# New, Noncommutative Mathematics

This website teaches new, noncommutative mathematics consisting of a system of six numbers constructed from 24 polarities, 12 ↓ and 12 ↑ ordered into six rows of four columns. Each row consists of four ordered polarities, 2 ↓ and 2 ↑ and each column consists of  six ordered polarities, 3 ↓ and 3 ↑. In one of the four columns, the polarities oscillate ↓ ↑ ↓ ↑ ↓ ↑. In the other three columns, the polarities oscillate ↑ ↑ ↑ ↓ ↓ ↓.

The first three rows are the mirror images of the second three rows. The ordering of the polarities shown in the diagram is not the only possible ordering. There are 4 x 3 x 2 x 1 = 24 ways to permute the four columns, and hence the six rows.

Each of the six numbers is defined by the combination of its unique polarity string of 2 ↓ and 2 ↑ and its unique position relative to the other five polarity strings. This associates a colour with each number. Through the colour, the polarities can be related physically to the eight opposed corners and six opposed faces of a cube consisting of the two interpenetrating enantiomers of a chiral tetrahedron: the corners of one enantiomer having one of the polarities (↑ or +) and the corners of the other enantiomer having the opposite polarity (↓ or -). Rotation of the cube about an axis defined by opposed corners permutes the order of the four colours, which order defines each of the six faces. When these colours are sorted into a fixed order, it is the four polarities that are permuted.

The numbers interact pairwise with one another to produce two new numbers, each number acting on the other to turn it into a third number. This interaction corresponds physically with rotation of the cube about an axis defined by the column of polarities that does not change. The six numbers behave as two kinds of +1 (↓ ↑), two kinds of -1 (↑ ↓) and two kinds of zero: +0 (↓ ↓) and -0 (↑ ↑). The interaction between the numbers is noncommutative: -1 turns +1 into one kind of zero and +1 turns -1 into the other kind of zero. The properties of the signed integers and signed zeroes are determined by their interactions in a cycle:  A pattern giving rise to the cycle can be seen by looking at the rows of polarities in blocks of four rows, such that two diagonals of four like polarities are seen. The oscillation of the polarities through each cycle produces patterns that appear like footprints. Each time one number changes a second number into a third number, a corresponding change takes place in the mirror complements of the numbers. The change is like an exchange interaction wherein a number exchanges polarities with its mirror pair. When polarities are exchanged, it appears to the mind as if one polarity selected from ↓ and ↑ behaves like a physical token (a traditional 1) and the other polarity selected from ↓ and ↑ behaves like a non-physical empty space (a traditional 0), such that two physical tokens appear to move, one from each side to the other. Nothing actually moves. This exchange interaction results from the conservation of symmetry. It links the numbers together.