## hypersphereA 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.
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.## Related categories• GEOMETRY• SPACE AND TIME | |||||

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