im currently working on a Project about teaching symmetry to kids. Within that Project, i have to give a mathematical foundation of the concept of symmetry. However, thats where im running into a Problem.
I have trouble finding a Definition that connects all the different aspects of the topic. Let me explain what i mean: Consider two identical squares within in the Plane $\mathbb{R}^2$. Using orthogonal transformations in the context of linear algebra, or using metric spaces and isometries, one can define symmetry between two objects by finding a transformation that maps one object onto the other while preserving certain properties.
Now consider a single Square. Within a square, one can find 8 transformations, that map the Square onto itself. Using the tools of algebra, symmetry is defined with Permutations, automorphisms and Symmetry groups.
What i need is a definition of Symmetry that unites both of those aspects. Let me know if any aspect needs a more detailed explanation.
The most general definition of a symmetry in mathematics is that it is a set of transformations that leave some property of an object unchanged.
The "object" here is not necessarily a geometric object. For example, we can say that the expression $a+b+c$ is symmetric under any permutation of the symbols $\{a,b,c\}$ - the property of $a+b+c$ that is unchanged is its value. So we have
$a+b+c = b+c+a = c+a+b = b+a+c = a+c+b = c+b+a$
On the other hand $(a-b)(c-d)$ is only symmetric under four permutations of its symbols:
$(a-b)(c-d) = (b-a)(d-c) = (c-d)(a-b) = (d-c)(b-a)$
In fact, these four symmetries have the same structure as the four symmetries of a rectangle - we say that the symmetry "group" of $(a-b)(c-d)$ is the same as the symmetry "group" of a rectangle (or, more technically, they are "isomorphic"). The language of groups is the most natural context in which to study the abstract properties of symmetry families.
Even within geometry not all symmetries have to be distance preserving isometries. Dilations preserve ratios of distances rather than distance themselves. And conformal transformations preserve angles but not distances.