Definitions of Advanced ConceptsDate: 11/13/98 at 12:14:48 From: Kaelyn Morrison Subject: Simple Definitions Could you please give simple, clear-cut, understandable definitions of the following? Pythagorean Triplets Principle of Duality Euclid's Elements Cycloid Fermat's Last Theorem Sincerely, Kaelyn Morrison Date: 11/13/98 at 14:42:28 From: Doctor Rob Subject: Re: Simple Definitions Pythagorean Triplets are triples of natural numbers that can be the sides of a right triangle. They satisfy the Pythagorean Theorem, so a^2 + b^2 = c^2. There are lots of real numbers that satisfy this equation, but not so many in which all of a, b, and c are natural numbers. The smallest example is (3,4,5), then (5,12,13), (8,15,17), and so on. There are formulas that give all Pythagorean Triples in terms of three freely chosen parameters: Let r and s be natural numbers such that r > s > 0, r and s have no common factor, and one of them is odd and one even. Let d be a natural number. Then: a = 2*r*s*d b = (r^2-s^2)*d c = (r^2+s^2)*d form a Pythagorean Triple, and every Pythagorean Triple is of this form (or the form gotten by swapping a and b). The Principle of Duality states that for every true theorem in plane geometry, another true theorem can be obtained by replacing the word "line" by "point" and "point" by "line," and so on. Example: Two points can be joined by a unique line. Dual: Two lines meet at a unique point. Example: Two overlapping circles have two common points. Dual: Two overlapping circles have two common tangent lines. Often the proof of one can be transformed in the same way to a proof of the other. In solid geometry, the words "point" and "plane" have to be exchanged, instead. Example: A plane and a line not parallel to it share a unique point. Dual: A point and a line not passing through it line on a unique plane. Euclid's Elements is a book written by Euclid, the ancient Greek mathematician, about plane geometry. It is the foundation of the study of geometry throughout the ages. It introduced the method of beginning with a set of postulates and deducing the truth of geometric theorems from them. A cycloid is a curve in the plane traced by a point on the circumference of a circle rolling (without slipping) along a line. It resembles a series of wide arches. If the line is the x-axis, and the radius of the circle is r, then parametric equations are: x = r*[t-sin(t)] y = r*[1-cos(t)] where t is any real number. 0 <= t <= 2*Pi gives one arch of the cycloid. Here is the cycloid for 0 <= t <= 2*Pi: Here is the cycloid for 0 <= t <= 4*Pi: Fermat's Last Theorem is a statement made by the French Mathematician Pierre de Fermat in about 1640 in the margin of one of his books that, while a^n + b^n = c^n has infinitely many natural number solutions for n = 1 and 2 (see above under Pythagorean Triples), it has none if n >= 3. Despite heroic efforts for more than three centuries by the best minds, no proof of this fact was known until the 1990's, when Prof. Andrew Wiles of Princeton University finally produced one. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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