11 September 2024 AD
Sts. Protus & Hyacinth 257 AD; St. Adelphus 5th century AD; St. Paphnutius 356 AD
This article will be very short. I will describe the spiral angle and how to build a spiral of the Transcendental Fibonacci-like Sequences, Pater, Filius, and Spiritus. To determine this all we need is two consecutive terms of a sequence.
Let us start with the Pater Sequence.
For n>=0
Using 7th and 8th term.
n=7; P7+1 = P8 = 15866.4826151
And
n=6; P6+1 = P7 = 4268.41629973
First, the ratio of the nth term and the nth plus the (n+1)th term must be equal to the reciprocal of the (‘golden ratio’ plus 1):
(P7 = 4268.41629973) / [(P7 = 4268.41629973) + (P8 = 15866.4826151)] = 4268.41629973 / (4268.41629973 + 15866.4826151) = 4268.41629973 / 20134.8989148 = 0.211990947548 = 1 / 4.71718255694 = 1 / (1 + 3. 71718255693)
For n<=0
n=-1; P-1 = 0.435280824907
And
n=-2; P-2 = 0.117099663049
(P-2 = 0.117099663049) / [(P-2 = 0.117099663049) + (P-1 = 0.435280824907)] = (0.117099663049) / (0.552380487956) = 0.211990947548 = 1 / 4.71718255692 = 1 / (1 + 3. 71718255692)
The results are the same.
Calculation of an Angle of the Pater Spiral.
The Angle of the Pater Spiral = 1 / (1 + ‘golden ratio) = [1/ (1 + 3. 71718255693) = (0.211990947548)] * (360.0 degree = 2π) = 76.3167411173 degree = 1.33197840689 radians
The Pater Spiral Angle fits (1 + ‘golden ratio’) in a circle.
360.00 degree / 76.3167411173 degree = 4.71718255693 = 1 + 3.71718255693
How to ‘build’ a Transcendental Spiral?
One of the ways to make a spiral graph of a sequence is from squares, where the side is equal to the value of the term, that is, for example, P1 = 1.61801828971 will be the first part of a spiral, then P2 = 6.0144693633 will be another part of the spiral, then P3 = 22.3568806064, and so on.
Let us calculate the Filius Spiral.
The steps are as before for the Pater Spiral.
For n>=0
Using 9th and 10th term.
n=8; F8+1 = F9 = 1101.76951107
And
n=9; F9+1 = F10 = 2531.16941531
First, the ratio of the nth term and the nth plus the (n+1)th term must be equal to the reciprocal of the (‘golden ratio’ plus 1):
(F9 = 1101.76951107) / [(F9 = 1101.76951107) + (F10 = 2531.16941531)] = 1101.76951107 / 3632.93892638 = 0.303272235894 = 1 / 3.29736745288 = 1 / (1 + 2.29736745288)
And so on for the n<=0.
Calculation of an Angle of the Filius Spiral.
The Angle of the Filius Spiral = 1 / (1 + ‘golden ratio’) = [1/ (1 + 2.29736745287) = (0.303272235894)] * (360.0 degree = 2π) = 109.178004922 degree = 1.90551565664 radians
The Filius Spiral Angle fits (1 + ‘golden ratio’) in a circle.
360.00 degree / 109.178004922 degree = 3.29736745287= 1 + 2.29736745287
Let us calculate the Spiritus Spiral (Transcendental Fibonacci).
The steps are as before for the Filius and Pater Spiral.
For n>=0
Using 9th and 10th term.
n=8; S8+1 = S9 = 53.5308082347
And
n=9; S9+1 = S10 = 86.6138267866
First, the ratio of the nth term and the nth plus the (n+1)th term must be equal to the reciprocal of the (‘golden ratio’ plus 1):
(S9 = 53.5308082347) / [(S9 = 53.5308082347 ) + (S10 = 86.6138267866 )] = 53.5308082347 / 140.144635021 = 0.381968301724 = 1 / 2.6180182897 = 1 / (1 + 1.6180182897)
And so on for the n<=0.
Calculation of an Angle of the Spiritus (Transcendental Fibonacci) Spiral.
The Angle of the Spiritus (Transcendental Fibonacci) Spiral = 1 / (1 + ‘golden ratio’) = [1/ (1 + 1.6180182897) = (0.381968301724)] * (360.0 degree = 2π) = 137.508588621 degree = 2.39997762121 radians
The Spiritus (Transcendental Fibonacci) Spiral Angle fits (1 + ‘golden ratio’) in a circle.
360.00 degree / 137.508588621 degree = 2.6180182897 = 1 + 1.6180182897
A couple of links.
https://en.wikipedia.org/wiki/Golden_spiral
https://en.wikipedia.org/wiki/Golden_angle
https://en.wikipedia.org/wiki/Golden_rectangle
The next articles will be on sequences formed by addition/subtraction tables and proton and neutron g-factor.
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