TRIDIMENSIONAL HYPERSPHERE

 Cartesian equation: . Cartesian parametrization:  1) with  ??.  2) Stereographic coordinates with center (0, 0,0,-R):  (the point (0, 0,0,-R) is missing). 4-dimensional quartic hypervolume of the associated ball: ; 3-dimensional cubic hyperarea: .

The (tridimensional) hypersphere with center O and radius R is the locus of the points of the 4-dimensional space located at distance R from O.

It is a 3-dimensional manifold homeomorphic to the Alexandroff compactification of the usual tridimensional space R3, written S3. In other words, the hypersphere minus one point is topologically equivalent to the usual space.

 Just like the shadow of a sphere on a plane is a disk, the shadow of a hypersphere on a hyperplane (more precisely, its hyperplanar orthogonal projection) is a 3-dimensional ball. However, its sections by hyperplanes are spheres.

The section by the hypercylinder is the Clifford torus.

Here are ways of thinking about the hypersphere:

 Just like the sphere is topologically equivalent to the reunion of two closed disks the boundaries of which have been glued (representing the hemispheres), the hypersphere is topologically equivalent to the reunion of two closed balls glued along their boundaries (the two hemihyperspheres). Just like the sphere is the reunion of circles with radii increasing from 0 to R and then decreasing back to 0, the hypersphere is the reunion of spheres with radii increasing from 0 to R and then decreasing back to 0.