A mathematical structure, called Zero-Totality (ZT), appears to be the most suitable mathematical architecture for the representation of duality concepts, dualistic relationships, and duality models.

In ZT a fundamental equation that is at the "foundation of physics" called the nilpotent Dirac equation is derived. A "universal alphabet and rewrite system" is a simple concept to understand but its implications are far-reaching and truly profound.

The requirement of of zero-totality allows for the creation of well-defined structures which have characteristics of an algebra, including countable numbers, even without a prior assumption of the existence of any mathematics at all.

Mathematics can then be regarded as an emergent property of a system which privileges the zero.

This website generalizes the concept of zer0, utilizing the the symbol Z to represent several types of symmetries.

- a numeric value
- an equilibrium point
- a stability condition
- a steady state condition
- a homestasis condition in cells
- an oscillatory dynamic balance.

Symmetry is a tool used to analyze and understand particle physics and the fundamental interactions. Classical mechanics, electrodynamics, relativity, quantum mechanics can be understood using symmetry models. The symmetries involve the fundamental parameters space, time, mass and charge. Charge represents the sources of the three nongravitational interactions: weak, strong, and electric. Space has properties required of any parameter used in making a measurement: it is real, nonconserved, and countable.

The discipline of physics is based on systems in which conserved quantities remain fixed while nonconserved quantities vary. A conserved quantity is defined in relation to changes in a nonconserved quantity. The conserved quantities include energy, momentum and angular momentum, as well as mass and charge. The nonconserved ones are space and time. Physical systems can be described in terms of combined conservation and nonconservation principles. Physicists study how mass and charge, or quantities such as energy, momentum, and force remain constant, zero, or a maximum or a minimum while the space and time coordinates alter arbitrarily. ZT unravels the many contradictory relationships found in both physics and biology.

## The Mathematics of Duality: A Universal Alphabet and Rewrite System

In Section 1.2 of ZT, The Genesis of Nothing, Rowlands writes that "since 'nothing comes from nothing', we are left with the question of how we preserve the total nothingness in the presence of the seemingly 'something' which we call 'the universe'. Both mathematics and physics suggest

that the answer lies in the concept of duality, the idea that to every fundamental concept there exists some kind of partnership or relationship with another that is its 'dual'. The pairing of positive and negative numbers being only the most obvious example. So we might well expect that the scheme required for preserving the conceptual nothingness of mathematical physics, is essentially a dualistic one. But we still have to identify the origin of such processes, and...

### EXAMPLE I: Statistical Mean

Consider the three numbers 5, 7, 21.

The mean is (5 + 7 + 21)/ 3 = 33/3 = 11

Elements in the Statistical Alphabet are:

5 - 11 = -6 = element1

7 - 11 = -4 = element2

21 - 11 = 10 = element3

SUM of 3 elements = 0 = Zero-Totality

### EXAMPLE II: Zero-Sum Game

#### Payoff Matrices:

A | B | ||
---|---|---|---|

1 | 40, -40 | -20, 20 | 30, -30 |

2 | 20, -20 | 50, -50 | -10, 10 |

Red chooses action 2 and Blue chooses action B. When the payoff is allocated, Red gains 50 points and Blue loses 50 points.

### EXAMPLE III: Dirac Equation

The Dirac Equation can be derived from a zero-totality quarterion algebra along with associated conjugate variables.

(± kE ± ii p + ij m) (± kE ± ii p + ij m) = 0.