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A non-Euclidean geometry is any geometry that contrasts the fundamental ideas of Euclidean geometry, especially with the nature of parallel lines. Any geometry that does not assume the parallel postulate or any of its alternatives is an absolute geometry (Euclid's own geometry, which does not use the parallel postulate until Proposition 28, can be called a neutral geometry). The first non-Euclidean geometries arose in the exploration of disputing Euclid's notorious Fifth Postulate, which states that if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles. Critics of the "parallel postulate" do not argue that it is a mathematical fact. Instead, they do not find it as brief, simple, and self-evident as postulates are supposed to be. Furthermore, the converse of the parallel postulate, corresponding to Proposition 27, Book I, of Euclid's Elements, has a proof, which fueled the argument that the parallel postulate should be a theorem.

Many logically equivalent statements include, but are not limited to:

1. Through a given point not on a given line, only one parallel can be drawn to the given line. (Playfair's Axiom)

2. A line that intersects one of two parallel lines intersects the other also.

3. There exists lines that are everywhere equidistant from one another.

4. The sum of the angles of a triangle is equal to two right angles.

5. For any triangle, there exists a similar noncongruent triangle.

6. Any two parallel lines have a common perpendicular.

7. There exists a circle passing through any three noncollinear points.

8. Two lines parallel to the same line are parallel to each other.

For two thousand years, geometers attempted to prove the parallel postulate, but every proof failed due to an assumption made similar to the ones above or just faulty thinking. Probably the most interesting of these are the proofs of the 17th-18th century Italian geometer Girolamo Saccheri. He tried to prove it using a reductio ad absurdum argument. By proving that the sum of the angles of a triangle cannot be greater than or less than 180 degrees, he would have achieved his goal. He successfully proved that they cannot be greater that 180 degrees, but could not find a contradiction of the latter case. He ended his proof and denied himself the opportunity to be history's first non-Euclidean geometer. This honor would be saved for two later mathematicians, Janos Bolyai and Nicolai Lobachevsky.

Both contemporaries of Carl Gauss, Lobachevsky and Bolyai did pioneering work in hyperbolic geometry, which keeps Euclid's other four postulates in tact, but supposes that through any given point not on a given line, infinitely many lines can be drawn parallel to that given line. As opposed to Euclidean geometry, which asserts that the distance between any two lines is constant, hyperbolic geometry visually means that lines curve toward each other. They discovered this to be logically coherent and a feasible alternative to Euclidean geometry. It is safe to assume that these facts were known to previous mathematicians such as Gauss and Adrien-Marie Legendre, both contributing much to elliptic functions and having conducted experiments that led them to conclude that the sum of the angles of a triangle can be less than 180 degrees. Sadly, Legendre did this in an attempt to prove the parallel postulate (hence disposing of his chance as first non-Euclidean geometer), and Gauss never published his findings in order to avoid controversy (Immanuel Kant, a prominent German philosopher of the late 1700's, in his "Critique of Pure Reason", stated the Euclidean geometry is the true geometry of the universe and to contradict it is to contradict thought itself.) Gauss did, however, discover much of differential geometry and potential theory.

Bernhard Riemann, a student of Gauss, in a famous lecture in 1854, established Riemannian geometry and discussed modern concepts such as curvature, manifolds, and (Riemannian) metrics. By giving a formula for a family of Riemannian metrics on the unit ball in Euclidean space, Riemann constructed infinitely many possible non-Euclidean geometries and provided the logical foundation for elliptic geometry, which states that through a given point not on a given line, no parallel lines exist. Visually, we can interpret this as lines curving toward each other. We cannot call Riemann, however, the sole inventor of elliptic geometry since his theory extends to all geometries, including the default Euclidean n-space. The ideas for elliptic and, mainly, hyperbolic geometry continued to develop by mathematicians of the later half of the century, such as Eugenio Beltrami, Felix Klein, and Henri Poincare. Such geometries have proven useful to the development of topology in the 20th century and to physics, notably in Albert Einstein's theory of general relativity.

