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 ²r ⁴/2, and a surface volume (analogous to the sphere's surface area) of 2π ²r ³. A solid angle of a hypersphere is measured in hypersteradians, of which the hypersphere contains a total of 2π². 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.