The quaternions, invented by the mathematician Hamilton, provided a crucial stepping stone in the development of chiralkine from intuition about what it is to complete understanding about how it works as a relational logical system under the mirror symmetry of Order 4.
The quaternions constitute a hypercomplex system of real and imaginary numbers interconnected through non-commutative multiplication.
A paper authored by Capozziello and Lattanzi describes how the enantiomers of chiral tetrahedral molecules (regarded as absolute objects) behave the same as spinors (regarded as relational objects) under quaternionic algebra. Yours truly had the privilege of meeting with these scientists at Salerno in Italy to discuss the paper. Collaboration followed, but could not find a way to map the four ordered objects (1, 2, 3, 4) that define a chiral tetrahedral molecule with the intuition about how a chiralkine system should work.
The authors had noted in their paper that a group of Order 4 has six generators and had calculated algebraically that these correspond with six eigenvalues, two real, two imaginary and two complex. We recognised that this should be isomorphic with the powers of the cube root of minus 1.
The solution, which has taken several years to find, is to reconfigure each of the four objects as relational (a, A, b, B, c, C and d, D). This now aligns the six generators for Order 4 with the six different ways in which four different objects can be ordered in rings. Each lower case letter is treated as 0 and each upper case letter as 1, keeping the letters in alphabetical order.
Spinors can now be mapped intuitively to molecules.
This changes everything! It comes at the expense of giving up using equations. In the relational symmetry of Order 4:
.