A formal definition of symmetry in cartesian and polar coordinates

Question

I know that symmetry in mathematics is a deep and very well studied area. However, I'm looking specifically to define a rule by which we might define symmetry of graphs of functions in 2D/3D Cartesian coordinates and in (2D) polar coordinates.

In this sense, symmetry implies some form of periodicity of the graph about some line or set of lines, but I'd like something more clear and formal than that. Are there any such useful definitions?

Answer 1

A graph has a symmetry when you can apply a transformation to it, and end up with the same graph again. For example, a graph is symmetric about the line $x=2$ if you can apply the transformation $x\mapsto 4-x$ to its equation, and have it simplify to the same thing:

$\begin{align} y=x^2-4x+12 \mapsto\,\,\, &y = (4-x)^2 - 4(4-x) + 12\\ &y = 16 - 8x + x^2 - 16 + 4x + 12\\ &y = x^2 - 4x + 12 \end{align}$

Similarly, a graph is symmetric about the line $y=x$ if you can apply the transformation $x\mapsto y, y\mapsto x$, and have the equation come out the same:

$xy=2\mapsto\,\,\, yx=2\\$

A graph like $y=\sin x$ has translational symmetry, because you can apply the transformation $x\mapsto x+2\pi$, and end up with the same graph:

$\begin{align} y=\sin x \mapsto\,\,\, &y = \sin(x+2\pi)\\ &y = \sin x\cos 2\pi + \cos x\sin 2\pi\\ &y = \sin x \end{align}$

Does this help?

Answer 2

In general, an object is symmetric if it is invariant under some transformation.

What makes the concept of symmetry so powerful is that this can be applied to many objects and many transformations. Here are some examples of objects and their corresponding transformations:

• Regular polygons; rotations, reflections (Rotating a regular $n$-gon $2\pi/n$ degrees results in an identical-looking shape, so the $n$-gon is symmetric with respect to that rotation)
• Matricies; transposition (A matrix that is invariant under transposition, i.e. $A^T = A$, is called symmetric)
• Numbers; adding 0 or multiplying 1 (0 is called the additive identity, 1 is called the multiplicative identity)
• and functions.

Your question--if I understood it correctly--is what sort of symmetries functions can have, and how they can be formally defined. Here are some examples.

Reflection about the $y$-axis. This is easiest to write down in rectangular coordinates. A function $f(x)$ is symmetric about the $y$-axis if and only if $f(x) = f(-x)$ for all $x$. Examples include the function $y = x^2$ and the function $y = \cos(x)$. (These are also called even functions.)

Symmetry with respect to the origin.

In rectangular coordinates: A function $f(x)$ is symmetric about the origin if and only if $f(x) = -f(-x)$.

In polar coordinates: A function $r(\theta)$ is symmetric about the origin if and only if $r(\theta) = r(\theta + \pi)$.

Translation symmetry This one is also easiest to write down in rectangular coordinates and captures the concept of periodicity.

A function $f(x)$ is periodic if and only if there exists some $k \in \Bbb R$ with $k>0$ such that for all $x$ we have $f(x) = f(x+k)$. As long as $f$ isn't constant, there exists a "smallest" $k$ that works. This is the period of $f$. So if $f(x) = \cos(x)$, then $k = 2 \pi$.