# Logical Systems

## arithmetic, geometry, algebra

• Phi emerges at the outset when exploring the simplest of logical relationships

$\large \mathbf{\Phi^2 \, - \, \Phi \, - \, 1= \, 0 }\\$

$\large \mathbf{roots\; of\; \phi \, := \, \left\{ \left(\frac{-\sqrt{5} + 1}{2}, 0 \right), \left(\frac{\sqrt{5} + 1}{2}, 0 \right) \right\} }$

$\large \mathbf{\color{#C90}{\phi} \, = \, \frac{\sqrt{5} + 1}{2} } \approx 1.618033989\\ \large \mathbf{\color{#C90}{\varphi} \, = \, \frac{\sqrt{5} - 1}{2} } \approx .618033989\\$

$\large .5 \times 5^.5 + .5$

## The Definition

It is known by many names. The Divine Proportion The Golden Ratio, Mean, Section

Euclid Extreme and Mean Ratio

A proportion is a relationship between two values. Such as 1:2 or a:b or a/b

Two values are in The Divine Proportion when the ratio of the lesser value over the greater value is equal to the greater value over the sum of the lesser and greater value.

In other words, when a segment is sectioned into the Divine Proportion, the parts are in a harmonic relationship to the whole. Setting up a Harmonic Rhythm.

The very nature of the Golden Ratio is harmonic resonance

# Extreme and Mean Ratio

## The Golden Ratio is also a Golden Mean

$\large \frac{a}{b} = \frac{b}{a+b}$

# Identities

## Algebraic Expressions related to PHI

A study of the algebraic attributes and identities of the Divine Proportion

# Golden Sections

## Geometric constructions which derive the Golden Ratio

A Golden Section is the ratio of two segments within a construction where the proportion of their measures is PHI.

# Series

• Fibonacci Series
• Lucas Numbers
• Binet's Forumal