Nine-dimensional space

The ninth dimension is the ninth in the sequence of dimensions that arises when the rules we know and apply to familiar spaces like two and three are generalised. When for example three dimensional geometry is extrapolated from two dimensions an extra dimension is added, so $(x, y)$ in two dimensions is replaced with $(x, y, z)$ in three. If this is done a further six times the result is a nine-dimensional space.

In physics and mathematics, a sequence of n numbers can also be understood as a location in n-dimensional space. When n = 9, the set of all such locations is called 9-dimensional Euclidean space.

Abstract nine-dimensional space occurs frequently in mathematics, and is a perfectly legitimate construct. Whether or not the real universe in which we live is somehow nine-dimensional is a topic that is debated and explored in several branches of physics, including astrophysics and particle physics.

Formally nine-dimensional Euclidean space is generated by considering all real 9-tuples as 9-vectors in this space. As such it has the properties of all Euclidian spaces, so it it linear, has a metric and a full set of vector operations. In particular the dot product between two 9-vectors is readily defined, and can be used to calculate the metric. 9 × 9 matrices can be used to describe transformations such as rotations which keep the origin fixed.

Three-dimensional space is the space we live in, the fourth dimension can be interpreted as spacetime a model for the universe used in special relativity, while the fifth dimension can be used to model the curvature of spacetime as required in general relativity. The sixth dimension is not commonly used for modelling our universe, but in most string theories, six dimensions have curled up below the Planck length, making them unobservable. In M-theory, an extra dimension is added, making seven extra dimensions. In superspace, there are eight dimensions, four of ordinary spacetime and the other four of "superspace". However, there are no theories in nine dimensions that describe our universe.

9-polytope
A polytope in eight dimensions is called an 9-polytope. The most studied are the regular polytopes, of which there are only three in nine dimensions.

9-sphere
The 9-sphere or hypersphere in nine dimensions is the eight dimensional surface equidistant from a point, e.g. the origin. It has symbol $S^{8}$, with formal definition for the 8-sphere with radius r of


 * $$S^8 = \left\{ x \in \mathbb{R}^9 : \|x\| = r\right\}.$$

The volume of the space bounded by this 7-sphere is


 * $$V_9\,= ?$$

which is ? × r8, or ? of the 8-cube that contains the 7-sphere.

In philosophy
A website exists that talks about the first nine dimensions of human consciousnessStationlink.com