# Seven-dimensional space

In physics and mathematics, a sequence of n numbers can also be understood as a location in n-dimensional space. When n = 7, the set of all such locations is called 7-dimensional Euclidean space. Seven-dimensional elliptical and hyperbolic spaces are also studied, with constant positive and negative curvature.

Abstract seven-dimensional space occurs frequently in mathematics, and is a perfectly legitimate construct. Whether or not the real universe in which we live is somehow seven-dimensional (or indeed higher) is a topic that is debated and explored in several branches of physics, including astrophysics and particle physics, but it does not matter for mathematics.

Formally, seven-dimensional Euclidean space is generated by considering all real 7-tuples as 7-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 7-vectors is readily defined, and can be used to calculate the metric. 7 × 7 matrices can be used to describe transformations such as rotations which keep the origin fixed.

A distinctive property is that a cross product can be defined only in three or seven dimensions (see seven-dimensional cross product). This is due to the existence of quaternions and octonions.

## Geometry

### 7-polytope

A polytope in seven dimensions is called a 7-polytope. The most studied are the regular polytopes, of which there are only three in seven dimensions: the 7-simplex, 7-cube, and 7-orthoplex. A wider family are the uniform 7-polytopes, constructed from fundamental symmetry domains of reflection, each domain defined by a Coxeter group. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram. The 7-demicube is a unique polytope from the D7 family, and 321, 231, and 132 polytopes from the E7 family.

Regular and uniform polytopes in seven dimensions
(Displayed as orthogonal projections in each Coxeter plane of symmetry)
A6 BC7 D7 E7

7-simplex

7-cube

7-orthoplex

7-demicube

321

231

132

### 6-sphere

The 6-sphere or hypersphere in seven dimensions is the six dimensional surface equidistant from a point, e.g. the origin. It has symbol S6, with formal definition for the 6-sphere with radius r of

$S^6 = \left\{ x \in \mathbb{R}^7 : \|x\| = r\right\}.$

The volume of the space bounded by this 6-sphere is

$V_7\,=\frac{16 \pi^3}{105}\,r^7$

which is 4.72477 × r7, or 0.0369 of the 7-cube that contains the 6-sphere.

## Applications

### Physics

#### Non-associative Ashketar gravity

There is a paper on Non-associative Ashketar gravity in seven dimensions.[1][2]

#### String theory and M-theory

A key feature of string theory is that, though it is an attempt to model our physical universe, it takes place in a space with more dimensions than the four of spacetime that we are familiar with. In particular a number of string theories take place in a ten dimensional space, adding an extra six dimensions. These extra dimensions are required by the theory, but as they cannot be observed are thought to be quite different, perhaps compactified so they form a six dimensional space with a particular geometry too small to be observable. In M-theory, which unifies the five types of string theory, there is a seventh dimension involved.

### Mathematics

#### Cross product

As mentioned above, a cross product in seven dimensions analogous to the usual three can be defined, and in fact a cross product can only be defined in three and seven dimensions.

#### Exotic sphere

In 1956, John Milnor constructed an exotic sphere in 7 dimensions and showed that there are at least 7 differentiable structures on the 7-sphere. (The number is now known to be 28.)

## Notes

Seventh dimension discussions are less common than fifth dimension and eleventh dimension theories popularized in science at the moment by mathematicians in quantum physics under quantum mechanics, string theory, and quantum gravity. This is possibly due in part to stigma of time travel becoming a popular topic in science fiction, or simply that research has not needed to move in that direction lately.