Dimensionality of Space (3D)

In a closed group of operators, like [1 i j k], the result of the combination of any number of the basal elements is also a member of the same group. The result of any such combination can be known only if all the possible binary combinations of the elements are first defined in terms of the basal elements i, j and k themselves (besides, of course, the identity operator, 1). Let there be n basal elements (excluding the unit operator 1) in a group. Then the number of unique binary combinations of these elements, in which no element occurs twice, is n(n-1)/2. We can readily see that a group becomes self-sufficient (finite) only if the number of binary combinations of the basal elements is equal to the number of those basal elements themselves, that is

$$\frac{n(n−1)}{2} = n$$

The only definite solution for n is 3. (Zero and infinity are other solutions.) Therefore if we regard space (time) as a group of orthogonal rotations, its dimensionality has to be three in order to make it self-sufficient.

Cite: KVK Nehru, Some Thoughts on Spin, Section 9

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