Star Math(s)

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STAR MATH(S)

The Polygonal Puzzle in the Stars

Recently Published Math on this hypothesis: The Powers of Three (OEIS: A003462), Generalized Fibonacci Numbers (OEIS: A015441), 3^n - 2^n (OEIS: A001047), 7^n - 6^n (OEIS: A016169), Union of Cantor-Square and Koch-Snowflake Fractal (OEIS: A016153),

What if there was a shape that could represent all other shapes?

And what if this shape could be mathematically defined and demonstrated to be a 'Master Template' for all other shapes? And what if this shape was already known to the world as an internationally recognized icon and symbol? And what if this shape could be seen all throughout nature?

Could such a shape exist?

Could this shape have passed under the meticulous scrutiny of countless scientists and mathematicians for centuries? Could this shape have been smuggled under the noses of all the world through the long march of history and yet still somehow be present for thousands of years in the very first verse of the Bible?

This is not actually a what-if question.

This is a true story. This is the story of the Star Numbers, numbers that form the shape of a Star of David, numbers that can be found in a verifiable multitude in Genesis 1:1. These numbers have been studied by a small cadre of amateur mathematicians from around the world for decades, but only now are they really beginning to reveal their true secrets... .

This webpage has been time-stamped @ http://truetimestamp.org/ 11/04/2020

Public Key: 10568277ff25a5fe6df2791eabbc7df58057e9468eb8208c322a97fad9a099d7

One of the very early contentions in the short and modern history of biblical mathematics was that was something special about the star numbers. This theory can be summed up like so: The first verse of the Bible and many other significant names and identities directly related, appear to have a certain preoccupation with centered dodecagonal (12-gonal) polygonal numbers. The basis for this contention can be seen here and here and elsewhere and on other researcher's websites.

This contention, at first seemed scientifically baseless, but somehow theologically mandatory. It is undeniable that the primary constituents of the Bible (OT&NT) are found rooted in this pattern. There are too many examples to cite, but a large number of them can be seen here in the Map of the Three-Seven Code. The two that stand out and the two that convinced me (and should convince you) is the first verse of the Bible and the name Jesus Christ.

If it wasn't for these two, I wouldn't have ever even considered looking any deeper into the mathematics of the Bible. This is because those two pillars are theologically and logically necessary to even begin any sort of deeper analysis of a systemic pattern in the original text. In other words without Jesus Christ—2368 and Genesis 1:1—2701 there is no such thing as Biblical mathematics.

The scientific study of shape is called geometry. It is an ancient discipline. The scientific study of the shape of number, is called figurative geometry. Anything you can do in line geometry you can do in figurative geometry. It's just more rigorous and difficult, but can (unlike line geometry) be represented in the real world by real world objects.

The above image is my discovery of the relationship between the star numbers & the Koch Snowflake fractal & and the powers of 3 and 2, see OEIS A001047

POLYGONAL TESSELLATION (PART I)

All polygonal numbers (number collections that reproduce regular geometry) are classified in either centered or non-centered varieties. This is a fundamental feature of the requirements of space itself. Objects in space can either be aligned in straight rows or have an alternating alignment in alternating rows.

These polygonal numbers are very regular with very regular formulas. And there are an infinite number of them. So every number and all number has its own shape and all these shapes can be defined mathematically. And all these numbers, and all their shapes and their definitions (formulas) all revolve around a singular 'master' shape.

That master shape is the star number.

The Star Numbers (centered dodecagonals) are the master template for all other c. polygonal numbers. They do this in both a 2D & 3D geometry, as the Star Formation (hexagram/Star of David) which carries over into 3D prisms, and is perfectly symmetrical as a 3D rhombic dodecahedron bi-pyramid (polygonal diamond formation) and as the continuous summation of all star numbers (known as partial sums) where it is a series of alternating nesting cubes, (see animation below at end of section.)

In three dimensions Star Numbers show unilateral symmetry on a 12-faced figure.

