I have read from the book "Mathmatics and Its History" by John Stillwell. In Section 18.6 it is about complex interpretations of geometry. The book says:
The triangle and hexagon tessellations have similar group of motions, generated by
$z \mapsto z+1 ,z \mapsto z+\tau,z \mapsto z\tau$, ($z=x+iy$)
and more generally any motion of the Euclidean plane can be composed from translations $z \mapsto z+a$ and rotations $z \mapsto ze^{i\theta}$.
(For example, the unit square pattern is mapped by the rotation of $\pi/2$ about the origin, and these three motions generate all motions of the tessellation onto itself. Then these generating motions are given by the transformations $z \mapsto z+1 ,z \mapsto z+i,z \mapsto zi$.)
My question is why the rotation must be of the form $z \mapsto ze^{i\theta}$? Why it must be $ze^{i\theta}$? Can it be any other forms? How do you conclude this form?
Thanks in advance.
Any complex number $z$ has the form $z=re^{i\phi}=r\cos\phi+ir\sin\phi$.
$r$ is its distance from the origin, and $\phi$ is the angle between $z$ and the positive real axis. When you multiply by $e^{i\theta}=\cos\theta+i\sin\theta$, the product is $$r( \cos\phi+i\sin\phi)(\cos\theta+i\sin\theta)$$ You should check that equals $$r\cos(\phi+\theta)+ir\sin(\phi+\theta)$$
So the new point is the same distance from the origin, but the angle to the positive real axis is increased by $\theta$. That is why multiplying by $e^{i\theta}$ is a rotation by $\theta$.