Antipodal point

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. Each line through the centre intersects the sphere in two points, one for each ray emanating 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 ${\displaystyle S^{n}}$ to ${\displaystyle \mathbb {R} ^{n}}$ maps some pair of antipodal points in ${\displaystyle S^{n}}$ to the same point in ${\displaystyle \mathbb {R} ^{n}.}$ Here, ${\displaystyle S^{n}}$ denotes the ${\displaystyle n}$-dimensional sphere and ${\displaystyle \mathbb {R} ^{n}}$ is ${\displaystyle n}$-dimensional real coordinate space.

The antipodal map ${\displaystyle A:S^{n}\to S^{n}}$ sends every point on the sphere to its antipodal point. If points on the ${\displaystyle n}$-sphere are represented as displacement vectors from the sphere's center in Euclidean ${\displaystyle (n+1)}$-space, then two antipodal points are represented by additive inverses ${\displaystyle {\mathbf {v}}}$ and ${\displaystyle -{\mathbf {v}},}$ and the antipodal map can be defined as ${\displaystyle A({\mathbf {x}})=-{\mathbf {x}}.}$ The antipodal map preserves orientation (is homotopic to the identity map)[2] when ${\displaystyle n}$ is odd, and reverses it when ${\displaystyle n}$ is even. Its degree is ${\displaystyle (-1)^{n+1}.}$

If antipodal points are identified (considered equivalent), the sphere becomes a model of real projective space.

• Cut locus

• "Antipodes", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "antipodal". PlanetMath.