The golden and Silver ratios are the only metallic ratios which can be expressed as ratios between the side-lengths and/or diagonal lengths of regular polygons. We did this finding the cyclotomic zeros of some polynomials that, in a way, matched up with metallic ratios. If you're interested in the details, the full paper is here: https://arxiv.org/abs/1910.10325 It was also published in "Integers": http://math.colgate.edu/~integers/v40/v40.pdf