Lucas Numbers

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The Lucas Numbers have a perfect graphing curve (monocurve) on a triangular grid. Monocurves are extremely rare and have only been seen in Fibonacci related number series. This new Lucas Number cycle connects strongly with the exact lettering of the first verse of the Bible in the Hebrew ordinal.

The first five words of the Bible is also the first complete thought from the Word of God.

If we fill-in the 'hold' we get an equilateral triangle which will always be a square number. Here is it is the sum of the first complete thought of the Bible.

The real enigma of this configuration is how the 92 capture (red triangles) perfectly matches the geometry of octagonal numbers.

Lucas Numbers are intrinsically related to the Fibonacci Numbers in many ways. Perhaps the most compelling is the fact that the powers of phi 1.618.... (the Golden Ratio) become Lucas Numbers. The Fibonacci Numbers also weave together to form the Lucas Numbers.

Every other Fibonacci sums to a Lucas Number

The Powers of Phi: Decimal Expansions of the powers of the Golden Ratio, the Lucas Numbers

Phi^37 & Phi^73 Spreadsheet

Similarly, the terminal repeat cycle of The Fibonacci Numbers also display a monocurve, here on a square graph.

The number of encapsulated squares is exactly 298

the sum of Genesis 1:1 in its alphabetic order (ordinal).

The Fibonacci Numbers have a third repeat cycle that is either very little known or new to science. This is the Nth Root cycle. Like 24-Digit digital root cycle, any series that has a digital root cycle has a corresponding Nth root cycle, which is the nth sum of that digital root cycle.

The Fibonacci Nth Sum graphs on a triangular grid:

The 138 sum of the Nth Root cycle is the equivalent of Ben Elohim, Hebrew for the Son of God.

The sum of these numbers however is not 117 (the sum of both the Fibonacci and Lucas digital root cycles) but 138.

Fibonacci Nth Root 24-Digit repeat Cycle:

(1, 2, 4, 7, 3, 2, 6, 9, 7, 8, 7, 7, 6, 5, 3, 9, 4, 5, 1, 7, 9, 8, 9, 9 ) = 138

These numbers connect to the digital root cycle, summing the digits of the digital root cycle itself will produce the Nth Root cycle numbers.

The Nth Root cycle of the Lucas Numbers also graph on a triangular grid;

Lucas Nth Root 24-Digit repeat Cycle:

(2, 3, 6, 1, 8, 1, 1, 3, 5, 9, 6, 7, 5, 4, 1, 6, 8, 6, 6, 4, 2, 7, 1, 9 ) = 111

37 x 3 = 111

Most repeat cycles tested in this way (e.g. polygonal and polyhedral repeat cycles) either quarter-peat, requiring four cycles to repeat, or veer off on a severe southwest bias. Only Fibonacci related repeat cycles appear to have this graphing curve cycle. Numbers like (A013655) a Fibonacci Series from the Wythoff Array row 8, which is the sum of the Fibonacci and the Lucas Numbers, will reproduce the same graphing curves exactly, but starting and ending at different points. Indeed it appears the full infinity of the Wythoff Array will always display perfect graphing curves in these manners.

Wythoff Array 8 Repeat Cycle:

[2,5,7,2,9,1,0,1,1,2,3,5,8,3,1,4,5,9,4,3,7,0,7,7,4,1,5,6,1,7,8,5,3,8,1,9,0,9,9,8,7,5,2,7,9,6,5,1,6,7,3,0,3,3,6,9,5,4,9,3]

The primary metric for this new type of math, is the 'Capture' number, which represents the number of cells that the monocurve surrounds. The secondary metric is the 'Hold' the total footprint of the cycle (here on a triangular grid it is the total triangle in which the monocurve resides, which will always be a square number). The final metric is the difference between the two.

The Lucas Number Nth Root Cycle displays a strong connection to featured numbers in biblical mathematics. The capture being the sum of Jesus Christ (English ordinal) the hold being 22 squared and the remainder being three times the sum of the Lucas Nth Root cycle itself (111 x 3 = 333).

This has a strong resemblance to the way in which the Greek Jesus Christ—2368 coordinates with Genesis 1:1.

2368 + 333 = 2701

151 + 333 = 484

In both instances we have Jesus Christ (151 & 2368) plus the 37th and 73rd semiprimes equalling a triangular whole.

37th semiprime = 115

73rd semiprime = 218

This number coordinates with many other features of biblical mathematics.

The number 127 connects with the square graphing of the Fibonacci series in a number of mystifying ways. As this number is as close as whole numbers can get to representing the square root of phi (the Golden Ratio) I thought important to note it.

Performing the basic arithmetic of digital summation

we miraculously arrive at the remainder of the container rectangle.

How do the numbers 'know' that

the remainder of their rectangle has 127 cells?

These same Fibonacci numbers produce another repeating pattern that can be graphed using a triangular graph. Employing the 60 digit terminal repeat cycle, which repeats every three cycles on the triangular graph.

