Define p( t) to be the polynomial p( t) = 8 t 3 − 6 t − 1. Then by the triple-angle formula, cos π / 3 = 4 x 3 − 3 x and so 4 x 3 − 3 x = 1 / 2. If 60° could be trisected, the degree of a minimal polynomial of cos 20° over Q would be a power of two. This implies that a 60° angle cannot be trisected, and thus that an arbitrary angle cannot be trisected.ĭenote the set of rational numbers by Q. The argument below shows that it is impossible to construct a 20° angle. The angle π / 3 radians (60 degrees, written 60°) is constructible. 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. Every irrational number that is constructible in a single step from some given numbers is a root of a polynomial of degree 2 with coefficients in the field generated by these numbers. This equivalence reduces the original geometric problem to a purely algebraic problem.Įvery rational number is constructible. It follows that, given a segment that is defined to have unit length, the problem of angle trisection is equivalent to constructing a segment whose length is the root of a cubic polynomial. The triple-angle formula gives an expression relating the cosines of the original angle and its trisection: cos θ = 4 cos 3 θ / 3 − 3 cos θ / 3. From a solution to one of these two problems, one may pass to a solution of the other by a compass and straightedge construction. The problem of constructing an angle of a given measure θ is equivalent to constructing two segments such that the ratio of their length is cos θ. However, Wantzel published these results earlier than Évariste Galois (whose work, written in 1830, was published only in 1846) and did not use the concepts introduced by Galois. Wantzel's proof, restated in modern terminology, uses the concept of field extensions, a topic now typically combined with Galois theory. Pierre Wantzel published a proof of the impossibility of classically trisecting an arbitrary angle in 1837. The displayed ones are marked - an ideal straightedge is un-marked Compasses The problem of angle trisection reads:Ĭonstruct an angle equal to one-third of a given arbitrary angle (or divide it into three equal angles), using only two tools: Three problems proved elusive, specifically, trisecting the angle, doubling the cube, and squaring the circle. 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. Background and problem statement Bisection of arbitrary angles has long been solved. These "solutions" often involve mistaken interpretations of the rules, or are simply incorrect. Other techniques were developed by mathematicians over the centuries.īecause 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. 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. It is possible to trisect an arbitrary angle by using tools other than straightedge and compass. However, some special angles can be trisected: for example, it is trivial to trisect a right angle (that is, to construct an angle of 30 degrees). In 1837, Pierre Wantzel proved that the problem, as stated, is impossible to solve for arbitrary angles. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass. The example shows trisection of any angle θ> 3π / 4 by a ruler with length equal to the radius of the circle, giving trisected angle φ= θ / 3.Īngle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. Construction of an angle equal to one third a given angle Angles may be trisected via a neusis construction using tools beyond an unmarked straightedge and a compass.
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