n-gon Polynomials
n-gon Polynomials
{1-4x-4+8,-1-4x+4+8}
2
x
3
x
2
x
3
x
Consider these polynomials:
5-gon: 7-gon: = 17-gon:
1-12+16
2
x
4
x
-1+24-80+64
2
x
4
x
6
x
(1-4x-4+8)(-1-4x+4+8)
2
x
3
x
2
x
3
x
1-144+3360-29568+126720-292864+372736-245760+65536
2
x
4
x
6
x
8
x
10
x
12
x
14
x
16
x
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.
{-ϕ
1-ϕ
ϕ-1
ϕ
ϕ
The -gon polynomial is (x), a Chebyshev polynomial of the second kind. The graphs shown are of the factors of this Chebyshev polynomial.
n
U
n-1