The thing about the Fibonacci sequence and the golden ratio (PHI 1.618) is that I don’t believe in many precise constants in a chaotic elliptically wobbling Universe of numbers that simply cannot perfectly scale up to infinity as whole integers, because of the precision of large numbers compared to the ambiguity of a single or double digit integers. In a thermodynamic fluidity of energy in a wavy wobbling Universe entropy and chaos are also factors to consider. At first glance one may not notice any pattern in the products of the standard Fibonacci sequence…

However it is a predictable spiral that begins a second iteration after 68 rotations with the 67th number adding up to 44945570212853 and the 68th being 72723460248141 before the 2nd iteration of the rotation begins again with the 69th number being “the second 1” again however it is now orders of magnitude greater and rounds to 117.669 Trillion to create the foundation for the sequence to begin again on the scale of quadrillions and quintillions and so on to your favorite countable or uncountable infinity.

Hence if the beginning of the first rotation was in fact a rounding error that caused Fibonacci to begin with 1 as the first two numbers in the sequence for the sake of simplicity, then consider that the first two numbers are actually the decimal form of the 68th and the 69th typical Fibonacci numbers such that the first two numbers are slightly less than and just greater than 1 respectively.

The result now is that the 11th and 12th numbers are now 89.4439… then 144.7233… not simply 89 and 144 and PHI is either 1.6180339887498947 or 1.618033988749895 not simply 1.618.

**The Data **

deciPHI.r created by Ryan Bagnulo ( @iryanb ) on 10.9.2016 Β©

if π # 1 = 0.72723460248141 & π # 2 = 1.17669030460994 then

π # 3 is 1.90392490709135 then deciPHI.r is 1.6180339887498947

if π # 4 is 3.08061521170129 then deciPHI.r is 1.618033988749895

if π # 5 is 4.984540118792641 then deciPHI.r is 1.618033988749895

if π # 6 is 8.06515533049393 then deciPHI.r is 1.618033988749895

if π # 7 is 13.04969544928657 then deciPHI.r is 1.6180339887498947

if π # 8 is 21.1148507797805 then deciPHI.r is 1.618033988749895

if π # 9 is 34.16454622906707 then deciPHI.r is 1.618033988749895

if π # 10 is 55.27939700884757 then deciPHI.r is 1.618033988749895

if π # 11 is 89.44394323791464 then deciPHI.r is 1.6180339887498947

if π # 12 is 144.7233402467622 then deciPHI.r is 1.618033988749895

if π # 13 is 234.16728348467683 then deciPHI.r is 1.618033988749895

if π # 14 is 378.89062373143906 then deciPHI.r is 1.6180339887498947

if π # 15 is 613.0579072161158 then deciPHI.r is 1.618033988749895

if π # 16 is 991.9485309475549 then deciPHI.r is 1.6180339887498947

if π # 17 is 1605.0064381636707 then deciPHI.r is 1.618033988749895

(The code actually computes until it reaches infinity after 1478 iterations, only showing the first 17 for brevity.)

**The js source code for math:**

<code>

window.addEventListener(‘load’, function(e) {

document.querySelector(‘#deciPHI’).innerHTML = ‘deciPHI.r created by Ryan Bagnulo ( @iryanb ) on 10.9.2016 Β© ‘;

}, false);

var x = 0.72723460248141;

var y = 1.17669030460994;

var n = 1;

var f = 1;

var start = Date.now();

document.writeln(“<br /> if π # 1 = 0.72723460248141 & π # 2 = 1.17669030460994 then” + “<br />”);

for(n = 3; n < 1479; n++){

f=x+y;

r=y/x;

document.writeln(” <br />”);

document.writeln(“if π # ” + n + ” is ” + f + ” then deciPHI.r is ” + r );

x=y;

y=f;

document.writeln(” <br />”);

}

var speed = “average”;

t=Math.abs(Date.now() – start);

if (t < 24) { speed = “fast”;}

if (t > 50) { speed = “slow”;}

document.writeln(” <br />”);

document.writeln(“Compute Time = ” + t + ” (ms) . ” + speed);

<!doctype html>

<html>

<head>

<meta charset=”UTF-8″>

<meta name=”viewport” content=”width=device-width” />

<title>deciPHI.r</title>

<link rel=”stylesheet” href=”style.css” />

</head>

<body>

<h1>deciPHI.r</h1>

</body>

</html>

</code>

The Stylesheet:

@charset “UTF-8”;

#deciPHI {

color: #111975;

}