5-gon:
7-gon:
=
17-gon: ![](img/desc7816793598506127378.png)
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.
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