Origins of Chaos, part 1.

The most fundamental level of chaos may be found in the infinite set of Cartan-like matrices
with (1,1,1,...) in the super and subdiagonals and (-1,0,0,0,...) in the main diagonal. For example,
[-1, 1, 0;
1, 0, 1;
0, 1, 0]
.
with charpoly x^3 + x^2 - 2x - 1 = 0, has e-vals/roots relating to the Heptagon,
2 * 2Cos (k)*2Pi/7; k = 1, 2, 3.; with constants as f(x), x^2 - 2 cyclic with a 3 period..
The charpolys corresponding to the foregoing matrices = A065941, http://tinyurl.com/4zq4q;
as follows, with signs: (++--++...):
.
1;
1, 1;
1, 1, 1;
1, 1, 2, 1;
1, 1, 3, 2, 1;
1, 1, 4, 3, 3, 1;
1, 1, 5, 4, 6, 3, 1;
...
getting:
1;
x + 1;
x^2 + x - 1;
x^3 + x^2 - 2x - 1;
x^4 + x^3 - 3x^2 - 2x + 1;
x^5 + x^4 - 4x^3 - 3x^2 + 3x + 1;
...with the following trigonometric identities:
(given x = 2 Cost 2A), then
.
sin A / sin A = 1;
sin 3A / sin A = (x + 1);
sin 5A / sin A = (x^2 + x - 1)
sin 7A / sin A = (x^3 +x^2 - 2x - 1);
...etc. above; continued.