= Angle trisection
Angle trisection is the construction of an angle one-third as large as a given angle. Some angles can be trisected by <straightedge and compass construction>, but there is no such construction for every angle.
For a concrete obstruction, trisecting $60^\circ$ would construct $20^\circ$. Set $x=2\cos20^\circ$. The triple-angle identity gives
$$
x^3-3x-1=0.
$$
The only possible rational roots of this monic cubic are $\pm1$, and neither is a root. The cubic is therefore irreducible over $\mathbb Q$, giving $x$ degree three. A <constructible number> has degree a power of two, so this angle cannot be constructed from a unit length and the given constructible $60^\circ$ angle using those tools.
In contrast, $90^\circ$ can be trisected by constructing $30^\circ$ from an equilateral triangle. The failure concerns a general construction procedure; it does not prohibit all individual trisections, numerical approximations, or constructions with other tools.
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