You’ve been hired to design a new logo for Riddler City. The mayor is a little eccentric and has requested that the logo have at least two lines of symmetry that intersect at an angle of precisely 1 radian, or 180/𝜋 (approximately 57.3) degrees.
What sorts of logos meet the mayor’s requirements? (You can give one example or describe what all possible logos have in common.)
Consider a shape to have order $n$. Its magnitude is $\dfrac{360}{n}$. The angle between any pair of lines of symmetry can be expressed as $\dfrac{360k}{n}$, where $2 \le k \le n-1$.
Since $\dfrac{180}{\pi}$ is irrational, there exists no shape of order $n$.
It follows that the logo has not only two lines of symmetry, but an infinite number of lines of symmetry. The simplest such logo is a circle.
Any logo that is created from a set of concentric circles of varying widths and colors will fit the mayor's request.
Although the mayor is eccentric, the logo is not.