hypersphere (thing)

A four-dimensional analog of a sphere. Just as the circle is defined by

x^2+y^2=r^2

and the sphere is defined by

x^2+y^2+z^2=r^2,

the hypersphere is defined by

x^2+y^2+z^2+w^2=r^2.

One cannot visualize the hypersphere; however, one can visualize the intersection of a hypersphere with a hyperplane: a sphere. Also see n-sphere.