Nice overview. FYI the bronze mean appears in the tridecagon. 5th mean appears in a 29-gon. 6th mean appears in a tetracontagon (40-gon). 8th mean appears in a heptadecagon. Not as a ratio of side to diagonal but as a ratio of line segments. The resultant trigonometric expressions are shown on the wikipedia page (https://en.wikipedia.org/wiki/Metallic_mean) and https://tellerm.com/math/metal_ngons/.