View Full Version : The 60 digit pattern of the Fibonacci sequence

02-05-2019, 09:53 PM
I thought I'd put this up for anyone who might be interested.

Quick background.. as you "iterate" through the fibonacci sequence (adding the last two numbers together for the next number), a pattern emerges involving the left most digit. It repeats after every 60 iterations. In the graphic below, the artist (JoeDubz) attributes it to Lucien Khan, but it's been known for centuries:

"The sequence of final digits in Fibonacci numbers repeats in cycles of 60. The last two digits repeat in 300, the last three in 1500, the last four in 15000, etc. The number of Fibonacci numbers between n and 2n is either 1 or 2."


As well shown in the above is that taking the digital root of each result (adding digits together until you are left with a single digit result) leads to a 24 digit pattern. But we aren't dealing with that one right now.

Also above, it highlights how the 5/0 cuts the circle into 12 equal divisions. A facebook friend pointed this out to me, how numbers opposite each other (thru the center) are pairs that equal "10":


Now, this is a "thing" in arithmetic (number theory).. the "marriages" or partners among the primary digits that equal 10. 1-9, 2-8, 3-7, 4-6, with the pivot being 5. It might be mentioned in Aesch Mezareph.. I'm sure I saw it in other text being talked about explicitly but I can't remember it lately. I also seem to have found some support in Kircher's Arithmologia sive De abditis numerorum mysterijs.. this is 4 parts of different pages stuck together:


And i realized that if you used the same color to connect the same pair, patterns emerge:


It's very strange. 0 by itself forms a 90 degree axis, while 5 is two 90 degree axes, offset. The even numbers have 4 connections.. with 4-6 being a 90 degree rotation of the 2-8 pattern.
0 would seem connected to the evens, with evens doubling the connections of 0 (2 to 4).

The odds would seem to be connected to 5 (which is of course odd) by the number of connections.. 5 having 4, and the odds having double that at 8. And again, the 1-9 pattern is a 90 degree rotation.

the 0 pattern alone allows you to draw the square inside the circle, and half of the points of 0+5 (middle top) would allow you to draw the hexagon. These are also important points on the "unit circle (https://en.wikipedia.org/wiki/Unit_circle)" and related trigonometric functions.

These points also possibly connect it back to the standard "internet sacred geometry":


To be honest, you have all the points you need after the first 6 circles (the flower). The nodes where the circles intersect gives you the other 6 equal angles. What is special about the hexagon and the related grid it forms as the structure progresses has to do with its chord length (the side of the hexagon). It's the same as the radius of the circle. and the circumference = 2*pi*radius. The spacing ends up being equal everywhere.

What i wonder if this is somehow connecting Pi and Phi.