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. Nine dimensional elliptical space and hyperbolic spaces are also studied, with constant positive and negative curvature.

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 is 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.

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

8-sphere
The 8-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 and centre at the origin of


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

The volume of the 9-ball bounded by this 8-sphere is


 * $$V_9\,= \tfrac{32}{945}\pi^4r^9,$$

which is about 3.2985 × r9, or a fraction of 0.00644 of the volume of the smallest 9-cube that contains the 8-sphere.

Superstring theory
Nine dimensional space was commonly used by physicists in models exploring Superstring theory - it was posited that 6 imperceptible spatial dimensions exist in addition to the standard 3 dimensions of length, width, and depth. Later, an extra spatial dimension was added once the concept of membranes was introduced and study of superstrings evolved to M-theory. To both the 9 spatial dimensions of superstring theory and the 10 spatial dimensions of m-theory, the temporal dimension of time was added requiring equations that handled 10 dimensional geometry for superstrings and 11 dimensional geometry for m-theory.

Material science
Nine dimensional space is used as a conceptual tool in analysis studying plastic deformation in solids. To calculate the combined force that points in a material object are suffering under load, two 3rd order tensors can be used, where the tensor product will be nine dimensional. Nine dimensional space has been used for studying stress and strain in materials from as early as 1932.