Function with 4-Fold Symmetry

Puzzle: Find a function \(f: \R \rarr \R\) with 4-fold rotational symmetry.

• \(f\) must be a total function
• When you rotate the graph of the function, it should appear unchanged:
  • \(f(x) = y \lrArr f(y) = -x \lrArr f(-x) = -y \lrArr f(-y) = x\)

Solution

There are many solutions, but here's a nice animated family of them. Click the bottom right to view+edit. $$ f(x) = \begin{cases} 0 & x = 0 \\ x \cdot c & \lfloor\log_{c}\lvert x\rvert\rfloor < 1 \pmod{2} \\ \frac{-x}{c} & \text{otherwise} \\ \end{cases} $$