Antipodal point

The word Antipodes (pronounced an TIP uh deez) is also used in the United Kingdom to refer to Australia and New Zealand.

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 Greek expression means "opposite feet" or "opposing feet". When the Earth was thought to be flat, one theory held the some people lived on the opposite side from ours, and of course their feet pointed in the opposite direction from ours; their "down" was our "up". This caused great consternation in the church because there was the issue of whether Christ had saved these people as well. This idea is deeply explored in Umberto Eco's brilliant Island of the Day Before as well as in Rennaissance encyclopedias.

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 vectorss 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 : Sⁿ → Sⁿ, 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).

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