Visualization of the root structures of Vieta-Fibonacci polynomials (see Vₙ below). Being rather special, the polynomials share many of their roots despite apparent differences in form. The relative popularity of each root corresponds to the brightness of the "stars" in the image — zero and ±1 being the brightest (across the horizontal). The duplication of the "root stars" across the vertical axis visualizes the periodicity of the sequences at that root location.

The Vieta-Fibonacci polynomials are very much related to my earlier explorations in continued fraction fractals. In fact, the formula used for the continued fraction images happens to generate a sequence of rational polynomials where both the numerator and divisor are adjacent Vieta-Fibonacci polynomials. Because of this relation their root structures are identical.

All in all, the topic is a rather deep rabbit hole which somehow manages to tie together continued fractions, trigonometry, diagonals of regular polygons, perfect proportions of arbitrary degree (golden ratio being just the first example), orthogonal polynomials à la Chebyshev, generalized/interleaved Fibonacci sequences, combinatorics, and even quantum spin mechanics in a way that I still don't quite understand .. and probably never will.