The fact that these numbers loop here is not that amazing. Many repeat cycles will loop twice or thrice but only Fibonacci/Lucas numbers have been seen to repeat singularly in a monocurve. What is amazing, however, is how this pattern corresponds to the Three-Seven Code both in its 'hold' and 'capture', the total geometric footprint of the pattern and the individual cells that it captures.

On a triangular grid any equilateral triangle will always be a perfect square. So the hold of any pattern here will also always be a perfect square. The outline, then, of the entire configuration is an equilateral triangle of 77x77 cells or 77 squared, which is also the 49th (7x7) heptagonal number (7-shaped).

77 x 77 = 5929 = (7^2 x 11^2)

The 2359 captured cells produce a further 3373 sum when we add in the six missing c. hexagons (7 x 337 + 6 x 169 = 3373).

Plotting the 60-Digit Fibonacci repeat cycle on a triangular graph, we find it returns to origin in three cycles — most fitting for triangles. The total number of cells (including the large central area) comes to 2359 or 7 x 337, a perfect Three-Seven Code factoring. The total number of rooms is the same number that keeps cropping up in the square graphing curve, an analog approximation of the square root of phi. Even more curious, is when group each of the three sections into the large clusters of triangles (25 rooms) and the other various triangle clusters (17 rooms) the exact dimensions of the square graphing curve (25 x 17).

A fascinating connection to this number 2359 (as pointed out by RobSlattery of Biblegematria.com)

is that it is the sum of this rather pivotal and prophetical phrase of from the NT: Mark 13:21 & Matthew 24:23

And then if any man shall say to you,

Lo, here is Christ; or, lo, he is there; believe him not:

ωδε ο χριστος = 2359 = Here is Christ

If we subtract the central captured cells (2359) we get a triangular remainder of 3570.

3570 = T(84)

(3 x 7) + (3 x 7) + (3 x 7) + (3 x 7) = 84

I AM that I AM = 84 ordinal

John 20:5 = 3570

And he stooping down, and looking in, saw the linen clothes lying; yet went he not in.

This verse is one of the very few if not the only plain reference in the Bible to the Shroud of Turin, here referenced as the Linen Clothes. Should we put the two cryptic references together we have a complete puzzle:

2359 & 3570

(Here is Christ) on the (Shroud of Turin)

The Star-Rose

I've tested many of the repeat cycles in many of the most important number series in figurative geometry, to see if this is a common feature or somehow unique to the Fibonacci related series. The basic outline of this research is that only Fibonacci/Lucas numbers repeat in a single cycle monocurve. All other repeat cycles only 'graph' in double, triple or quadruple cycles or will wander right off the page. Of those that do repeat, but not singularly repeat, there appears only at best 'weak' connections.

There are only three types of graphing curves possible, (numerical cycles that can be plotted perfectly on segmented graph paper), being that there are only two types of tessellating grids (graph paper) the square and the triangular. Where the square graph can produce only one type of cycle, the triangular can house both a triangular vectoring and a hexagonal vectoring. All vectoring on both graphs must conform to the unilateral parallel lines of the grids and only three such unilateral tessellations exist, namely: squares, triangles and hexagons.

Although the research has shown that uniformly no other repeat cycles replicate the Fibonacci/Lucas monocurves, there is one series of cycles that reveals both biblical encoding and how these lesser graphing curves 'can' coordinate with other related numbers series. The triangular numbers are of course centrally important to all of mathematics and biblical mathematics in particular. So it is significant that they were the only sample to yield anything worth recording.

Here the triangular terminal repeat cycle coordinates with both the hexagonal repeat and the centered nonagonal repeat cycle. The reason for this is simply because both hexagonal and centered nonagonal numbers are subsets of triangular numbers. Hence the numbers that produce the graphic patterns are the part of the same numbers series of the larger pattern of the triangular numbers.

What should be noted however, is that hexagonal numbers only coordinate on a triangular grid and the centered nonagonals only coordinate on a hexagonal grid. Why this is, and if this is a universal feature is currently unknown.

This cycle I call a 'Star-Rose' pattern as its central element is of a Star of David surmounted by hexagonal petalling.

The nonagonal pattern (in blue) displays self-similarity to the Star-Rose pattern and complements the centralized array.

The Star-Rose of the triangular numbers has exactly 37 rooms in the capture of its cells:

The exact capture of the Star-Rose is 360 cells, exactly twice the sum of the three angles of any triangle. The hold of the graphing curve is 900 (30x30) which corresponds very well to the ordinal Hebrew of Genesis 1:1, as it quarters into 225's.

The hexagonal numbers do not graph in any meaningful or significant way on a hexagonal vector. They do chart however with the triangle numbers (of which they are a subset) on a triangular grid.

The capture of the triangular number pattern comes to 211, Hebrew for 'The Word'. The hold of the graph is 676 (26x26) the Tetragrammaton squared:

The Star-Rose pattern of the triangular numbers repeat cycle, appears at this point to be a 'one-off.' It doesn't surprise me terribly, that of all the galaxy of figurative numbers series in the mathematical universe, that only the triangular numbers appear special in this way. That's just how special the triangular numbers are.

