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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).

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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