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Random Maths

Page history last edited by PBworks 13 years, 4 months ago

Unusual Mathematical Concepts

Some interesting things about maths...

 

Non-Euclidean Geometry

The term non-Euclidean geometry (also spelled: non-euclidian geometry) describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. In Euclidean geometry, if we start with a line l and a point A, which is not on l, then we can only draw one line through A that is parallel to l. In hyperbolic geometry, by contrast, there are infinitely many lines through A parallel to l, and in elliptic geometry, parallel lines do not exist. (See the entries on hyperbolic geometry and elliptic geometry for more information.)

 

Another way to describe the differences between these geometries is as follows: Consider two lines in a two-dimensional plane that are both perpendicular to a third line. In Euclidean and hyperbolic geometry the two lines are then parallel. In Euclidean geometry the lines remain at a constant distance, intersecting only in the infinite; while in hyperbolic geometry they "curve away" from each other, increasing their distance as one moves further from the point of intersection with the common perpendicular. In elliptic geometry the lines "curve toward" each other and eventually intersect. Therefore no parallel lines exist in elliptic geometry.

 

Divine Proportion

1.618033989

The term "golden number" is also used for an unrelated concept: see golden numbers. The term "golden mean" also has an unrelated meaning: see golden mean (philosophy).

 

Since the sixteenth century, shapes proportioned according to the golden ratio have been considered aesthetically pleasing in Western cultures; the golden ratio is still frequently used in art and design. The golden ratio has attracted a large following for its supposed aesthetic, psychological, historical, mystical, natural, and metaphysical properties, in addition to its mathematical properties.

 

The most common other names used for or closely related to the golden ratio are golden section (Latin: sectio aurea), golden mean, golden number, and phi.

 

Infinity

The word infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts which arise in theology, philosophy, mathematics and everyday life. Popular or colloquial usage of the term often does not accord with its more technical meanings.

 

In theology, for example in the work of theologians such as Duns Scotus, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity. In philosophy, infinity can be attributed to space and time, as for instance in Kant's first antinomy. In both theology and philosophy, infinity is explored in articles such as the Ultimate, the Absolute, God, and Zeno's paradoxes.

 

In mathematics, infinity (∞) is relevant to, or the subject matter of, limits, aleph numbers, classes in set theory, Dedekind-infinite sets, large cardinals, Russell's paradox, hyperreal numbers, projective geometry, extended real numbers and the absolute Infinite.

 

In popular culture, we have Buzz Lightyear's rallying cry, "To infinity — and beyond!", which may also be viewed as the rallying cry of set theorists considering large cardinals.

 

Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. It is one of many things named after Bernhard Riemann. The key difference between the two is that on a pseudo-Riemannian manifold the metric tensor need not be positive-definite. Instead a weaker condition of nondegeneracy is imposed.

 

Arguably, the most important type of pseudo-Riemannian manifold is a Lorentzian manifold. Lorentzian manifolds occur in the general theory of relativity as models of curved 4-dimensional spacetime. Just as Riemannian manifolds may be thought of as being locally modeled on Euclidean space, Lorentzian manifolds are locally modeled on Minkowski space.

 

Pseudo-Riemannian metrics of signature (p,1) (or sometimes (1,q), see sign convention) are called Lorentzian metrics. A manifold equipped with a Lorentzian metric is naturally called a Lorentzian manifold. After Riemannian manifolds, Lorentzian manifolds form the most important subclass of pseudo-Riemannian manifolds. They are important because of their physical applications to the theory of general relativity. A principal assumption of general relativity is that spacetime can be modeled as a Lorentzian manifold of signature (3,1)

 

Arbelos

In geometry, an arbelos is a plane region bounded by a semicircle of radius 1, connected to semicircles or radius r and (1 − r), all oriented the same way and sharing a common baseline. (The baseline of a semicircle is a straight line forming the diameter which connects the ends of the arc.)

 

"Arbelos" means "shoemaker's knife" in Greek.

