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Geometric space with five dimensions
mathematical spaces having five dimensions
A five dimensional space is a space with five dimensions. In mathematics, a sequence of N numbers can represent a location in an N dimensional space. If interpreted physically, that is one more than the usual three spatial dimensions and the fourth dimension of time used in relativistic physics. Whether or not the universe is five dimensional is a topic of debate.

Much of the early work on five dimensional space was in an attempt to develop a theory that unifies the four fundamental interactions in nature: strong and weak nuclear forces, gravity, and electromagnetism. German mathematician Theodor Kaluza and Swedish physicist Oskar Klein independently developed the Kaluza–Klein theory in 1921, which used the fifth dimension to unify gravity with electromagnetic force. Although their approaches were later found to be at least partially inaccurate, the concept provided a basis for further research over the past century. While not detectable, it would indirectly imply a connection between seemingly unrelated forces. The Kaluza–Klein theory experienced a revival in the 1970s due to the emergence of superstring theory and supergravity: the concept that reality is composed of vibrating strands of energy, a postulate only mathematically viable in ten dimensions or more. Superstring theory then evolved into a more generalized approach known as M theory. M theory suggested a potentially observable extra dimension in addition to the ten essential dimensions which would allow for the existence of superstrings. The other 10 dimensions are compacted, or "rolled up", to a size below the subatomic level. M theory would explain the weakness of gravity relative to the other fundamental forces of nature, as can be seen, for example, when using a magnet to lift a pin off a table—the magnet overcomes the gravitational pull of the entire earth with ease. The main novelty of Einstein and Bergmann was to seriously consider the fifth dimension as a physical entity, rather than an excuse to combine the metric tensor and electromagnetic potential. But they then reneged, modifying the theory to break its five dimensional symmetry. Their reasoning, as suggested by Edward Witten, was that the more symmetric version of the theory predicted the existence of a new long range field, one that was both massless and scalar, which would have required a fundamental modification to Einstein's theory of general relativity. Minkowski space and Maxwell's equations in vacuum can be embedded in a five dimensional Riemann curvature tensor. In 1993, the physicist Gerard 't Hooft put forward the holographic principle, which explains that the information about an extra dimension is visible as a curvature in a spacetime with one fewer dimension. For example, holograms are three dimensional pictures placed on a two dimensional surface, which gives the image a curvature when the observer moves. Similarly, in general relativity, the fourth dimension is manifested in observable three dimensions as the curvature path of a moving infinitesimal (test) particle. 'T Hooft has speculated that the fifth dimension is really the "spacetime fabric". Recent research suggests several alternative interpretations of the 5D extension of spacetime, most of them generalizing the earlier Kaluza Klein theory. The first approach is space time matter, which utilizes an unrestricted group of 5D coordinate transforms to derive new solutions of the Einstein's field equations that agree with the corresponding classical solutions in 4D spacetime. Another 5D representation describes quantum physics from a thermal space time ensemble perspective and draws connections with classical field theory as limiting cases. Yet another approach, spacekime representation, lifts the ordinary time from an event ordering positive real number to complex time (kime), which effectively transforms longitudinal processes from time series into 2D manifolds (kime surfaces).

According to Klein's definition, "a geometry is the study of the invariant properties of a spacetime, under transformations within itself." Therefore, the geometry of the 5th dimension studies the invariant properties of such space time, as we move within it, expressed in formal equations. Fifth dimensional geometry is generally represented using 5 coordinate values (x,y,z,w,v), where moving along the v axis involves moving between different hyper volumes.

A hypersphere in 5 space (also called a 4 sphere due to its surface being 4 dimensional) consists of the set of all points in 5 space at a fixed distance r from a central point P, that is rotationally symmetrical. The hypervolume enclosed by this hypersurface is: