Angle trisection

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A few angles
A few angles
Rulers. The displayed ones are marked — an ideal straightedge is un-marked
Rulers. The displayed ones are marked — an ideal straightedge is un-marked
A compass
A compass

The problem of trisecting the angle is a classic problem of compass and straightedge constructions of ancient Greek mathematics. Two tools are allowed

  1. An un-marked straightedge, and
  2. a compass,

Problem: construct an angle one-third a given arbitrary angle.

With such tools, it is generally impossible. This requires taking a cube root, impossible with the given tools; see below.

Contents

[edit] A common misunderstanding

It is common to hear "It is impossible to trisect an angle! Q.E.D." Leaving aside the lack of proof, this statement is false: it is only impossible to solve in general using only an un-marked straightedge and a compass, i.e. it may be done by using other tools, and, some angles may be trisected with a straightedge and a compass.

[edit] Perspective and relationship to other problems

Bisection of arbitrary angles has long been solved.
Bisection of arbitrary angles has long been solved.

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:

[edit] Angles may not in general be trisected

Denote the rational numbers \mathbb{Q}.

Note that a number constructible in one step from a field K is a solution of a second-order polynomial; again, see constructible number. Note also that π / 3 radians (60 degrees, written 60°) is constructible.

However, the angle of π / 3 radians (60 degrees) cannot be trisected. Note \cos(\pi/3) = \cos(60^\circ) = 1/2.

If 60° could be trisected, the minimal polynomial of \cos(20^\circ) over \mathbb{Q} would be of second order. Note the trigonometric identity cos(3α) = 4cos3(α) − 3cos(α). Now let y = \cos(20^\circ).

By the above identity, \cos(60^\circ) = 1/2 = 4y^{3} - 3y. So 4y3 − 3y − 1 / 2 = 0. Multiplying by two yields 8y3 − 6y − 1 = 0, or (2y)3 − 3(2y) − 1 = 0. Now substitute x = 2y, so that x3 − 3x − 1 = 0. Let p(x) = x3 − 3x − 1.

The minimal polynomial for x (hence \cos(20^\circ)) is a factor of p(x). If p(x) has a rational root, by the rational root theorem, it must be 1 or −1, both clearly not roots. Therefore p(x) is irreducible over \mathbb{Q}, and the minimal polynomial for \cos(20^\circ) is of degree 3.

So an angle of 60^\circ = \pi/3 radians cannot be trisected.

[edit] Some angles may be trisected

However, some angles may be trisected. Given angle θ, angle 3θ trivially trisects to θ. More notably, 2π / 5 radians (72°) may be constructed, and may be trisected, [1]. Also there are angles, while non-constructable, but (if somehow given) are trisectable, for example 3π / 7

[edit] One general theorem

Again, denote the rational numbers Q:

Theorem: The angle θ may be trisected if and only if q(t) = 4t3 − 3t − cos(θ) is reducible over the field extension Q(cos(θ)).

Proof. The proof would take us afield, but it may be derived from the above trig identity.[1]

[edit] Means to trisect angles by going outside the Greek framework

[edit] Origami

Trisection, like many constructions impossible by ruler and compass, can easily be accomplished by the more powerful (but physically easy) 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). See mathematics of paper folding.

[edit] With a marked ruler

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. The next construction is originally due to Archimedes, called a Neusis construction, i.e., that uses tools other than an un-marked straightedge.

This requires three facts from geometry (at right):

  1. Any full set of angles on a straight line add to 180°,
  2. The sum of angles of any triangle is 180°, and,
  3. Any two equal sides of an isosceles triangle meet the third in the same angle.
Angle a to be trisected

Look to the diagram at right; note angle a left of point B. We trisect angle a.

First, a ruler has two marks distance AB apart. Extend the lines of the angle and draw a circle of radius AB.

"Anchor" the ruler at point A, and move it until one mark is at point C, one at point D, i.e., CD = AB. A radius BC is drawn as obvious. Triangle BCD has two equal sides, thus is isosceles.

That is to say, line segments AB, BC, and CD all have equal length. Segment AC is irrelevant.

Now: Triangles ABC and BCD are isosceles, thus by Fact 3 each has two equal angles. Now re-draw the diagram, and label all angles:

Again, trisect angle a with a marked straightedge

Hypothesis: Given AD is a straight line, and AB, BC, and CD are all equal length,

Conclusion: angle b = 1 / 3a.

Proof:

Steps:

  1. From Fact 1) above, e + c = 180°.
  2. Looking at triangle BCD, from Fact 2) e + 2b = 180°.
  3. From the last two equations, c = 2b.
  4. From Fact 2), d + 2c = 180°, thus d = 180° − 2c, so from last, d = 180° − 4b.
  5. From Fact 1) above, a + d + b = 180°, thus a + (180° − 4b) + b = 180°.

Clearing, a − 3b = 0, or a = 3b, and the theorem is proved.

Again: this construction stepped outside the framework of allowed constructions by using a marked straightedge. There is an unavoidable element of inaccuracy in placing the straightedge.

[edit] With a string

Hutcheson published an article in Mathematics Teacher, vol. 94, No. 5, May, 2001 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.

Hutcheson constructed a cylinder from the angle to be trisected by drawing an arc across the angle, completing it as a circle, and constructing from that circle a cylinder on which a, say, equilateral triangle was inscribed (a 360-degree angle divided in three). This was then "mapped" onto the angle to be trisected, with a simple proof of similar triangles.

For the detailed proof and its generalization, see the article cited: Mathematics Teacher, vol. 94, No. 5, May, 2001, pp. 400-405.

There are other constructions (references).

[edit] See also

[edit] Notes

  1. ^ Stewart, Ian (1989). Galois Theory. Chapman and Hall Mathematics, pg. 58. ISBN 0412345501. 

[edit] External references

[edit] Other means of trisection