I was in the middle of doing a writeup for another thread when I noticed that among exact trigonometric constants every single value listed has a residue of 3. Dipping into 5 residue at least generates the pentagon and the golden ratio. That second 3, however, just seems to be a useless artifact of trying to match the days in a year to degrees of a circle. But once we get into approximations, we might as well use radians, so that's a lost cause as well.
What's worse, this page even has fractional degrees, with 22.5-degrees describing the humble octagon. I found a page describing the exact value of the sine of 1-degree, and it just struck me as extravagant to the point of absurdity. I was mildly curious about whether a seventh would do anything for the idea of degrees but it seems it's already fat with an extra factor of three.
The use of the seventh and ninth of a circle is a question bound up with the use of the regular heptagon and enneagon. These are the two most "basic" non-constructible polygons in that the equations that determine the sine and cosine of pi/7 and pi/9 have degree 3.
It is perhaps instructive to note that the paucity of Greek mathematics concerning the regular heptagon stems directly from construction being their only means of exploring the mathematical universe - a light that flickers out after quadratic irrationals. Only Archimedes, he who was almost two millennia ahead of his time, seems to have explored them: he constructed the heptagon with a marked straightedge (neusis), and composed a lost treatise On The Heptagon in the Circle.
There is a bunch of interesting mathematics that comes from generalising how the golden ratio τ is bound up with the pentagon (a unit pentagon has a diagonal of length τ), by performing the same construction with the unit heptagon, enneagon, hendecagon, and so on. Here are some interesting papers: one, two, three.