Angle trisection

Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics.

In 1837, Pierre Wantzel proved that the problem, as stated, is impossible to solve for arbitrary angles.

For example, neusis construction, also known to ancient Greeks, involves simultaneous sliding and rotation of a marked straightedge, which cannot be achieved with the original tools.

Because it is defined in simple terms, but complex to prove unsolvable, the problem of angle trisection is a frequent subject of pseudomathematical attempts at solution by naive enthusiasts.

These "solutions" often involve mistaken interpretations of the rules, or are simply incorrect.

[1] Using only an unmarked straightedge and a compass, Greek mathematicians found means to divide a line into an arbitrary set of equal segments, to draw parallel lines, to bisect angles, to construct many polygons, and to construct squares of equal or twice the area of a given polygon.

Three problems proved elusive, specifically, trisecting the angle, doubling the cube, and squaring the circle.

[2] Wantzel's proof, restated in modern terminology, uses the concept of field extensions, a topic now typically combined with Galois theory.

Therefore, any number that is constructible by a sequence of steps is a root of a minimal polynomial whose degree is a power of two.

[6] For any nonzero integer N, an angle of measure 2π⁄N radians can be divided into n equal parts with straightedge and compass if and only if n is either a power of 2 or is a power of 2 multiplied by the product of one or more distinct Fermat primes, none of which divides N. In the case of trisection (n = 3, which is a Fermat prime), this condition becomes the above-mentioned requirement that N not be divisible by 3.

[5] The general problem of angle trisection is solvable by using additional tools, and thus going outside of the original Greek framework of compass and straightedge.

Some of these methods provide reasonable approximations; others (some of which are mentioned below) involve tools not permitted in the classical problem.

The mathematician Underwood Dudley has detailed some of these failed attempts in his book The Trisectors.

[1] Trisection can be approximated by repetition of the compass and straightedge method for bisecting an angle.

[7] Trisection, like many constructions impossible by ruler and compass, can easily be accomplished by the operations of paper folding, or origami.

Huzita's axioms (types of folding operations) can construct cubic extensions (cube roots) of given lengths, whereas ruler-and-compass can construct only quadratic extensions (square roots).

There are a number of simple linkages which can be used to make an instrument to trisect angles including Kempe's Trisector and Sylvester's Link Fan or Isoklinostat.

[8] In 1932, Ludwig Bieberbach published in Journal für die reine und angewandte Mathematik his work Zur Lehre von den kubischen Konstruktionen.

Now the right triangular ruler is placed on the drawing in the following manner: one leg of its right angle passes through O; the vertex of its right angle is placed at a point S on the line PC in such a way that the second leg of the ruler is tangent at E to the circle centered at A.

With a similar construction, one can improve the location of E, by using that it is the intersection of the line SE and its perpendicular passing through A.

On the other hand, the triangle PAE is isosceles, since all radiuses of a circle are equal; this implies that

There are certain curves called trisectrices which, if drawn on the plane using other methods, can be used to trisect arbitrary angles.

[10] Examples include the trisectrix of Colin Maclaurin, given in Cartesian coordinates by the implicit equation and the Archimedean spiral.

Another means to trisect an arbitrary angle by a "small" step outside the Greek framework is via a ruler with two marks a set distance apart.

Hypothesis: Given AD is a straight line, and AB, BC, and CD all have equal length, Conclusion: angle b = ⁠a/3⁠.

Thomas Hutcheson published an article in the Mathematics Teacher[11] that used a string instead of a compass and straight edge.

A string can be used as either a straight edge (by stretching it) or a compass (by fixing one point and identifying another), but can also wrap around a cylinder, the key to Hutcheson's solution.

Trisection is executed by leaning the end of the tomahawk's shorter segment on one ray, the circle's edge on the other, so that the "handle" (longer segment) crosses the angle's vertex; the trisection line runs between the vertex and the center of the semicircle.

As a tomahawk can be used as a set square, it can be also used for trisection angles by the method described in § With a right triangular ruler.

1 A regular polygon with n sides can be constructed with ruler, compass, and angle trisector if and only if