Antipodal point

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Antipodal points on the surface of a sphere are diametrically opposite - so situated that a line drawn from the one to the other passes through the centre of the globe and forms a true diameter. 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, and the inhabitants of these countries are sometimes referred to as antipodeans. 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.

Any two places on Earth having this relation must be distant from each other by 180° of longitude, and the one must be as many degrees to the north of the equator as the other is to the south, in other words, the latitudes are numerically equal, but one is north and the other south. Noon at the one place is midnight at the other, the longest day corresponds to the shortest, and midwinter is contemporaneous with midsummer.

In the calculation of days and nights, midnight on the one side may be regarded as corresponding to the noon either of the previous or of the following day. If a voyager sail eastward, and thus anticipate the sun, his dating will be twelve hours in advance, while the reckoning of another who has been sailing westward will be as much in arrear. There will thus be a difference of twenty-four hours between the two when they meet. To avoid the confusion of dates which would thus arise, it is necessary to determine a meridian at which dates should be brought into agreement, i.e. a line the crossing of which would involve the changing of the name of the day either forwards, when proceeding westwards, or backwards, when proceeding eastwards. See International Date Line.

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).

References

External links

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