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For (n + 1)-dimensional Euclidean space $ \mathbb{R}^{n+1} $ the hypersphere (n-sphere) $ S^n $ of radius r and center c is the set of all points that are at the same distance r from point c:

$ S^n = \left\{ x \in \mathbb{R}^{n+1} | d(x,c) = r \right\} $

where $ d(x,c) $ is euclidean metric for (n+1)-dimensional space

$ d(x,c)=\sqrt{\sum_{i=1}^{n+1} (x_i - c_i)^2} $

Examples:

  • For 2-dimensional Euclidean space the hypersphere is circle (1-sphere),
  • For 3-dimensional Euclidean space the hypersphere is sphere (2-sphere).
  • For 4-dimensional Euclidean space the hypersphere is glome (3-sphere).

See Also

Dimensionality Zero One Two Three Four Five Six Seven Eight Nine Ten ... Aleph null
Hyperbolic space

$ \mathbb H^{n} $

Hyperbolic plane

$ \mathbb H^{2} $

Hyperbolic realm

$ \mathbb H^{3} $

Hyperbolic flune

$ \mathbb H^{4} $

Hyperbolic pentrealm

$ \mathbb H^{5} $

Hyperbolic hexealm

$ \mathbb H^{6} $

Hyperbolic heptealm

$ \mathbb H^{7} $

Hyperbolic octealm

$ \mathbb H^{8} $

Hyperbolic ennealm

$ \mathbb H^{9} $

Hyperbolic decealm

$ \mathbb H^{10} $

... Hyperbolic omegealm

$ \mathbb H^{\aleph_0} $

Euclidean space

$ \mathbb R^{n} $

Point

$ \mathbb R^{0} $

Euclidean line

$ \mathbb R^{1} $

Euclidean plane

$ \mathbb R^{2} $

Euclidean realm

$ \mathbb R^{3} $

Euclidean flune

$ \mathbb R^{4} $

Euclidean pentrealm

$ \mathbb R^{5} $

Euclidean hexealm

$ \mathbb R^{6} $

Euclidean heptealm

$ \mathbb R^{7} $

Euclidean octealm

$ \mathbb R^{8} $

Euclidean ennealm

$ \mathbb R^{9} $

Euclidean decealm

$ \mathbb R^{10} $

... Euclidean omegealm

$ \mathbb R^{\aleph_0} $

Hypersphere

$ \mathbb S^{n} $

Point pair

$ \mathbb S^{0} $

Circle

$ \mathbb S^{1} $

Sphere

$ \mathbb S^{2} $

Glome

$ \mathbb S^{3} $

Tetrasphere

$ \mathbb S^{4} $

Pentasphere

$ \mathbb S^{5} $

Hexasphere

$ \mathbb S^{6} $

Heptasphere

$ \mathbb S^{7} $

Octasphere

$ \mathbb S^{8} $

Enneasphere

$ \mathbb S^{9} $

Dekasphere

$ \mathbb S^{10} $

... Omegasphere

$ \mathbb S^{\aleph_0} $

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