Though interesting, much of non-Euclidean geometry is far too advanced to be taught in high school (or even at the undergraduate level in college!) along with basic Euclidean geometry. In order to grasp it fully and do original work in it, one must have a good working knowledge of multivariable calculus, linear and abstract algebra, real and complex analysis, and topology.

Many logically equivalent statements include, but are not limited to:

1. Through a given point not on a given line, only one parallel can be drawn to the given line. (Playfair's Axiom)

2. A line that intersects one of two parallel lines intersects the other also.

3. There exists lines that are everywhere equidistant from one another.

4. The sum of the angles of a triangle is equal to two right angles.

5. For any triangle, there exists a similar noncongruent triangle.

6. Any two parallel lines have a common perpendicular.

7. There exists a circle passing through any three noncollinear points.

8. Two lines parallel to the same line are parallel to each other.

For two thousand years, geometers attempted to prove the parallel postulate, but every proof failed due to an assumption made similar to the ones above or just faulty thinking. Probably the most interesting of these are the proofs of the 17th-18th century Italian geometer Girolamo Saccheri. He tried to prove it using a reductio ad absurdum argument. By proving that the sum of the angles of a triangle cannot be greater than or less than 180 degrees, he would have achieved his goal. He successfully proved that they cannot be greater that 180 degrees, but could not find a contradiction of the latter case. He ended his proof and denied himself the opportunity to be history's first non-Euclidean geometer. This honor would be saved for two later mathematicians, Janos Bolyai and Nicolai Lobachevsky.

Both contemporaries of Carl Gauss, Lobachevsky and Bolyai did pioneering work in hyperbolic geometry, which keeps Euclid's other four postulates in tact, but supposes that through any given point not on a given line, infinitely many lines can be drawn parallel to that given line. As opposed to Euclidean geometry, which asserts that the distance between any two lines is constant, hyperbolic geometry visually means that lines curve toward each other. They discovered this to be logically coherent and a feasible alternative to Euclidean geometry. It is safe to assume that these facts were known to previous mathematicians such as Gauss and Adrien-Marie Legendre, both contributing much to elliptic functions and having conducted experiments that led them to conclude that the sum of the angles of a triangle can be less than 180 degrees. Sadly, Legendre did this in an attempt to prove the parallel postulate (hence disposing of his chance as first non-Euclidean geometer), and Gauss never published his findings in order to avoid controversy (Immanuel Kant, a prominent German philosopher of the late 1700's, in his "Critique of Pure Reason", stated the Euclidean geometry is the true geometry of the universe and to contradict it is to contradict thought itself.) Gauss did, however, discover much of differential geometry and potential theory.

Bernhard Riemann, a student of Gauss, in a famous lecture in 1854, established Riemannian geometry and discussed modern concepts such as curvature, manifolds, and (Riemannian) metrics. By giving a formula for a family of Riemannian metrics on the unit ball in Euclidean space, Riemann constructed infinitely many possible non-Euclidean geometries and provided the logical foundation for elliptic geometry, which states that through a given point not on a given line, no parallel lines exist. Visually, we can interpret this as lines curving toward each other. We cannot call Riemann, however, the sole inventor of elliptic geometry since his theory extends to all geometries, including the default Euclidean n-space. The ideas for elliptic and, mainly, hyperbolic geometry continued to develop by mathematicians of the later half of the century, such as Eugenio Beltrami, Felix Klein, and Henri Poincare. Such geometries have proven useful to the development of topology in the 20th century and to physics, notably in Albert Einstein's theory of general relativity.

Though interesting, much of non-Euclidean geometry is far too advanced to be taught in high school (or even at the undergraduate level in college!) along with basic Euclidean geometry. In order to grasp it fully and do original work in it, one must have a good working knowledge of multivariable calculus, linear and abstract algebra, real and complex analysis, and topology.

Other examples of a non-Euclidean geometry include affine geometry, the modern projective geometries of Girard Desargues, Blaise Pascal, Michel Chasles, Jean-Victor Poncelet, and Jakob Steiner, the line geometry of Julius Plucker, the algebraic geometry of Frederigo Enriques and Francesco Severi, the enumerative geometry of Hermann Schubert, and the taxicab geometry of Hermann Minkowski.

by some punk kid
October 18, 2006