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
Get the Non-Euclidean Geometrymug. Solo:
Melle Mel
Rakim
Nas
KRS-One
Grandmaster Caz
The Notorious B.I.G.
Big Daddy Kane
Ice Cube
Common
Kool Moe Dee
2Pac
Jay-Z
LL Cool J
Chuck D
Slick Rick
Big Pun
GZA
Scarface
Mos Def
Busy Bee
Eminem
Lauryn Hill
Method Man
Queen Latifah
Kool G. Rap
Canibus
Ras Kass
Talib Kweli
Andre 3000
Coke La Rock
Groups:
Run-D.M.C.
Wu-Tang Clan
Cold Crush Brothers
Masta Ace Incorporated
The Beastie Boys
Public Enemy
Company Flow
Salt-N-Pepa
A Tribe Called Quest
Crash Crew
Grandmaster Flash and the Furious 5
De La Soul
Dead Prez
Black Moon
Diggin' In The Crates
Melle Mel
Rakim
Nas
KRS-One
Grandmaster Caz
The Notorious B.I.G.
Big Daddy Kane
Ice Cube
Common
Kool Moe Dee
2Pac
Jay-Z
LL Cool J
Chuck D
Slick Rick
Big Pun
GZA
Scarface
Mos Def
Busy Bee
Eminem
Lauryn Hill
Method Man
Queen Latifah
Kool G. Rap
Canibus
Ras Kass
Talib Kweli
Andre 3000
Coke La Rock
Groups:
Run-D.M.C.
Wu-Tang Clan
Cold Crush Brothers
Masta Ace Incorporated
The Beastie Boys
Public Enemy
Company Flow
Salt-N-Pepa
A Tribe Called Quest
Crash Crew
Grandmaster Flash and the Furious 5
De La Soul
Dead Prez
Black Moon
Diggin' In The Crates
" So all you hip-hops get on up,
And let's take it to the top where we belong,
Cause the age of the Beat Street wave is here.
Everybody let's sing along. Now come on."
-Melle Mel, "Beat Street Breakdown"
And let's take it to the top where we belong,
Cause the age of the Beat Street wave is here.
Everybody let's sing along. Now come on."
-Melle Mel, "Beat Street Breakdown"
by some punk kid March 17, 2005
Get the World's Best Rappers Listmug. Just hanging around. Modeled after the term used to describe the curve/shape assumed by a suspended cord or cable.
by some punk kid March 20, 2005
Get the catenarymug. The 25th anniversary of something. It is called the silver anniversary because the traditional anniversary gift for a marriage lasting 25 years is silver.
by some punk kid September 06, 2005
Get the silver anniversarymug. The fundamental theorem of arithmetic states that {n: n is an element of N > 1} (the set of natural numbers, or positive integers, except the number 1) can be represented uniquely apart from rearrangement as the product of one or more prime numbers (a positive integer that's divisible only by 1 and itself). This theorem is also called the unique factorization theorem and is a corollary to Euclid's first theorem, or Euclid's principle, which states that if p is a prime number and p/ab is given (a does not equal 0; b does not equal 0), then p is divisible by a or p is divisible by b.
Proof: First prove that every integer n > 1 can be written as a product of primes by using inductive reasoning. Let n = 2. Since 2 is prime, n is a product of primes. Suppose n > 2, and the above proposition is true for N < n. If n is prime, then n is a product of primes. If n is composite, then n = ab, where a < n and b < n. Therefore, a and b are products of primes. Hence, n = ab is also a product of primes. Since that has been established, we can now prove that such a product is unique (except for order). Suppose n = p sub1 * p sub2 * ... * p subk = q sub1 * q sub2 * ... * q subr, where the p's and q's are primes. If so, then p sub1 is divisible by (q sub1 * ... * q subr) by Euclid's first theorem. What is the relationship between p sub1 and one of the q's? If the r in q subr equals 1, then p sub1 = q sub1 since the only divisors of q are + or - 1 and + or - q and p > 1, making p = q. What about the other factors in the divisor? If p does not divide q, then the greatest common denominator of p and q is 1 since the only divisors of p are + or - 1 and + or - p. Thus there are integers m and n so that 1 = am + bn. Multiplying by q subr yieds q subr = amq subr + bnq subr. Since we are saying that p is divisible by q, let's say the q sub1 * q subr = cp. Then q subr = amq subr + bnq subr = amq subr + bcm = m(aq subr + bc). Therefore, p is divisible by q sub1 of q sub2 * ... * q subr. If p sub1 is divisible by q sub1, then p sub1 = q sub 1. If this does not work the first time, then repeat the argument until you find an equality. Therefore, one of the p's must equal one of the q's. In any case, rearrange the q's so that p sub1 = q sub1, then p sub1 * p sub2 * ... * p subk= p sub1 * q sub2 * ... * q subr and p sub2 * ... * p subk = q sub2 * ... * q subr, and so on. By the same argument, we can rearrange the remaining q's so that p sub2 = q sub2. Thus n can be expressed uniquely as a product of primes regardless of order, making the fundamental theorem of arithmetic true.
by some punk kid September 06, 2005
Get the Fundamental Theorem of Arithmeticmug. by some punk kid May 01, 2005
Get the karmamug. The act of speaking in rhyme. Not to be confused with rap, which is music. Rapness is just a manner of speech. Beats, skills, natural talent, or even basic linguistic intelligence aren't necessary.
Ayo Jo! Woah! What it be like? Please don't mind my chapped lips.
Gimme yo math baby. Don't you love my rapness?
Gimme yo math baby. Don't you love my rapness?
by some punk kid April 27, 2005
Get the rapnessmug.