Trisecting an Angle Problem

History of the Problem

Trisection of an angle means dividing a given angle into three smaller angles with the same measure. This is one of the three geometric problems of antiquity that had puzzled mathematicians since the time of Euler, more than 2000 years ago. This problem was found by the ancient Greek mathematicians, who could not solve it using the rigid Greek rules of constructions – using only a compass and an unmarked straightedge. While there are some angles which can be trisected easily like 180°, 90°, 27° etc., the solution cannot be extended to any arbitrary angle (O’Connor ; Robertson 1)

(http://mathworld.wolfram.com/AngleTrisection.html)

Mathematicians working on the Problem

Trisecting an angle came to be known as a geometric problem when Hippocrates was working to find its solution. His method was to use a marked straightedge and draw perpendiculars to the angle to form a rectangle. Then he chose points such that one forms segments twice as long as other segments. Ultimately, a trisected angle can be drawn. Although, he did manage to find a solution to the problem, it was not popular. The Greeks did not like this, as they were more interested in the solutions from a purely mathematical point of view, which they did not find here (O’Connor & Robertson 7)

Another early mathematician who worked on the problem of angle trisection was the Greek Hippias, whose main contribution to Greek geometry a curve called the quadratrix. The curve allows to and it allows dividing an angle into any number of equal parts and can hence be used for angle trisection (Loy 35).

One popular solutions, though it was again a mechanical solution like one given by Hippocrates, was found by Archimedes. According to him angle trisection was possible if two marks were allowed on the straightedge (Baum 4). Like Hippocrates, he also used a marked straight edge and a compass to perform the trisection. Archimedes created a curve that he called the spiral, and used this to find the trisection of an angle. This spiral was generally used to for cutting an angle contained by straight lines in a given ratio, and was not just limited to angle trisection (O’Connor ; Robertson 12)

Another Greek mathematician who studies the trisection of an angle was Nicomedes, who lived about the same time as Archimedes and produced his famous conchoid curve, which forms the basis for an angle trisection. However, his work was more was more of a theoretical than practical interest (Jackter 12). This theory was discussed by Pappus in his famous book Mathematical Collection, where he also wrote about how the problem of trisecting an angle was solved by Apollonius using conics. Pappus himself gave two solutions for this problem, both of which involved the drawing of a hyperbola (Jackter 15).

Mathematical Solution

All of these solutions given above did not provide the accurate construction for the trisection of angles. This problem remained a puzzle to mathematicians until the 19th century. Carl Friedrich Gauss, in 1800s realized that the problem could not be solved using only lines and circles. Unfortunately, Gauss gave no proofs of his theory. The first person to give the proof of impossibility was Pierre Laurent Wantzel, who published his proofs in a paper called “Research on the Means of Knowing If a Problem of Geometry Can Be Solved with Compass and Straight-edge”, in 1837. In this paper the impossibility of the solution under Euclidean restrictions was proved by regarding the magnitudes involved not as geometric segments, but as numerical lengths, via analytic geometry.

Wantzel showed that the two problems of trisecting an angle and of solving a cubic equation are equivalent, and in fact very few cubic equations can be solved using the straight-edge-and-compass method (Turner 2001). The prrof of impossibility if a quadrature was lgiven by He thus deduced that most angles cannot be trisected. Hence, he used algebra rather than pure geometry to prove the impossibility..

Figure below traces the time line of the problem from its origins to its solution:

(Ringle, Schweiger ; Smith 2)

References

Professor Baum, “Mathematical Impossibilities: Angle Trisection is Impossible”, 4th

December 2006, http://www.clubs.psu.edu/up/math/minutes/2006fall/Dec%204.doc

Jackter A, “The Problem of Angle Trisection in Antiquity”, 2000,

http://www.math.rutgers.edu/~cherlin/History/Papers2000/jackter.html

Loy J, “Trisection of an Angle”, 1997-2003, http://www.jimloy.com/geometry/trisect.htm

O’Connor J.J, Robertson E.F, “Trisecting an Angle”, April 1999,

http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trisecting_an_angle.html

Ringle R, Schweiger J, Smith A, “Trisecting an Angle”, 2007,

http://www.cst.cmich.edu/users/piate1kl/MTH_341_S07/Trisecting.ppt#9

Turner M, “How to trisect a line”, 23rd December 2001,

http://www.everything2.com/index.pl?node_id=1186835

http://mathworld.wolfram.com/AngleTrisection.html

http://mathforum.org/dr.math/faq/faq.impossible.construct.html