A four-dimensional extension of a sphere. Similar to a hypercube.

Just as the shadow cast by a sphere is a circle, the shadow cast by a hypersphere is a sphere.

Because we cannot actually even conceptualize what a fourth dimension would be lilke, it's impossible to actually describe what one would look like.

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.
Though it cannot be visualized (at least not without several accompanying headaches), the hypersphere can be dealt with mathematically.

A couple of mathematical properties of the hypersphere:

It has a hypervolume (measured in distance^4) of π2r4/2, and a surface volume of 2π2r3.

A solid angle of a hypersphere would be measured in hypersteradians, of which the hypersphere would contain 2π2 in total. (It would seem that 2π radians in a circle and 4π steradians in a sphere were setting up a nice simple pattern and that 8π hypersteradians would be next, but actually the N-volume, N-area, and number of N-radians of an N-sphere are all related to the Gamma function and the way it can cancel out powers of π halfway between integers.)

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