A tapering Star Diamond can be reconfigured into a regular 3D Rhombic Dodecahedron

Click to ENLARGE

The 12-gonal star number is the master template for all centered polygons and the 8-gonal star number (octagonal) is the master template for non-centered polygons. Both of these stellated polygons are intimately knitted together in a variety of ways and tessellate like three dimensional puzzle pieces. For the non-centered polygons see the next section Polygonal Tessellations Part II.

This animation shows how centered polygons first create a centered hexagon at maximum compression and then continue to form regular hexagram-like formations forever after.

The illustration below demonstrates how all central polygonal numbers, beginning with the monogon (CPI) and onto the digon (CPII) and then into the more familiar territory of the centered triangle, all fill out the pattern of the hexagram. Polygonal numbers larger than centered 12-gonals simply add another triangle per polygon.

The first two categories may be unfamiliar, even to those familiar with figurative mathematics in general, but they are important number series in their own right.

What's really incredible here is that these shapes also have a figurative tessellation all their own. Let's just call this micro-modularity, where in addition to all the patterns of the star puzzles above, you can also add these micro-puzzles which are like puzzles within puzzles.

The two-dimensional frames of these configurations also perpetually formulate in a regular and rather 'spot-on' formula.

Polygonal numbers are nothing new and are basic to mathematics in general, so in other words largely explored. This means that their formulas have been well known and established for time in memorial. That being said, I could find no regular formula for the polygonal and centered polygonal diamonds. These three-dimensional objects are however very regular and have very regular formulas listed below in the two far right columns.

By looking at the polygonal numbers as a whole (family) including the oddball cousins that are seldom invited over for holidays, we find a remarkable consequence of the natural logic of the polygon: All polygonal numbers appear to come in two varieties, centered and non-centered. BUT the monogon and the digon are only of the centered variety.

This begs the question: What is a non-centered monogon and digon?

The answer to this question can be deduced mathematically from how the way all centered and non-centered polygons relate to one another. The difference between any and all centered and non-centered polygons of the same variety is always a square. Hence, triangles: 3, 6, 10, 15... . and centered triangles (4, 10, 19, 31... .): 3+1^2 = 4, 6+2^2 = 10, 10+3^2 = 19, 15+4^2 = 31, etc.. All polygonal numbers systems behave this way. So all we have to do is subtract a square from the centered monogon and digon to get the non-centered version and with a similar arithmetic can do the same for the diamond versions.

What this means is that the non-centered digonal polygons and polygonal diamonds are the natural numbers (1,2,3,4...) and the squares, respectively. What this also means is that the first non-centered polygon is a negative number, as is its polygonal diamond. Put that through the mathematical translator and you get: The first shape is a negative number non-entity. Or you could say, it is only the shadow of a number or the shadow of a thing, an imaginary construction impossible to create in the real world!

The real beauty of viewing all the polygonal numbers as a star configuration is in the way they tessellate (fit together). There are an infinite variety of an infinite number of iterations of these puzzles. Here is but a small sample:

Infinite Addition... .

Partial Sums (infinite addition) is the term mathematicians use to refer to the continuous summation of a series of formulated numbers. The best example of partial sums (and how important and ubiquitous they are in math) is the partial sums of the natural counting numbers, which is of course non-other than the triangular numbers.

1 + 2 = 3: T(2) triangular number 2

1 + 2 + 3 = 6: T(3)

1 + 2 + 3 + 4 = 10: T(4); etc.... .

Having seen that all centered polygonal numbers have a single unifying pattern, exemplified by the Star of David 'star numbers,' what then happens when we start adding all those shapes and stars and pseudo-stars and almost stars and extra-shiny stars together again and again and again forever?

Yes they will make star shaped towers, but the Logos, the guiding hand of nature, is exceptionally efficient and concise. Often these sorts of perpetual additions create a transformation into something new. So it is a real revelation that these same shapes fold up inside their own cubes to make a series of cubic nesting shells, where the shell is in checkerboard alternation.

Since the centered hexagons always add up into a cube, the n+6th polygon (Star Numbers) add up into a series of perfect nesting checkerboard cubes. All polygonal partial sums after 12-gonal wrap truncated tetrahedrons around their surface endlessly.

