# How do you trisect an angle?

Mar 19, 2016

Using neusis or origami.

#### Explanation:

If you are talking about classical construction with unmarked straight edge and compasses, then this is impossible. Ruler and compasses allow the construction of sums, differences, products, quotients and square roots, including simple ways to bisect an angle.

Trisecting an angle involves taking a cube root. You can do this using a construction the Greeks frowned upon, called "neusis" (meaning "incline towards"), involving a marked ruler. You can also do it in origami (See https://plus.maths.org/content/trisecting-angle-origami).

One of the interesting things about this is that whereas you can construct a regular triangle, square, pentagon or hexagon using ruler and compasses, the regular heptagon (or nonagon or tridecagon) cannot be so constructed. It can be constructed using neusis or origami, because it essentially involves being able to solve a cubic equation.

Constructing a heptagon is equivalent to constructing the Complex $7$th roots of $1$, which means solving ${x}^{7} - 1 = 0$ apart from the obvious $x = 1$ solution.

We find:

${x}^{7} - 1 = \left(x - 1\right) \left({x}^{6} + {x}^{5} + {x}^{4} + {x}^{3} + {x}^{2} + x + 1\right)$

So we want to solve:

${x}^{6} + {x}^{5} + {x}^{4} + {x}^{3} + {x}^{2} + x + 1 = 0$

If you divide this equation through by ${x}^{3}$ you get:

${x}^{3} + {x}^{2} + x + 1 + \frac{1}{x} + \frac{1}{x} ^ 2 + \frac{1}{x} ^ 3 = 0$

which is symmetric in $x$ and $\frac{1}{x}$.

Substituting $t = x + \frac{1}{x}$ it becomes:

${t}^{3} + {t}^{2} - 2 t - 1 = 0$

..a cubic in $t$

If we can solve this cubic in $t$, then we are left with quadratics to solve to find $x$.

Being able to trisect an angle (using neusis or origami) allows us to solve the cubic and hence construct a regular heptagon.