n-sphere
![2-sphere wireframe as an orthogonal projection](/Images/godic/202501/19/Sphere_wireframe.svg1317.png")
![Just as a stereographic projection can project a sphere's surface to a plane, it can also project the surface of a 3-sphere into 3-space. This image shows three coordinate directions projected to 3-space:
parallels (red), meridians (blue) and hypermeridians (green).
Due to the conformal property of the stereographic projection,
the curves intersect each other orthogonally (in the yellow points) as in 4D.
All of the curves are circles: the curves that intersect <0,0,0,1> have an infinite radius (= straight line).](/Images/godic/202501/19/Hypersphere_coord1317.PNG")
![(Note that n refers to the dimension of the ambient Euclidean space, which is also the intrinsic dimension of the solid whose volume is listed here, but which is 1 more than the intrinsic dimension of the sphere whose surface area is listed here.) The curved red arrows show the relationship between formulas for different n. The formula coefficient at each arrow's tip equals the formula coefficient at that arrow's tail times the factor in the arrowhead. If the direction of the bottom arrows were reversed, their arrowheads would say to multiply by 2π/n − 2. Alternatively said, the surface area Sn+1 of the sphere in n + 2 dimensions is exactly 2πR times the volume Vn enclosed by the sphere in n dimensions.](/Images/godic/202501/19/N_SpheresVolumeAndSurfaceArea1317.png")
![A set of uniformly distributed points on the surface of a unit 2-sphere generated using Marsaglia's algorithm.](/Images/godic/202501/19/2sphere-uniform1317.png")
In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in (n + 1)-dimensional Euclidean space which are at distance r from a central point, where the radius r may be any positive real number. Thus, the n-sphere centred at the origin is defined by: