There are six logic gates: OR, AND, NAND, NOR, XOR and XNOR, each defined by a truth table constructed from two inputs and an output. In accordance with De Morgan’s laws, these can be sorted into three mirror opposite pairs: OR/AND, NOR/NAND and XOR/XNOR. When the truth tables are sorted strictly into mirror opposite pairs, such that the inputs and outputs are constructed from four ordered polarities, it can be seen that the truth tables differ from one another only in the areas shown in colour below.
When the truth table for one member of a mirror pair is placed over the truth table for the other mirror pair member, it can be seen that one column of polarities oscillates + – + – + – and the other three columns of polarities oscillate – – – + + +. In other words, the truth tables for the pair members can be interconverted by sliding all the columns by three spaces.
The six truth tables can be interconverted by sliding the third or fourth columns of polarities relative to the other columns.
An isomorphism can be identified between between the construction of the six logic gates and the new mathematics.
The new mathematics is constituted in oscillation between XOR and XNOR.