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Encyclopedia > Antipodal point

In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Diameter is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ...


An antipodal point is sometimes called an antipode, a back-formation from the Greek loan word antipodes, which originally meant "opposite the feet." In etymology, the process of back-formation is the creation of a neologism by reinterpreting an earlier word as a compound and removing the spuriously supposed affixes. ... A loanword (or a borrowing) is a word taken in by one language from another. ...


Theory

In mathematics, the concept of antipodal points is generalized to spheres of any dimension: two points on the sphere are antipodal if they are opposite through the centre; for example, taking the centre as origin, they are points with related vectors v and −v. On a circle, such points are also called diametrically opposite. In other words, each line through the centre intersects the sphere in two points, one for each ray out from the centre, and these two points are antipodal. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... A sphere (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ... In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ... Ray may refer to: A ray, or half-line in geometry or physics. ...


The Borsuk-Ulam theorem is a result from algebraic topology dealing with such pairs of points. It says that any continuous function from Sn to Rn maps some pair of antipodal points in Sn to the same point in Rn. Here, Sn denotes the sphere in n-dimensional space (so the "ordinary" sphere is S3). The Borsuk-Ulam theorem states that any continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...


The antipodal map A : SnSn, defined by A(x) = −x, sends every point on the sphere to its antipodal point. It is homotopic to the identity map if n is odd, and its degree is (−1)n+1. The two bold paths shown above are homotopic relative to their endpoints. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... This article is about the term degree as used in mathematics. ...


If one wants to consider antipodal points as identified, one passes to projective space (see also projective Hilbert space, for this idea as applied in quantum mechanics). In mathematics, a projective space is a fundamental construction from any vector space. ... In mathematics and the foundations of quantum mechanics, the projective Hilbert space P(H) of a complex Hilbert space is the set of equivalence classes of vectors v in H, with v ≠ 0, for the relation given by v ~ w when v = λw with λ a scalar, that is, a... For a non-technical introduction to the topic, please see Introduction to Quantum mechanics. ...


References

  • This article incorporates text from the Encyclopædia Britannica Eleventh Edition, a publication now in the public domain.

Encyclopædia Britannica, the 11th edition The Encyclopædia Britannica Eleventh Edition (1910–1911) is perhaps the most famous edition of the Encyclopædia Britannica. ... The public domain comprises the body of all creative works and other knowledge—writing, artwork, music, science, inventions, and others—in which no person or organization has any proprietary interest. ...

External links


  Results from FactBites:
 
Antipodal point - Wikipedia, the free encyclopedia (315 words)
In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter.
In mathematics, the concept of antipodal points is generalized to spheres of any dimension: two points on the sphere are antipodal if they are opposite through the centre; for example, taking the centre as origin, they are points with related vectors v and −v.
In other words, each line through the centre intersects the sphere in two points, one for each ray out from the centre, and these two points are antipodal.
Antipode - Wikipedia, the free encyclopedia (120 words)
Antipode - 1 : the parts of the earth diametrically opposite -- usually used as pl. 2 : the exact opposite or contrary
The word antipode is a back-formed singular from the word antipodes, which originally was not plural.
For the abstract mathematical concept that arose from the geographical usage, see antipodal point.
  More results at FactBites »

 
 

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