Alphabet of Lines: Geometric Construction Essay

806 Words Jan 15th, 2013 4 Pages
In antiquity, geometric constructions of figures and lengths were restricted to the use of only a straightedge and compass (or in Plato's case, a compass only; a technique now called a Mascheroni construction). Although the term "ruler" is sometimes used instead of "straightedge," the Greek prescription prohibited markings that could be used to make measurements. Furthermore, the "compass" could not even be used to mark off distances by setting it and then "walking" it along, so the compass had to be considered to automatically collapse when not in the process of drawing a circle.

Because of the prominent place Greek geometric constructions held in Euclid's Elements, these constructions are sometimes also known as Euclidean
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Call the upper endpoint of this perpendicular diameter . For the pentagon, find the midpoint of and call it . Draw , and bisect , calling the intersection point with . Draw parallel to , and the first two points of the pentagon are and . The construction for the heptadecagon is more complicated, but can be accomplished in 17 relatively simple steps. The construction problem has now been automated (Bishop 1978).

Simple algebraic operations such as , , (for a rational number), , , and can be performed using geometric constructions (Bold 1982, Courant and Robbins 1996). Other more complicated constructions, such as the solution of Apollonius' problem and the construction of inverse points can also accomplished.

One of the simplest geometric constructions is the construction of a bisector of a line segment, illustrated above.

The Greeks were very adept at constructing polygons, but it took the genius of Gauss to mathematically determine which constructions were possible and which were not. As a result, Gauss determined that a series of polygons (the smallest of which has 17 sides; the heptadecagon) had constructions unknown to the Greeks. Gauss showed that the constructible polygons (several of which are illustrated above) were closely related to numbers called the Fermat primes.

Wernick (1982) gave a list of 139 sets of three located points from which a triangle was to be

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