for different algebras. For example what groups exist in a structure that makes up vector spaces1. The groups we are talking about contain internal symmetries whereby when they act on a object in a structure leave it invariant with respect to the operation. Let me try with a simple example. We start with a vector in a unit circle, $\mathbf{v} = \frac{1}{\sqrt{2}} \mathbf{x} + \frac{1}{\sqrt{2}} \mathbf{y}$ with magnitude $\lvert \mathbf{v} \rvert = 1$, now if I rotate2 the vector by any angle but do so such that the magnitude says the same the vector will always be relatable or transferable back to the original vector by some rotation that conserves the magnitude. This is what constitutes a symmetry. For our vector in the unit circle it has an continuous symmetry operation, i.e., I can define an infinitely small rotation angle and then sum up an infinite number of those rotations to get back to the original state. But there are other groups where the symmetry is discrete, which show up in physical systems such as crystals or shapes.
Group Theory in Science
The areas of discrete and continuous symmetry groups is called Lie theory and finite group theory, respectively. Lie theory is the mathematics that leads to modern quantum physics, in the context of what our universe "consist" of, it provides the language to discuss/represent internal symmetries. These internal symmetries are associated with fundamental fields that are consistent with the principals of quantum mechanics and special relativity. At the moment this is all I can say about Lie theory since I only have a limited understanding. It is a very powerful approach in quantum physics.
Intuitively its much easier to think about finite symmetry groups, these are every so familiar to use as I described in the first paragraph. Think of a geometric shape like an equilateral triangle or square. For the triangle, which is a 2D object, we can ask what operations leave the appearance of the system invariant? Well there is a grouping of 6 operations. There are 2 rotation operators, counter and clockwise rotations by 120 degrees about the axis perpendicular to the center, that leave leave the appearance of the triangle the same. Although if we were to add some kind of "label" to the sides or corners we could then say that the appearance is invariant but the system has been reoriented. This is how I think of the laws of physics and the quantities we observe in nature. The laws of physics are always invariant, they are the same and fully encompassing within our universe. However, the quantities energy, matter, spacetime, etc. that we measure or relate one another to can differ depending on how we are "oriented". In addition to the rotations there are mirror/reflection operations, 3 of them, that would leave the appearance of the triangle the same but produce unique orientations if we labeled the corners or sides. There is also the operation that does nothing, the identity operation, this can literally be though of as doing nothing such as multiplying a number by one does nothing, or can be thought of as the operation that brings the object back onto itself. For our triangle that would be the rotation of 180 degrees.
The operations described above are called a set which has the unique property that is is closed under any combination of operators in the set, this is what is referred to as a group. To discuss this more specifically, think of the identity operation. We could write this as the combination of 3 clockwise 90-degree rotations. $^\dagger$
$^{\dagger}$ This post is incomplete.
1I want to be a bit clear here, the term vector spaces is used in the most abstract way possible. We shouldn't think of a spatial vector in 3D, we should think of a vector in some N-dimensional space that can represent essentially anything
2 The rotation of a vector in a unit circle can be done using a matrix, $\mathbf{M} = \begin{bmatrix} \cos{\theta} & \sin{\theta} \\ -\sin{\theta} & \cos{\theta} \end{bmatrix}$ where the $\mathbf{x}$ and $\mathbf{y}$ components are adjusted based on the angle $\theta$. The magnitude will be unchanged and therefore this type of matrix acting on this this 2D vector space is refereed to an orthogonal matrix.
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