Escher-Like Spiral Tilings
The article mentions a lot of interesting properties of the complex exponential map. I find it especially neat that horizontal and vertical translation map to uniform scaling and rotation. A mapping that shuffles transformations around like that can be used in creative ways.I'd like to add another interesting property, that naturally stems from the above: the mapping transforms regularly repeating shapes into self-similar shapes. Any geometry that repeats along the x axis (with a given period P) will be mapped to a geometry that is self-similar under scaling by the same value P. You can create mappings in more dimensions that have the same properties, for example in 3 dimensions: https://www.osar.fr/notes/logspherical/