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7次元は結構特殊なんだ・・
http://en.wikipedia.org/wiki/Seven-dimensional_space
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.