Definitions of Advanced Concepts

Date: 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/