Fibonacci Sequence

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The Fibonacci Pyramids

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The Fibonacci pyramids are of course, not actually pyramids, nor tetrahedrons (stacks of consecutive triangles); they are rather, the surface points of said tetrahedrons or any four consecutive triangular numbers. These geometries appear in the Fibonacci family of numbers in the repeat cycles, but only when coupled with the dual number sequence of the Lucas Numbers, (termed L.E.F.T.s for Lucas Equalized Fibonacci Tetrahedrons for short).

Of the various LEFTs, the dual LEFT 27300, is the most important to the discipline of biblical mathematics and the most stunning in its orchestration of 3D geometry. As evidenced on this website 37x73.com and many others besides, the protoverse, Genesis 1:1 is the primary evidence for the mathematical engineering of the Judeo-Christian Scriptures, known commonly as the Bible. So finding a 3D triangle (tetrahedron shell) highlighted in the infinity of the Fibonacci Sequence, is very much like finding the proverbially 'diamond in the rough'.

Tetrahedron(73) & (93)

Equalizing the Fibonacci and Lucas double digit cycles we find an astonishing geometry of tetrahedron shells. A geometry that is found many times in these numbers (see below). That this sum (14,800 + 12,500 = 27,300) is a double tetrahedron, is important in its own right, but it is the way that these two geometries connect that is most telling.

In fact, it is a bit of a puzzle.

In order to fit the Fibonacci sum onto the larger tetrahedron you have add 2130. In order to fit the Lucas Equalized sum you have subtract 2130. The only way to do this, so that each side of the tetrahedron has an equivalent sum is to include the edges of each tetrahedron shell in the equation. When one does so, the math works out perfectly, with two triangular bases, both of which are features of the math of Genesis 1:1.

Above are the only two tetrahedron shells (surface points) that can be formed from the equalized double digit cycles of both.

The sum (27,300) was something of mystery to me at first. The tetrahedron shell of 17,300 (see below) made from the unequalized double digit cycles was certainly worthy of note and an important find, but the mysterious equalization sum, must be (I reasoned) to be the more important. It remained a mystery until I watched Bill Murray's comedy, The Man Who Knew Too Little, whereupon the solution came to me by going back to the first principle of biblical mathematics, which is that everything always relates to the first verse, Genesis 1:1.

I mention this anecdote, because it is an important part of the discovery. I applied the basic hypothesis of biblical mathematics, let's call it The Protoverse Theorem, and it proved correct. This process is known by a more famous name —science.

Because the Fibonacci and Lucas numbers cycle at alternating frequencies (an innate aspect of their twin nature) the Lucas cycles must be equalized. As stated elsewhere, the operative numbers and ratios of the two is the prime number 5 and the highly composite number 12. These integers hold the key that unlocks the symphony of mathematics hidden in their repeat cycles. And it is simple arithmetic. The Lucas Numbers cycle first at 12 terminal digits while the Fibonacci's cycle at 60, where 5 x 12 = 60. This relationship between them is infinite.

Fibonacci-Lucas Equalization

Fibonacci-60 = Lucas-12 x 5

Fibonacci-300 = Lucas-60 x 5

Fibonacci-1500 = Lucas-300 x 5

The above graphic displays how these numbers correspond to the operative triangles involved in these two tetrahedron shells. The only way these numbers work is by first highlighting the edges of the tetrahedrons. This is to say, in order to create full rotational symmetry in both tetrahedrons, the edges must be included in the calculation.

Fibonacci 300 cycle = 14,800

Lucas 60 (Equalized) = 12,500

Tetrahedron 93 (Shell) = 16,930

Tetrahedron 73 (Shell) = 10,370

14,800 + 2130 = 16,930

12,500 - 2130 = 10,370

We can say from this then that the number 2130 is the operative sum. However, 2130 is not divisible by four (tetrahedrons are four sided shapes). Only when we subtract the edge-sum of these tetrahedrons does the sum become divisible by four:

2130 - 430 Tt.(73) = 1700

2130 - 550 Tt.(93) = 1580

1700/4 = 425

1580/4 = 395

Both of these distributed remainders (425 & 395) are bases (plinths) of relevant triangles. And both of them point directly to one of the guiding principles of biblical mathematics, that being Jenkins Containment seen in Genesis 1:1, where T37 is contained by T73. The 425 sum points to the 37th triangle (703), while the number 395 is the sum of the 5th word of the Bible, Hashamayim, The Heavens, the last word in that containment.

