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 of radius 
in 
-dimensional Euclidean space,
for integer 
,
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 of the enclosed ball 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).
