# hypersphere

A hypersphere is a four-dimensional analog of a sphere; also known as a **4-sphere**.
Just as the shadow cast by a sphere is a circle,
the shadow cast by a hypersphere is a sphere, and just as the intersection
of a sphere with a plane is a circle, the
intersection of a hypersphere with a hyperplane is a sphere. These analogies
are reflected in the underlying mathematics.

*x*^{2} + *y*^{2} = *r*^{ 2} is the Cartesian equation of a circle of radius *r*;

*x*^{2} + *y*^{2} + *z*^{2} = *r*^{ 2} is the corresponding equation of a sphere;

*x*^{2} + *y*^{2} + *z*^{2} + *w*^{2} = *r*^{ 2} is the equation of a hypersphere, where *w* is measured along a fourth dimension at right angles to the *x-*, *y*-, and *z*-axes.

The hypersphere has a **hypervolume** (analogous to the volume
of a sphere) of π^{2}*r*^{ 4}/2, and a surface volume
(analogous to the sphere's surface area) of 2π^{2}*r*^{ 3}. A solid angle of a hypersphere is measured in **hypersteradians**,
of which the hypersphere contains a total of 2π^{2}. The apparent
pattern of 2π radians in a circle and
4π steradians in a sphere does not
continue with 8π hypersteradians because the *n*-volume, *n*-area,
and number of *n*-radians of an *n*-sphere are all related to
gamma function and the way it can cancel out powers of π halfway between
integers. In general, the term "hypersphere" may be used to refer to any *n*-sphere.