In the beginning God created the heavens (and the earth)

The Heavens (השמים) = 395

Both numbers are also at play in the make up of the 93rd triangle, the large sided triangle of the 93rd tetrahedron:

The 595 number here is also of some importance, here being the completion of the inset triangle, the 44th, 990. This number is a nonagonal (9 polygon), numbers which are always triangles and always produce hexagrams, i.e. the Star Numbers.

793 = 12th Star

397 = 12th Centered Hexagon

595 = 12th Nonagonal Triangle

A pattern is clearly discernible in these constituents of the hexagram, and that same pattern is reiterated in the 37th Star (7993, 3997 & 5995 respectively).

More importantly, however, 595 and the 44th triangle are primary patterns found implicitly, within the Fibonacci 300 repeat cycle. This quartet of tetrahedron shells or tetrahedron of tetrahedrons, is found patterned in the Fibonacci Series, Ad Nauseum, if you will. The following graphic shows only a few of the ways in which 3700 (Tt.(44) surface points) is found in the infinite repeat cycle of the Fibonacci Series.

The above shows how the 395 integrates with the quadrilateral LEFT 3700, a pattern established in the columnar matrix in at least six different ways, i.e. the lower left example can be iterated in two other fashions.

These also connect with the 24 Digital Root Cycle and the newly discovered (by 37x73.com) 24 Nth Root Cycle. Both cycles cover the entirety of the Fibonacci Sequence. Digital Root cycles are found by decomposing the digits of any number to its digital root, hence, 12 becomes 1+2 = 3, and 12,345 = 1+2+3+4+5 = 15 = 1+5 = 6. Nth Root cycles employ the same math while continuously summing them together, e.g. 1+1 = 2, 1+1+2 = 4, 1+1+2+3 = 7... .

This same triangle (1128 or T(47)) shows an amazing coordination with both basic biblical mathematics, in the highly important name Eheyeh Asher Eheyeh or in English I Am that I Am AND once again the 24 digital root cycle of 117.

If you look carefully at the spacing of the white stars, of which there are 45 (13 x T(9)) you can see it is, what you might call a 'perfect fit'. The number 585 is highly consistent with the Fibonacci FIVEness, being a fivefold cycle of the 24 digital root. But it is not just T(47) that does this, it is the entire complex of T(93), which we see manifested in the dual tetrahedron (LEFT 27300).

These patterns also show strange and unexplained connections to the decimal expansion of phi (the Golden Ratio).

The Halo Phenomenon

One of the features of the Protoverse Theorem (which states that the patterns seen in Genesis 1:1 inform the rest of the mathematics of the Bible, or that the first verse is the 'Key' to the rest of the Book) is that the border or perimeter of the various figurative geometries also contributes to the overall design and meaning. This is fitting, in that the perimeter in figurative geometry represents the pure geometry of the figure.

The image above, showing the star numbers 37 & 73, is a perfect example of this, but now with the 'halo-effect', where the highlighted perimeter draws attention to the interior contents and vice-versa. The numbers involved here are very much within the precincts of the Three-Seven Code, being the prime number 37 and 73 with a dual surround of (3 x 7) x (7 - 3) = 84. It also follows a similar theology seen throughout all biblical mathematics, where titles/divine-phrases, like 'I AM that I AM' act as a unifying feature between what might be considered two opposites. This, in a very grand metaphysical sense, is the concept of the triangle and theologically the Trinity, where a third element balances the opposition of two.

But in order for any new concept to be considered part of the canon of biblical mathematics, it must prove itself by connecting to both important elements in science and fundamental patterns already seen within the canon itself.

The Halo Phenomenon, does so phenomenally.

