Quaternions are composed of four ordered elements (1, i, j, k) that are governed by the following multiplication rules:
and also by the familiar sign multiplication rules:
Many applications have been found for them, a very important one being the control of rotation in 3D.
One consequence of the multiplication rules is that the quaternions can form a six-membered number cycle, for example:
-k.-i = j; -i.j = -k; -j.-k = i; -k.i = -j; -i.-j = k; -j.k = -i
Each quaternion in the cycle is turned into its polar opposite in three multiplication steps. The cycles rotate in one direction (order yellow, blue, green) if the quaternions are multiplied sequentially by +j, +i and +k and in the opposite direction (order blue, yellow, green) if they are multiplied sequentially by -i, -j and -k.
Chiralkine numbers are composed of four ordered polarities: ± 1, ±i, ±j, ±k, two of each polarity. Each behaves under quaternion multiplication rules as a unitary quaternion: ±i, ±j or ±k.
When each term in a chiralkine number is multiplied by the same quaternion, all of the terms re-order themselves in a unitary manner.
This multiplication method is different from how quaternions are multiplied as wholes normally. For example, the product of -1, i, -j, k and 1, i,-j,-k under normal quaternion multiplication is -2, 2i, 2j, 2k. It has one negative and three positive numbers. Chiralkine numbers behave as pure, unitary quaternions. When they are multiplied together, this corresponds with rotation only. There is no addition or subtraction (negation) of coefficients of 1, i, j and k in the multiplication of chiralkine numbers.
Provided that these rules are followed, the multiplication of chiralkine numbers works like quaternion multiplication. Thus, each chiralkine number can be identified with a positive or negative quaternion: ± i, ± j or ± k.
Each ordered combination of four quaternions codes one face of a cube.
Multiplication effects rotation of a cube about an axis between two opposed vertices.
Each multiplication step corresponds with exchanging one of the two positive polarities with one of the two negative polarities in a chiralkine number, leaving the others unchanged. It is like exchanging coins of different kinds.
Multiplication of a chiralkine number by itself gives -1 (-1, -i, -j, -k) and by its mirror opposite gives 1 (+1, +i, +j, +k). This is equivalent to multiplying a quaternion by itself or its’ signed opposite.
Multiplication of a chiralkine number by another kind of chiralkine number gives a different chiralkine number. This is equivalent to multiplying two quaternions of different kinds.
Multiplication of three different chiralkine numbers gives -1 (-1, -i, -j, -k) or +1 (+1, +i, +j, +k). This is equivalent to multiplying three different quaternions together.
Multiplication corresponds with rotation of a cube in 3D, just as with quaternion multiplication, and each chiralkine number can be read from the face of a cube as shown below, where +1, +i, -j, -k (unitary quaternion: -k) is facing the viewer.
There are 6 x 4 = 24 combinations of (+/-) i, j and k that afford (+/-) i, j, k (see table). This is the number of possible Fischer projections for a chiral tetrahedral molecule. The relationship with Fischer projections can seen by looking at this diagram, and identifying that it contains black and white tetrahedrons. Each face of the cube can be read as a Fischer projection of the black tetrahedron where a positive number (+) is projecting up towards the viewer and a negative number (-) is projecting away from the viewer.
The relationship between the quaternions and the faces of a cube extends to a relationship with the corners of a chiral cube as well.
The cube rotates as a unitary whole. The corners are coded with three of the four ordered polarities the same and one different. Taken together the codes for the corners and faces control the axis of rotation from each face to another: which polarity is fixed and which is changed common to both codes.
Taken together, the unitary transformations of the codes for each corner and face control which axis the cube rotates about whenever a unitary quaternion is multiplied by another.
Each quaternion in a multiplication cycle is turned into its polar opposite in three multiplication steps. The cycles rotate in one direction (order yellow, blue, green) if the quaternions are multiplied sequentially by +j, +i and +k and in the opposite direction (order blue, yellow, green) if they are multiplied sequentially by -i, -j and -k. Each multiplication step corresponds with a rotation about an axis defined by opposed corners of a cube, which corners are identified by + + + and – – – triplets that define XOR and XNOR truth tables.
In a chiralkine exchange, the positions of members of the exchange cycle through six states, two of which function as the poles. The creation, exchange and redemption of units under a contract correspond with rotation about corners, transferring ownership of a good or service in steps, while the coupled exchange of units corresponds with a 180 degree rotation about the poles, transfers ownership of a good or service in one step.
The significance of the maths appears to be this.
In conventional maths, 0 = -1 + 1. We use three numbers: 0, +1 and -1. We take 0 to be an absolute (nothing). +1 and -1 mutually negate.
What has been found is a very simple, relativistic number system. In this number system + and – are opposites defined relative to one another. They do not mutually negate. Their order matters. They exchange with one another in order according to clear rules, and each time they do so a complementary exchange takes place in a mirror opposite configuration. These rules operate at all scales, from sub-atomic particles to spiral galaxies. It is a chiral accounting system.
At a neuronal level, you can think of information being encoded in ordered combinations of neurons firing (+) and neurons not firing (-). The encoding is chiral, just like information is encoded in the genetic code.