Antipodal point

Antipodal points on the surface of a sphere are diametrically opposite; on the other side of a globe. For example, "Spain and New Zealand lie in antipodal regions."

An antipodal point is sometimes called an antipode, a back-formed word that originated from the mistaken idea that antipodes is the plural of antipode. In fact antipodes is an adapted Greek plural whose singular is antipous.

The word Antipodes was used in the Middle Ages to refer to far away countries where people allegedly walked 'on the other side of the flat Earth', i.e. to places like India. Today it is still used in the United Kingdom to refer to Australia and New Zealand.

The Antipodes Islands lie off the south coast of New Zealand, supposedly at the antipodal point to Great Britain. Their true antipodal point is near Cherbourg, France.

Generalization to more dimensions

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.

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 a 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 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.

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).

External links

de:Antipode it:Punto antipodale nl:Antipode pl:Antypody ru:Антипод


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