I don't like the implications of negative angles, but if you use them, thenSpoiler for
Spoiler for Regarding the shapes in the above post
As for irrational numbers...well I don't (want to) know.
I fully agree with your analysis Paralogic but now how do you identify the 'sides' of the shapes. In the case of 1.75-a-gon for example, where is the 1.75 sides? Also if a pentagon makes the same shape that a 1.25-a-gon makes (which i can confirm that it does -except the shape is flipped if created in the way described in the op) then is the shape the same? Does the shape have 5 sides or 1.25 sides or both?