TONY STEELE
Glastonbury Alignments
The Golden Ratio
Given: The human brain seeks (needs to create or absorb) patterns.
THE 38.2 ANGLE & THE KEPLER TRIANGLE
This enquiry is ongoing & the work (Patterns) produced reflect the flow of questions raised by the data.
GOLDEN RATIO (PHI:phi), FIBONACCI & LUCAS RELATIONSHIP
Although I decided to use the Extended Mile Algorithm I felt I needed to probe Lucas & Fibonacci further.
I investigated the mathematical connections between Fibonacci & Lucas.
I found that these sequences are purely indicators of underlying Golden Ratio Processing.
[ i.e. ADDITIVE Phi:phi (Fibonacci) or EXPONENTIAL Phi:phi (Lucas) ].
The number sequences (Fibonacci & Lucas) can be understood, therefore, to be ratios themselves, uniquely relative to the PHI:phi entity rather than 'free' integers.
[ A ratio minus or plus another ratio is still a ratio ]
Other indicators are possible. And they may be dismissed as arbitrary.
For example, the ratios measured from the Nautilus Shell appear arbitrary but they are indicators of underlying Double Phi:phi processing.
All shells naturally vary but on average some dimensions apply.
3.236 (The 3/4 spiral arc ratio) is, in fact, 2 x PHI 1.618.
1.08 (The chamber ratio) is, in fact, a sqrt sub-division of 3.236.
[ 3.236; 1.8; 1.34; 1.16; 1.08 ]
The above calculations appear convoluted but PHI is, by our best definition, convoluted:
Humankind defines PHI using 3 factors;
a) The Irrational sqrt5 [ https://apod.nasa.gov/htmltest/gifcity/sqrt5.1mil ],
b) a unit (+1), and
c) the doubling:halving ratio (2 : 0.5).
Phi = [ sqrt5 + 1 ] / 2
So, as per the Nautilus shell; double Phi = [sqrt5 + 1]
Although humankind's mathematics & geometry have reproduced a 'fair' mapping of the Phi ratio the formula we use does feel rather clumsy given that the primary factor (sqrt5) is a product of Phi.
sqrt5 = Phi + phi
[ sqrt5 (2.236068) = Phi (1.618034) + phi (0.618034) ]
[ phi = 1/Phi ]
i.e to produce Phi you will need Phi.
This begs the question - does Phi:phi express our number system?
e.g. 1 = Phi - phi, 5 = (Phi + phi)^2
[ c.2600BC Great Pyramid of Egypt uses harmonic 51/2 palm seked in construction - coincidence? ]
[ c. 300BC Euclid's 'Elements' describes the ratio (Phi:phi) as the extreme and mean ratio of a line AB. ]
[ The 'fibonacci' sequence was first described by the Indian mathematician Pingala (c.300BC–200BC). ]
[ 1202, Leonardo Bonacci, aka 'Fibonacci', publishes 'Liber Abaci' which introduces Europe to the Hindu-Arabic numeral system and cites the 'fibonacci' sequence as an example. ]
[ 1509, Fra Luca Pacioli writes about the Divine Proportion. ]
[ 1597 Kepler's teacher Maestlin achieves a value for the phi ratio. ]
[ Kepler (1571-1630) was among the first to give the converging ratio of the 'fibonacci' sequence a Phi value. ]
[ Kepler combines Pyrhagoras Theorem and Phi:phi to produce the Kepler Triangle. ]
[ The terms ' Golden section', 'Golden Ratio' appear in the 19th century. ]
[ 19th century, Edouard Lucas introduces the term 'Fibonacci Numbers' and presents his Lucas sequence. ]
CHARACTERISTICS OF FIBONACCI & LUCAS
Both Fibonacci & Lucas have PISANO PERIODS (repeating cycles) of 24 digits in Modulo 9.
Using Modulo 9 is a convenient way of discerning pattern and in this case relationship.
