There are other angles that can be trisected with relative ease. In fact, Hippocrates of Chios, who we have already seen was instrumental in finding solutions to the problem of squaring the circle, found a relatively simple method to trisect any given angle.

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Unfortunately at least for traditional Euclidean geometry , Hippocrates's method did not use only a straightedge and compass in its construction. Others succeeded in solving the problem, but never by using the plane methods that required only straightedge and compass. The methods that the Greek mathematicians did find to trisect an angle involved curves such as conic sections or more complicated curves requiring mechanical devices to construct.

The curve called the quadratix, which we have seen was used to square the circle, was also used to trisect an angle. Curves such as Nicomedes's c. The Greeks found several methods for trisecting an angle using curves called conic sections.

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A conic section is a curve obtained by the intersection of a cone and a plane. Examples of conic sections are circles, ellipses, parabolas, and hyperbolas. The method of solving the trisection problem by the use of conic sections lived on for many centuries. None of these curves, however, could be constructed using the restrictions required by traditional Greek geometry. The problem of trisecting an arbitrary angle using only a straightedge and a compass generated interest for centuries just as the problem of squaring the circle had.

If bisecting an arbitrary angle was so easy, and trisecting certain angles was also relatively simple, surely, thought some, the problem of trisecting an arbitrary angle could be solved. Mathematicians found that very close approximations to trisecting an angle could be made by continued bisections. In fact, if this process were repeated an infinite number of times, an exact trisection could be made. In addition to the straightedge and compass restrictions, however, the Greeks required that a process be accomplished in a finite number of steps to be valid.

Continued bisection, then, did not represent an acceptable solution to the trisection problem. Like the problem of squaring the circle, it had become evident to most trained mathematicians by the eighteenth century that a solution to the problem of trisection probably did not exist. In fact, in the Paris Academy of Sciences discontinued examination of angle trisection methods submitted by the public, much like they had done with solutions to the circle-squaring problem.

It was not until that Pierre Wantzel completed a proof that the problem was impossible using only a straightedge and a compass. Wantzel essentially showed that trisecting an angle could be reduced to solving a cubic equation. Since most cubic equations could not be solved with straightedge and compass, neither could the trisection problem. This put a stop to attempts by serious mathematicians to solve the problem, but unconvinced amateurs continue to seek fame by looking for methods to trisect an angle.

For a cube with a given volume, can one construct another cube whose volume is double the original using only a straightedge and compass? This is the third problem of ancient Greek geometry.

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Like the problems of squaring the circle and trisecting an angle, the origin of the problem of doubling a cube also referred to as duplicating a cube is not certain. Two stories have come down from the Greeks concerning the roots of this problem.

The first is that the oracle at Delos commanded that the altar in the temple which was a cube be doubled in order to save the Delians from a plague. After failing to solve the problem, the men from Delos questioned Plato as to how this might be done. Plato's response was that the command was actually a reproach from the gods for neglecting the study of geometry.

Needless to say, the plague at Delos continued. The problem of doubling a cube is often referred to as the Delian problem after the citizens of Delos who suffered for their ignorance. A second version of the origin of the cube-doubling problem relates that King Minos commanded that a tomb be erected for his son, Glaucus.

After its completion, however, Minos was dissatisfied with its size, as its sides were only feet He commanded that the cubic tomb be made twice as large by doubling each side of the tomb. Since the volume of a cube is the length multiplied by the width multiplied by the height, the volume of the original cube was. So if each side were doubled, the resulting volume would not be double the original, but eight times the original volume. Like the men of Delos, King Minos's subjects could not solve the problem of doubling the cube.

Although both of these stories may contain as much myth as fact, the problem of doubling a cube using only a straightedge and a compass became important in Greek geometry. Eratosthenes b. Interestingly, Plato is also credited with a mechanical solution to the problem, although he is said to have abhorred the use of mechanical devices in geometry. Archytas of Tarentum c.

Eudoxus b. In addition, Nicomedes c. Just as with the other two problems of Greek geometry, the problem of doubling a cube was solved using conic sections. Menaechmus c. In fact, it is said that Menaechmus discovered conic sections while attempting to solve the problem. Many other famous Greek mathematicians, including Apollonius, Heron, Philon, Diocles, Sporus, and Pappus, constructed their own solutions to the problem of doubling the cube.

None, however, succeeded in solving the problem using the straightedge and compass alone. Hippocrates of Chios, who was an important figure in the history of the problems of squaring the circle and trisecting an angle, also worked on the problem of doubling a cube. Hippocrates found that the problem of doubling a cube could be solved if the related problem of finding two mean proportionals between one line and its double was solved.

Although Greek mathematicians were able to find ways to perform this construction, none met the requirements of the exclusive use of straightedge and compass. Attempts to solve the problem of doubling the cube have led to many other important discoveries in mathematics, just as the same has happened with the other two problems of Greek geometry. Conic sections, discovered by Menaechmus while he was trying to solve the problem of doubling the cube, have been an extremely important part of mathematics throughout history.

The Persian mathematician and poet Omar Khayyam used the intersection of conic sections to solve cubic equations, a problem closely related to duplicating the cube. There is a very good reason that neither the Greeks nor anyone else were ever able to find a solution to the problem of doubling the cube using plane methods; such a solution does not exist. The final nail in the cube-doublers coffin came from Pierre Wantzel in Wantzel proved that a geometric construction to double a cube using only straightedge and compass could not exist, just as he had proved that a similar construction for trisecting an angle was impossible. Another physics based problem, the Yang Mills theories aim to tackle problems in our understanding of the fundamental forces of the universe.

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To explain these particles, Yang and Mills attempted to describe elemental particles by constructing a model based on geometric theories. Their theory, which says that certain quantum particles have a positive mass, has been verified by a number of computer simulations. Essentially, the resolution of this problem would solve many other problems, while for as long as it remains unsolved, so do many other problems in the fields of maths and computing.

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