This digital root sum connects to other cycles via the number 40. As all triangular numbers are the sum of consecutive integers (1+2+3+4 = 10), the perimeter of figurative triangles are the result of that summation minus a lesser summation. In this instance, 117 is the result of the sum of the numbers 1 through 40, minus the sum of the numbers 1 through 37. The number forty, besides being an important biblical number, is also highly integrated in the factorization of the repeat cycles (see 60):

Fibonacci Double-Digit Cycle, 1-300 = 14,800 = 40 x 370

1-60 = 2880 = 40 x 72

61-120 = 3040 = 40 x 76

121-180 = 3160 = 40 x 79

181-240 = 2760 = 40 x 69

241-300 = 2960 = 40 x 74

The math of a triangle's perimeter is very similar to the math that produces tetrahedron shells, the 3D perimeter of a tetrahedron, see below LEFTs).

The 37th triangle (703) is the heart and center of biblical math(s), and if it wasn't for the fact that Genesis 1:1 nests T(37) within T(73) then I wouldn't be writing this and you wouldn't be reading it. The above graphic (of the Fibonacci 24 cycle endlessly circling the T(37)) displays how this primary geometry is a pattern hidden within the entire infinity of all Fibonacci Numbers, the most important and studied number sequence in science.

The Fibonacci 24 cycle, is a pattern that runs the gamut. Take any consecutive 24 Fibonacci Numbers, add their digits together, and whatever sum you get, add those digits together, and continue this process until it becomes a single digit and you get the digital sum cycle of the Fibonacci Sequence, the sum of which is itself 117 a number which perfectly encapsulates the 37th triangle and the entire thesis of biblical mathematics.

Considering how important the Fibonacci Sequence is to the underlying math and physics of the universe, that is quite a statement.

So does this halo-effect confirm any other hypothesis of the field?

Naturally. It does.

A primary premise of the maths of the Bible is that the geometric deconstruction of the T(73) triangle, is the semantic split between the first five words and the final two. This can be seen in countless examples on this suite of websites and many, many others across the internet. We see a similar split in the ordinal version of the verse. The two versions are united in the triangular surround of T(73), the perimeter of the 76th triangle, alluded to in the very first word of the Bible (berashith).

Other patterns abound:

The final proof is how this phenomenon connects to the great secondary premise of biblical mathematics, being that there is a mathematical continuation between the patterns seen in the Old Testament, with those seen in the New. Foremost among them being the Genesis-John triangle T(112), where the John 1:1 sum (3627) completes a larger triangle with the Genesis 1:1 sum. A triangle that underscores the importance of primary divine names in the Bible (the 112th triangle is a reference to the Old Testament full Hebrew name of God, the Lord God, Yahweh Elohim = 112).

The above animation shows how these LEFTs are highly connected to the well-established patterns seen in biblical mathematics.

L.E.F.T.s

Lucas Equalized Fibonacci Tetrahedrons

One of the most interesting academic and mathematical outgrowth of these new discoveries in the Fibonacci family of numbers, is how these repeat cycles are inundated with a very specific, if not obscure form of figurative geometry. What makes this even more intriguing is this outpouring of geometry which can only be accessed with a very specific key, that key being the Lucas Numbers and their proper equalization. These configurations are so numerous and integrated in the mathematics of the Fibonacci that I have had to give them their own name for short LEFT or Lucas Equalized Fibonacci Tetrahedrons.

One of the most stunning is the LEFT-3700, a quadrilateral replication of the 44th tetrahedron shell (also called surface points). This pattern is integrated in the Fibonacci 300 repeat cycle in at least a half a dozen ways, where the columns of a 12 column matrix produce and reproduce this four-way split of the sum 14,800; itself a shadowing of the prime factorization 148 (2 x 2 x 37).

Whether or not other Fibonacci repeat sums do this is an open question.

Every other Fibonacci sums to a Lucas Number

Perhaps a more interesting question is why does equalizing the Lucas repeat cycles with the Fibonacci's produce a regular array of tetrahedron shells. The math governing such geometry is simple enough, being twice a square plus two (OEIS: A005893). And also from an elemental geometry, it is the sum of four consecutive triangles, one for each side.

