Addition

Addition is a binary operation with the following definition:

Using our geometrical definition of the operation T, we can demonstrate the addition of points on the diagonal.

 
Figure 4: Addition of diagonal points

Having defined addition, we can define a different algebraic structure called a loop. This is a non-associative group.

Proof:

1. For , by T2. Also, by T1.
2. If we are given a and b, then is unique. If we are given a and c, then and this has a unique solution by T5. If we are given b and c, then by T1. This system has a unique solution by T3.