Four-Dimensional Geometry

Four-dimensional geometry is Euclidean geometry extended into one additional dimension. The prefix "hyper-" is usually used to refer to the four- (and higher-) dimensional analogs of three-dimensional objects, e.g., hypercube, hyperplane, hypersphere. -dimensional polyhedra are called polytopes. The four-dimensional cases of general -dimensional objects are often given special names, such as those summarized in the following table.

The surface area of a hypersphere in dimensions is given by

(1)

where is the gamma function, giving the first few values as

(2) (3) (4) (5)

with coefficients 2, 2, 4, 2, 8/3, 1, 16/15, ... (OEIS A072478 and A072479).

The volume is given by

(6)

giving the first few values as

(7) (8) (9) (10)

with coefficients 2, 1, 4/3, 1/2, 8/15, 1/6, 16/105, ... (OEIS A072345 and A072346).