4D 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.

2-D	3-D	4-D	general
circle	sphere	glome	hypersphere
square	cube	tesseract	hypercube
equilateral triangle	tetrahedron	pentatope	simplex
polygon	polyhedron	polychoron	polytope
line segment	plane	hyperplane	hyperplane
square	octahedron	16-cell	cross polytope
polygon edge	face	facet	facet
area	volume	content	content

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).