Feeling a bit triumphant about constructing an enneagon from a hexagon, it seemed like an easy next step to construct pentagons from squares!
A pretty fundamental mistake was assuming that equal divisions of a line were also equal divisions of an angle. They are not! Everything (including enneagons) falls apart!
. . . . which leaves me falling back on the lowest common denominator of four and five being twenty, which is not really big news I guess?!?