Consider these polynomials:[more]
5-gon: 7-gon: = 17-gon:
The 5-gon polynomial has roots ,,, }, where is the golden ratio. Any one of these values can be used to construct a regular pentagon. The construction of a regular 17-gon (or heptadecagon) requires any root of the 17-gon polynomial. Gauss, as a teenager, showed that nested square roots can solve the 17-gon polynomial, making the 17-gon classically constructible. He also proved that roots of the 7-gon polynomial are not classically constructible. Curiously, any cubic equation can be solved with origami, making the heptagon origamically constructible.
The -gon polynomial is , a Chebyshev polynomial of the second kind. The graphs shown are of the factors of this Chebyshev polynomial.[less]