The 216 CODE: Pisano Function P(24) = 24.
An Original Jain 108 Discovery
Clock Mathematics Using Fibonacci Numbers.
Visualize a Clock of Numbers, not based on the normal counting numbers, going in natural consecutive order, as in 1-2-3-4-5-6 etc but rather a Clock that only uses Fibonacci Numbers: 1-1-2-3-5-8-13-21-34-55-89-144 etc.
We will be applying the common knowledge similar to converting 24 Hour time into 12 Hour time, eg: the time of 14:16 is really 2:16pm achieved by subtracting 12 from 14 hours.
Though what is of importance, what we are looking for, is that the count of how many Fibonacci Numbers that are being used is identical to the count of numbers Repeating which is its Cycle Length. This type of investigation is called the Pisano Function, a tribute to Fibonacci who was also known as Fibonacci of Pisa, as that is the place where he was born, near the well-known Leaning Tower of Pisa in Italy.
Pisano Cycles are therefore an investigation into Fibonacci Number Patterns on a Clock or Wheel, similar to Modulus Maths.
In this article, I will show that nothing happens when we say count with 8 Fib Nos, as the Repeating Cycle or Length is 12, summarized as Pisano 8 = 12 or P(8)=12.
But something very interesting happens at Pisano 24 ie: P(24) = 24. Which means the first 24 Fibonacci Numbers have a Repeating Cycle Length also of 24, which is shown in a diagram: Fibonacci Numbers Mod 24 Written in Circle
have a Repeat Cycle or Length of 24.
This is well known to Mathematicians, but what they overlooked, which is my Discovery or Revelation, is the Sum of all 24 Numbers in the Circle = 216 = 6x6x6 = 63 = 2×108.
nb: For Mod 24, we can only use the Fibonacci Numbers 0-1-1-2-3-5-8-13-21 only,
The Fibonacci Numbers Mod 24 are:
0, 1, 1, 2, 3, 5, 8, 13, 21, 10, 7, 17, 0, 17, 17, 10, 3, 13, 16, 5, 21, 2, 23, 1
which then repeats so the cycle has length 24: Pisano (24) = 24. But you may ask, how did I arrive at all these 24 numbers?
Observe the first 9 Fib Nos: 0, 1, 1, 2, 3, 5, 8, 13, 21,
observing that the next or 10th Fib No. is 34, but 34 is bigger than 24 on our Clock of 24, so we must always subtract 24 from the growing Fib Nos. until a number less than 24 is arrived at. This means that the Fib No. 34 less 24 = 10, so I have written the 10th number as 10 on this Wheel of 24.
The 11th Fib No. is the next bigger Fib No which is 55 less 24 less 24 = 55 less 48 = 7, so I write a 7 in the Wheel… This process keeps going on with a Repeat Cycle of 24, meaning that the 25thFib No. is a Zero, ie: it has no remainder after being divided by 24, in fact, the next 24 numbers are an exact copy of the first set of 24, and so on.
Observe, in this Wheel of 24, that there are 2 Zeros, and they are diametrically opposing one another!
The Sigma, or Sum of all the Numbers in the Wheel of 24 is 216 or 6 Cubed or 6x6x6… or 2 x sri 108. This means Pisano Functions are intimately connected to the Secret Code of the Golden Ratio 1:1.618… the Living Maths Of Nature.
The Sum of these 24 Numbers constitute “The 216 CODE”
nb: Are there other numbers like Pisano 24 that have the same Repeating Cycles? Yes. This is recorded in Sloane’s Sequences A235702 and they happen mysteriously at various multiples of 24.
These “Fixed Points” of the Pisano Function are Multiples of 24 x 5^n form and they form the series:
1, 24, 120, 600, 3,000, 15,000, 75,000, 375,000, etc.
Here are some numerical references that relate to the meaning of 216, extracted from my Dictionary of Anointed Numbers called “Harmonic Stairway”:
a)- The sum of the following 3 cubic numbers = 216
3^3 + 4^3 = 5^3 = 6^3
b)- Of interest, the first Cubic Number here, 3 Cubed, generates an important Harmonic Sequence when continually doubled, I call it “The Binary On 27” one of the numbers being 216: (as well as 6^3 = 216, 12^3 = 1,728, 24^3 = 13,824 etc).
27 – 54 – 108 – 216 – 432 – 864, 1,728, 3,456, 6,912, 13,824 etc
c)- A special Pythagorean Right-Angled Triangle:
(216, 288, 360) meaning:
216^2 + 288^2 = 360^2. This was highlighted by Bruce Cathie, who gave us the Leyline Grid around the Earth, emphasizing 216 as a Harmonic of the Moon’s Diameter 2,160 miles; 288 as double the Light Harmonic of 144, and 360 being the Harmonics of the Circle, all extremely important and potent numbers.
d)- A 3×3 Multiplication Magic Square of 3×3, where the multiplication of any 3 numbers in the square’s 3 columns or 3 rows or 2 diagonals sum to 216:
eg: the central horizontal row has 3 numbers:
(36, 6, 1) = 36 x 6 x 1 = 216.
e)- A 3-Frequency Star Tetrahedron has 216 smaller equilateral Triangles. A 1-F Star Tetra has 24 edges and 24 equilateral Triangles, and each of these triangles can be subdivided into 3-Frequency or 9 smaller triangles. Thus 24×9=216. This is shown in the diagram, a photo of a beautifully crafted glass Star tetrahedron hand-modelled by my colleague Asaf Zakay of Zakay Glass Creations.
f)- The Bayon Temple in Cambodia has 216 statues of smiling faces that resemble the enlightened Buddhist deity of Avalokitesvara.
g)- There are many other references to 216, like the height of the Temple of the Sun in Teotihuacan in Mexico, has a vertical height of 216 feet.
h)- The Speed Of Light is recorded as 186,624 mps (miles per second) and this incorporates the multiplication of 2 important harmonic numbers based on astronomical measurements. 216 x 864 = 186,624, where 216 is a Harmonic of the Moon’s diameter being 2,160 miles, and the Sun’s diameter being 864,000 miles. (In Harmonic Mathematics, we are permitted to slide the decimal left or right, to minimize or maximize, or to add or delete zeroes, to lessen and make greater, thus a harmonic of 864,000 is 864).
i)- 6^6 = 6x6x6x6x6x6 = 46,656 and we can also connect this number to the distance of the Earth from the Sun being approximately 93,000,000 miles. The Diameter of the Sun, 864,000 miles multiplied by the holy Indian Number Sri 108 gives 93,312,000 miles which is very close to the western understanding of the distance of the Sun to Earth being approximately = 93 million miles. The Radius of the Sun is half of this, being 432,000 miles. We could also say that 432,000 x 108 = 46,656,000 and as you can see, 6^6 = 46,656.
On a final note, to repeat the importance of Pisano(24) = 24 we can show a counter-example where the number of Fibonacci Numbers used, say 8, does not equal the Cycle Length that repeats, which is 12, as shown in the diagram.
For Mod 8, using Numbers 0-1-1-2-3-5 only, the Cycle or Length is 12.
Therefore Pisano 8 repeats every 12 Numbers. Therefore P(8) = 12.
Observe the 4 Zeroes in this Wheel of 24, meaning that there is no Remainder after division by 8. The 24th Fibonacci Number (aka F24) is the sum of 17,711 + 28,657 = 46,368 and also has no remainder when divided by 8.
December 2020, Mullumbimby Creek
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Did you know The Golden Ratio is part of our cellular memory? That it permeates all of creation like flowers, crystals and pyramids?