N-sphere
Generalized sphere of dimension n (mathematics)
N-sphere
In mathematics, an n-sphere or hypersphere is an n-dimensional generalization of the 1-dimensional circle and 2-dimensional sphere to any non-negative integer n. The circle is considered 1-dimensional and the sphere 2-dimensional because a point within them has one and two degrees of freedom respectively. However, the typical embedding of the 1-dimensional circle is in 2-dimensional space, the 2-dimensional sphere is usually depicted embedded in 3-dimensional space, and a general n-sphere is embedded in an n+1-dimensional space. The term hypersphere is commonly used to distinguish spheres of dimension n≥3 which are thus embedded in a space of dimension n+1≥4, which means that they cannot be easily visualized. The n-sphere is the setting for n-dimensional spherical geometry. Considered extrinsically, as a hypersurface embedded in (n+1)-dimensional Euclidean space, an n-sphere is the locus of points at equal distance (the radius) from a given center point. Its interior...
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