The concept of impossible constructions in mathematics draws in a unique interest by Mathematicians wanting to find answers that none have found before them. For the Greeks, some impossible constructions weren’t actually proven at the time to be impossible, but merely so far unachieved. For them, there was excitement in the idea that they might be the first one to do so, excitement that lay in discovery. There are a few impossible constructions in Greek mathematics that will be examined in this chapter. They all share the same criteria for constructability: that they are to be made using solely a compass and straightedge, and were referred to as the three “classical problems of antiquity”. The requirements of using only a compass and straightedge were believed to have originated from Plato himself. 1
The Greeks were able to a lot of things with only a compass and a straight edge (although these were not their sole tools, the Greeks in fact had access to a wide variety of tools as they were a fairly modern society). For example, they found means to construct parallel lines, to bisect angles, to construct various polygons, and to construct squares of equal or twice the area of a given polygon. However, three constructions that they failed to achieve with only those two tools were trisecting the angle, doubling the cube, and squaring the circle.
The first impossible construction to be examined is the trisection of an angle. Its purpose, to divide an arbitrary angle into three equal angles, could have proved useful for a variety of fields. However, mathematicians failed time after time to come up with a solution using only a compass and straightedge. It began to be pondered circa 5th century B.C. in Greece during the time of Plato. T...
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... impossible construction is the doubling of the cube. In this scenario you are given a cube and the goal is to construct a larger cube with a volume twice that of the first cube. If you were to assign the length of one side of the first cube the length x, and volume v, then x3 =v. If the first cube for example had a volume of 1, then the second cube would have a volume of 2. It’s length would therefore be the cubed root of 2, and as proven by Galois Theory, any root of a third degree polynomial is not a constructible number.
Works Cited
1. Anglin, W.S., Mathematics: A Concise History and Philosophy, Springer-Verlag, NewYork, 1994, pp. 75-80
2. Angle Trisection. Available: http://en.wikipedia.org/wiki/Angle_trisection. Last accessed 16th Sept 2013
3. Squaring the circle. Available: http://en.wikipedia.org/wiki/Squaring_the_circle. Last accessed 13 Sept 2013
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Geometry, a cornerstone in modern civilization, also had its beginnings in Ancient Greece. Euclid, a mathematician, formed many geometric proofs and theories [Document 5]. He also came to one of the most significant discoveries of math, Pi. This number showed the ratio between the diameter and circumference of a circle.
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Howard, Eves. An Introduction to the History of Mathematcis. Fifth Edition. New York: The Saunders Series, 1983. Print.
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"The Foundations of Geometry: From Thales to Euclid." Science and Its Times. Ed. Neil Schlager and Josh Lauer. Vol. 1. Detroit: Gale, 2001. Gale Power Search. Web. 20 Dec. 2013.
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Burton, D. (2011). The History of Mathematics: An Introduction. (Seventh Ed.) New York, NY. McGraw-Hill Companies, Inc.
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