There is another more important reason that these triangular numbers and patterns should be highlighted, by math, the universe and the Supreme Being, a synonymous metaphysical and philosophical trinity. The triangular numbers happen to perform one of the Herculean labors of mathematics, being that of the Sum Function, where each triangle number represents the sum of all the numbers up to that point.

With this understanding in mind, one could say that the 'Repeat-Cycle' of the integers, (the whole or natural counting numbers, i.e. the infinity of numbers that most people know and understand) is nothing more than the triangular numbers themselves. Take any number from here to infinity. Count all the numbers up to and including that number. That number will always be a triangular number. Those numbers repeat infinitely. The first and foremost repeat cycle of those numbers is the Star-Rose.

A newly discovered type of monocurve is the star-cycle monocurve. Here the repeat cycle follows a freeform geometric monocurve based on a star polygon figure. In the animated illustration below, we see that the double-terminal repeat cycle for the Lucas Numbers, has perfect equivalence in five columns. Since the Golden Ratio (phi) is based in the number five or the square root of five this not only makes sense but is perfectly appropriate considering how the pentagram is literally packed with the proportion of the Golden Ratio.

The fact of the matter is, is that the Lucas Numbers are in essence nothing more than the powers of phi.

What's really fascinating is that we can perform this very same trick with these very same numbers, but now employing their digital sum, which is also five-fold equivalent.

If we make a pentagram with these digital sum lengths, its area will be 3355.888. If we do this with the standard non-digitally sum lengths its area is 77567.675.

Every way you look at a 5-pointed star, the Golden Ratio is staring back at you.

What's amazing about these numbers is how they can seemingly endlessly combined to create similar patterns. By adding every two numbers of the double-terminal repeat cycle together we can star-cycle them once again:

Both of these star-cycle monocurves are trending toward the five-fold average of the double terminal repeat. This is of course a pentagram whose five basic lengths are 500 each. If you were inclined to see this as a perfected pentagram no one would blame you. Employing the rule of basic line geometry this pentagram displays some fascinating areas.

The area of the circle is of primary importance being that here it is a multiple of the Logos—373 and the prime number 97 which is the modern Hebrew for the Golden Ratio!

2 x 3 x 97 x 373 = 217,086

The biblical mathematics connection to this pattern is found in the sum of the double-terminal repeat cycle and at first it may seem like a bit of a mystery.

Lucas Double-Terminal Sum = 2500

The Only Begotten Son of God = 2500

o μονογενης υιος του θεου

This is a very exacting phrase from the New Testament and one that is filled with theological impact for any Christian.

But what does it mean? What do the Lucas numbers and the pentagram have to do with the Son of God?

The answer is core to the guiding philosophy of biblical mathematics, which is found in the Word, the New Testament word for the Logos. Each of the five columns that produce this fantastic geometry comes to 500, an important number in Greek Gematria, as it is the sum of both the Greek letter for phi (Φφ Greek Letter: Phi = 500) and the Law (The Law Ο ΝΟΜΟΣ ) and the Number (Ο ΑΡΙΘΜΟΣ).

John describes Christ as the Logos. Yet this word has lost most of its meaning and impact for the average human today; but in the day in which it was written it was a fountain of understanding. In that day Greek Thought, was thee state of the art, cutting edge technology. The only way to really capture it, is to call it the 'Science' of the ancient world and at its heart was the Logos. Indeed the science of today is not only indebted to the development of Greek Thought, but was born in its cradle.

Science was nursed to infancy by Plato, Euclid and in the schools of Pythagoras which taught that number and the relationships between numbers were the basis of the laws of the universe. This penultimate understanding is as true then as it is today. The relationships between numbers is what we call today mathematics. All of mathematics is nothing more than relationship between the pattern of one quantity to another.

This is math.

This essential understanding was called in the ancient world by a very specific name, translated today for us as 'Ratio,' and no other concept quite captures the whole of science and math better than ratio. This word in the ancient world was none other than Logos and today the word logos is synonymous with science in the study of human history.

The math textbook of the ancient world, up to the pre-modern world was Euclid's seminal piece of geometry the Elements. In it he describes the Golden Ratio, thusly:

The Extreme and Mean Ratio = ακρος και μεσος λογος

The ratio (logos) is in red.

Extreme (ακρος) = 391 = Jesus (Hebrew)

Mean Ratio (μεσος λογος) = 888 = Jesus (Greek)

More than any other pattern in science, the Golden Ratio is embedded in the engineering and architecture of creation itself, known today as the physical universe. Everywhere you look, from the smallest to the largest scale you will find the Golden Ratio. The numbers, the math and the ratios that produce this universally ubiquitous pattern, is consistently encoded with the names and numbers of Jesus the Son of God. This is indeed the proof of God.

The skeptic may ask, where is the proof of God's existence?

To which all the universe will reply, quite simply: Everywhere.