mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, antipodal points of a

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true diameter). This term applies to opposite points on a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

or any

n-sphere In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, cal ...

. An antipodal point is sometimes called an antipode, a

back-formation In etymology, back-formation is the process or result of creating a new word via inflection, typically by removing or substituting actual or supposed affixes from a lexical item, in a way that expands the number of lexemes associated with the c ...

from the

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

loan word A loanword (also loan word or loan-word) is a word at least partly assimilated from one language (the donor language) into another language. This is in contrast to cognates, which are words in two or more languages that are similar because the ...

''antipodes'', meaning "opposite (the) feet", as the true word singular is ''antipus''.

Theory

, the concept of ''antipodal points'' is generalized to

s of any dimension: two points on the sphere are antipodal if they are opposite ''through the centre''; for example, taking the centre as

origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...

, they are points with related

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

s v and −v. On a

, 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. The

Borsuk–Ulam theorem In mathematics, the Borsuk–Ulam theorem states that every continuous function from an ''n''-sphere into Euclidean ''n''-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in ...

is a result from

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

dealing with such pairs of points. It says that any

continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...

from ''S''^''n'' to R^''n'' maps some pair of antipodal points in ''S''^''n'' to the same point in R^''n''. Here, ''S''^''n'' denotes the ''n''-dimensional sphere in (''n'' + 1)-dimensional space (so the "ordinary" sphere is ''S''² and a circle is ''S''¹). The antipodal map ''A'' : ''S''^''n'' → ''S''^''n'', defined by ''A''(''x'') = −''x'', sends every point on the sphere to its antipodal point. It is

homotopic In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...

to the

identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...

if ''n'' is odd, and its

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...

is (−1)^''n''+1. If one wants to consider antipodal points as identified, one passes to

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

(see also

projective Hilbert space In mathematics and the foundations of quantum mechanics, the projective Hilbert space P(H) of a complex Hilbert space H is the set of equivalence classes of non-zero vectors v in H, for the relation \sim on H given by :w \sim v if and only if v = \ ...

, for this idea as applied in

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

Antipodal pair of points on a convex polygon

An antipodal pair of a convex polygon is a pair of 2 points admitting 2 infinite parallel lines being tangent to both points included in the antipodal without crossing any other line of the convex polygon.

References

External links

* * {{planetmath reference, urlname=Antipodal, title=antipodal Topology