Surface Points of a Tetrahedron Formula: 2*n^2+2

Basic Geometry: T(n) + T(n+1) + T(n+2) + T(n+3)

Below was the first major discovery of the now named LEFT-17300. This LEFT has a leading and mysterious role in the way it interacts with other LEFTs produced from both lesser and greater conjoined repeat cycles. As the illustration below demonstrates, the way in which these dual number systems (the double digit Fibonacci 300 repeat cycle and the double digit Lucas 60 repeat cycle) integrate in the construction of the tetrahedron, is positively mathematical and natural related to the number 5, being that the Golden Ratio is based on the square root of 5.

The number 17,300 has another strange connection to the Fibonacci's, that being it is the sum of the Fibonacci Numbers 8-20!

Fibonacci Numbers: 8-20 (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765)

21 + 34 + 55 + 89 + 144 + 233 + 377 + 610 + 987 + 1597 + 2584 + 4181 + 6765 = 17,300

The 'fine-tuning' of the integration of these number systems, where the very edges of the polyhedron are highlighted, is not a flourish or dalliance in the math, but surprisingly a common feature (see the dual LEFT 27300 above).

In a sense, one could say that this isn't actually a proper LEFT, because the Lucas cycle is not equalized. Nevertheless, this formation has a multitude of what you might call 'suspicious' connections to the geometry of the other LEFTs.

The animation displays the incredible tetrahedral modularity of the Fibonacci Numbers.

The configuration on the left above, shows the shell 17,300 perfectly containing the tetrahedron frame Tt.83 (490) with a base of T(84) through T(90). The second configuration on the right, shows something similar but now with the inclusion of whole new other tetrahedron shell, being tetrahedron-shell 90 (15,844) plus the triangular bases of T(84) through T(86) and the tetrahedron frame of Tt.83 (490) to cap it off. Is this a freak outlier result? It is unknown. BUT this is only one of an infinite number of untested Fibonacci repeat cycles. So if it is a freak, then there is something freakish about this cycle in particular.

What makes so much of this organization possible, is the making of the numbers themselves, being the result of the famous Fibonacci addition or rabbit multiplication, where one perpetually adds the former sum. This process of continual summation also encodes this numbers with a host of hidden properties, especially in the repeat cycles, which have been largely academically ignored even though they constitute a pattern that connects all Fibonacci numbers together.

The Lucas Numbers may be lesser known to the average person, but from a purely mathematical perspective, they are nothing less than the equivalent or twin of the Fibonacci's. Nature (the Logos) is never out of date nor ever under-read and has nor preference in popularity. It may be hard for us humans to appreciate this, but it is an important mindset, that every scientist and mathematician must adopt, regardless of what the publishing industry considers currently commercially viable.

So because these dual number systems are born of each other and born of perpetual addition, it makes perfect sense to add them together.

Equalizing the Lucas and Fibonacci cycles is the key to unlocking their power. Due to the intrinsic arrangement of these two number systems, they have varying period lengths in their repeat cycles. Naturally, these cycles are and must be modular.

By looking at LEFT 290 and LEFT 580, the tetrahedron made of the terminal one-digit cycle of both number sequences, we find they are integrated in the columnar matrix mathematics in a perfect harmony with the geometry itself.

For some time I struggled attempting to find a means by which these numbers could produce such patterns. Clearly, the Fibonacci family of numbers is based in addition and summation. In figurative geometry, summation is the triangle itself. So the fact that all tetrahedron shells are produced by four consecutive triangles, i.e. four consecutive overlapping summations, must be the function I was looking for.

Tetrahedron Shell Geometry: T(n) + T(n+1) + T(n+2) + T(n+3)

This pattern appears in the matrix mathematics in a regular fashion with a minor footnote. The four-way split in the upper right hand corner of the graphic, shows that the third triangle T(17) must intrude one cell into the first T(15). To the purist or the chronically OCD this may be a problem, but in reality nature and science are often this way, where complex patterns don't always perfectly fit the ideal realm. This same bleed over can be seen in the Fibonacci 300 cycle in the way that it replicates the geometry of an octagonal star number.

This rainbow pattern is no lark; it is nature's way of making organization, also known as 'geometry'.

The LEFT 27300, can be constructed in another fashion, other than the dual tetrahedrons and here being connected to the LEFT 580 formation, the math being governed by the dynamics of Hogben's Central Polygonal numbers, or the two halves of the tetrahedron shell.