Trisecting an Angle

Author:   Ishita Srivastava
B.Sc. (H) Mathematics

Angle trisection, as the name signifies, refers to dividing a given angle into three equal parts. Though this seems quite simple, here is a very trivial point, which seeks to ask: “Is it always possible to trisect any randomly chosen angle using a compass and an unmarked straight-edge only?” This has been a classical problem of ancient Greek mathematics and art of construction. This article aims to provide a basic insight into the concept of angular trisection.

It has already been proven geometrically by Pierre Wantzel in the \({19}^{th}\) century that it is not possible to trisect every angle. However, this, in no way, intends to say that no angle, whatsoever, can be trisected using a compass and an unmarked ruler. If we consider a compass-constructible angle \(\theta\), then an angle of measure \(3\theta\) can be trisected trivially by a compass. In fact, certain angles that are not constructible, can still be trisected with ease. For instance, given an angle \(\frac{6 \pi }{11} [\approx {98.18}^{\circ}]\) , on trisection yields an angle of \(\frac{2 \pi }{11}\). This angle cannot be constructed by a compass, but four such angles can be combined to form \(\frac{24 \pi}{11}\), which is a \(2 \pi \) radian circle, leaving an extra angle measuring \(\frac{2 \pi }{11}\) , which is equal to one-third of the original angle, and thus, its trisection. However, this method cannot be applied to all angles since all angles might not always follow a similar calculation. Angles as simple as \(\frac{ \pi }{6}\) cannot be trisected using this method, and hence are not ‘trisectible’ using a compass-straight edge combination. In addition to this, the trisection of this, which is \(\frac{ \pi }{18}\) radians or \({20}^{\circ}\) is not compass-constructible. After the preceding discussion, it has become clearly evident that we need a different process for trisecting angles. Apart from using highly advanced geometrical equipments or its electronic counterparts, there is a simpler way of achieving this seemingly difficult task. This is done by using Japanese Origami paper folding technique.

Origami[‘Ori’: folding, ‘kami/gami’: paper] It is the art of paper folding, which is often associated with Japanese culture. The goal is to transform a flat sheet of square paper into a finished sculpture through folding and sculpting techniques. A few Origami folds can be combined in a number of ways to make intricate designs. The best known model is the Japanese paper crane.

Follow these basic steps in order to trisect an angle:

Step 0: Take a piece of square sheet and create an arbitrary angle by folding the paper such that the edge of the paper acts as the base of the angle and the crease created by the fold (which meets the base edge at the bottom left corner) acts as the arm of the angle. Mark this as \(\theta\). Unfold it and proceed.

Step 1: Fold the paper into half horizontally, and call this crease \({L}_{1}\). Unfold it.

Step 2: Then, fold the lower edge upto L1 and call this second crease as \({L}_{2}\).

Step 3: Mark the lower left point of the paper (that is, the point where angle \( \theta \) originates) as O and the left end of crease \({L}_{1}\) as A.

Step 4: Fold the paper such that point O matches (or lies on) the lower crease, that is, \({L}_{2}\) and the point A matches the crease which defines the angle \(\theta\). Press the paper at this step to create another crease \({L}_{3}\).

Step 5: Mark the point on \({L}_{2}\) where point O matches the crease, during the creation of \({L}_{3}\) as B.

Step 6: Unfold the paper and locate the point where \({L}_{3}\) intersects \({L}_{2}\). Mark this point as C.

Step 7: Now, fold the paper so as to form two new creases; one which connects O and B, and the other which connects O and C. Call them \({L}_{4}\) and \({L}_{5}\) respectively.

Step 8: The newly formed creases \({L}_{4}\) and \({L}_{5}\) are the ones which trisect the angle \(\theta\).

Thus, we see that trisecting an arbitrary angle, which is not always possible using a compass and a straight-edge only, can easily be done through this simple paper folding technique.

Now, the careful and curious reader must have a very genuine question in mind, and without answering it, this article would be incomplete. You must have pondered upon ‘the secret super powers’ of Origami. Putting the question into simpler words, “What does Origami do to the paper, which a compass and a straight-edge are unable to, that allows trisecting an angle with such ease?” This property is owed to the methods that Origami provides for finding the solutions (or roots) not just of quadratic, but of cubic equations as well. The answer to this question lies in the fact that\({L}_{3}\) is a shared tangent of two parabolas (as mentioned earlier), combined with the methods of Origami for finding roots of quadratic and cubic equations. This is what makes an angle ‘trisectible’ using the techniques of Origami.