 

Riemann Sphere

 

Riemann sphere needs a drawing for the version being fermented in the discussion page. It should show a sphere, the top tangent plane (or both if it would be better), and a projection or two from the south pole through the sphere onto the top plane. It should clearly be shown that the point z = a + ib on the plane corresponds to the point on the sphere that lies on the same line. Also, the top plane should be labelled as the z plane

 

In mathematics, the Riemann sphere, named after Bernhard Riemann, is the unique way of viewing the extended complex plane (the complex plane plus a point at infinity) so that it looks exactly the same at the point infinity as at any complex number. The main application is to deal with extended complex functions (which may be defined at the point infinity and/or take the value infinity, in addition to complex numbers) in the same way at the point infinity as at any complex number, specifically with respect to continuity and differentiability.

 

From the geometrical view of the plane that deals with points, lines, circles and angles but not distances, the Riemann sphere is created by adding a point at infinity through which all lines cross, with parallel lines being tangent there and all other lines crossing at the same angle as they do at an existing point. This geometry is realized as a 2-dimensional sphere formed from the extended complex plane using the stereographic projection, where lines in the complex plane become circles through infinity. Angles in the Riemann sphere are identical to the corresponding angles in the complex plane (and the same is true at infinity with the natural choice of the angle between two lines at infinity).

 

Topologically, the Riemann sphere is the one-point compactification of the complex plane. This gives it the topology of a 2-dimensional sphere, preserving the topology of the complex plane. The Riemann sphere can be conveniently identified with a geometrical 2-dimensional sphere, in which lines become circles through infinity.

 

The 2-sphere admits a unique complex structure turning it into a Riemann surface (i.e. a 1-dimensional complex manifold). The Riemann sphere can be characterized as the unique simply-connected, compact Riemann surface, and may be taken to have the complex plane as a complex sub-manifold.

 

In all of these viewpoints, the point at infinity acquires an identical role to any point in the complex plane.

 

Pi

The mathematical constant π is an irrational real number, approximately equal to 3.14159, which is the ratio of a circle's circumference to its diameter in Euclidean geometry, and has many uses in mathematics, physics, and engineering. It is also known as Archimedes' constant (not to be confused with Archimedes number) and as Ludolph's number.

 

* March 14 (3/14 in U.S. date format) marks Pi Day which is celebrated by many lovers of π. Incidentally, it is also Einstein's birthday.

* On July 22, Pi Approximation Day is celebrated (22/7 - in European date format - is a popular approximation of π).

* 355/113 (~3.1415929) is sometimes jokingly referred to as "not π, but an incredible simulation!"

* Singer Kate Bush's 2005 album "Aerial" contains a song titled "π", in which she sings π to its 137th decimal place; however, for an unknown reason, she omits the 79th to 100th decimal places.7 She was preceded in this achievement by several years by a Swedish indie math lyrics artist under the moniker Matthew Matics, who loses track of the decimals at about the same point in the series.

* John Harrison (1693–1776) (famed for winning the longitude prize), devised a meantone temperament musical tuning system derived from π, now called Lucy Tuning.

* Users of the A9.com search engine are eligible for an amazon.com program offering discounts of (π/2)% on purchases.

* The Heywood Banks song "Eighteen Wheels on a Big Rig" has the singer(s) count pi in the final verse; they reach "eight hundred billion" before going into the chorus.

* In 1932, Stanisław Gołąb proved that, if a unit disk is defined using a non-standard norm as "distance", the ratio of circumference to diameter will always be between 3 and 4; these values are attained if and only if the unit "circle" has the shape of a regular hexagon or a parallelogram respectively. See unit disc for details.

* John Squire (of The Stone Roses) mentions π in a song written for his second band The Seahorses called "Something Tells Me". The song was recorded in full by the full band, and appears on the bootleg of the never released second-album recordings. The song ends with the lyrics, "What's the secret of life? It's 3.14159265, yeah yeah!!"

* Hard 'n Phirm's fourth track on Horses and Grasses is "Pi" (and is preceded by "An Intro", which discusses the topic like an educational television program). Many digits are recited through it, and a video appeared online inspired by it.

* There is a building in the Googleplex numbered 3.14159...

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