**ratios**of space and time, he was probably thinking in terms of the field of reals, . This is a mathematical object which contains the multiplication operation, and its inverse, division.

Division is required for motion (s/t, t/s) so its reasonable to say that any mathematical system which is able to express division will be useful for describing aspects of reality in a universe of motion. Thus, we should focus efforts towards the mathematical expression of such a universe in terms of the

**division algebras**. Specifically, we are interested in finite dimensional division algebras in order to describe finite aspects of the universe.

There is a well known mathematical theorem called the Frobenius theorem which states that there are 3 "representations" (isomorphisms) of finite-dimensional associative division algebras over the real numbers. These are the real numbers , the complex numbers , and the quaternions .

There are many examples where quaternions are important in the various physics fields. Bruce has discussed these in several posts. But while quaternions generalize the imaginaries and the reals (allowing for the representation of the noncommutative 3-d rotations), they still are not sufficient in describing more complex phenomena. In RS2 it is posited that Octonions are needed to describe

**life physics**. While the quaternions are associative but not commutative, the octonions are non-associative and non-commutative. This implies that the order of operations matters, even if they are displayed in the same order. In the Reciprocal System, and as recently discussed in a particle physics paper, the variables in this equation are both the states and the operators - in other words, it is all motion transferring between different types of motion. Thus, if the states are non-commutative, then they must also be non associative. We know that 3-d rotations are non commutative so this combined with the previous statement would imply that we are not generically interested in associative algebras. This is why octonions and potentially other algebras could be important.

There is another mathematical theorem that could provide guidance as to what other division algebras may be of import. This is Hurwitz's theorem, which states that

The key here isIn mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions and the octonions.

**unital**algebras, because physics often deals with conservation laws. Unitary transformations are useful for representing these. Also, a consequence of this theorem is that

. Hence, the octonions encompass the non-commutativeThe onlyEuclideanHurwitz algebras are the real numbers, the complex numbers, the quaternions and the octonions.

**and**non-associative possibilities of motion. Since it can be shown via the Cayley–Dickson construction that octonions can be constructed by composing quaternions in a special way (just as imaginaries can be constructed from reals, and quaternions from imaginaries and so on). This would seem to imply that determining the mathematical foundations of describing a universe of motion would necessitate the utilization of octonions.