This is the finest example of just how very, very special the centered 12-gonal star numbers are, because of all centered polygonal numbers, of that entire infinity, only the star numbers fold perfectly into a perfect nesting cube series. If you stop and think about it, this is a level of three-dimensional organization undreamed of and almost beyond imagining.

All centered polygonal numbers when continuously added together fold up into nesting cube patterns. The vast majority of these numbers (in consideration of the infinity of number itself) are cubic numerical configurations larger and more numerous than all the atoms in our universe and they always fold up into themselves into a perfect checkerboard series of nesting cubes (and very often with additional truncated pyramids wrapping around them).

There is no better example of this manifest complexity made simple, than found in the sum of the first 37 stars numbers, which fold themselves into a nesting cube of positively Old Testament and New Testament proportions:

The red cubes number 50,616 a number pregnant with many mathematical connections to the numbers of the Bible.

Perhaps what is most impressive about this already impressive cube-complex is what is missing. The total space occupied by this formation is exactly 73 x 73 x 73. When we subtract 101,269 from 73^3 we get 287,748, which is exactly thirty-six Star-37's. This also means that all odd cubes can always be produced by a regular series of star numbers, i.e. the accommodating odd cube will have always exactly the last star numbers value times its iteration (1+13 = 14. 3^3 - 14 = 1 x 13; 1+13+37 = 51. 5^3 - 51 = 2 x 37; 1+13+37+73 = 124. 7^3 - 124 = 3 x 73, etc. etc. ... .)

101,269 + (36 x 7993) = 73^3

Octagonal 2D tessellation & 3D paratessellation

POLYGONAL TESSELLATION (PART II)

We have seen above, that the basic pattern of the star numbers (centered 12-gonals) will accommodate the full array of centered polygonal numbers by either one or the other or both of its most basic features (the triangle and the hexagon). These patterns are endless and they all tessellate in two dimensions in innumerable ways. But what about the non-centered polygonals, the 'normal' polygons?

Martin Gardner, the mind that found the stars of God...

One of the quirks of the mathematics of polygons is that the 8-gonal numbers (octagonal) were once referred to as the Star Numbers, but that title then passed on to the 12-gonal numbers after Martin Gardner's discovery of the pattern in 1973 conferred a greater star status to these 6-pointed stars. It should be noted then, that both the old and the new Star Numbers, are the two templates for all polygonal numbers (centered and non-centered).

This is no accident, really, what it demonstrates is the basic build up of polygonal numbers, where the centered polygonals form triangles around a centered hexagon and the non-centered polygons form around the square. It should be obvious by now that nature and number employ triangular numbers as a basic building block to form larger and larger configurations.

The octagonal numbers are marvelous creatures. This is due to their inherit 8-ness. And it should be remembered that one of the most memorable sums in biblical mathematics is that of the name Jesus in Greek, eight-hundred and eighty eight (888). Eight is the first nontrivial cube and so octagonal numbers are highly connected to the geometry of the cube and the cube is a purely mathematical creation, as it is made simply by multiplying a number by itself by itself.

In three dimensions octagonal numbers create nesting cubes, alternating series of cube-frame upon cube-frame. This geometry can also be seen in the two dimensional versions of the number where the perimeters of the 4-pointed star configuration represents the next iteration of the cube-frame.

These nesting cubes are what you might call 'Big-Hitters' in the realm of mathematics, because of how they house n!

The exclamation mark in mathematics has a very specific (and huge) function called factorials. It represents the continuous multiplication of the natural numbers. These are numbers that get very big, very fast.

n! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9... .

This feat of massive multiplication has a home, if you will, in the recesses of the cubic octagonal nesting frames. By dividing n! by (n-3)! we get six tetrahedral numbers, which alternate in parity as stepped pyramids. These are a perfect fit for the octagonal cubes. But it also allows us to write an equation for all cubes. And any sort of new equation for something as simple as a cube (n^3) is an equation worthy of note.

n^3 = n!/(n-3)!+n*(3*n-2), n>2 See (A007531) Nexus Cubes or All-Pyramid geometry has a connection to factorials, the multiplication of all integers symbolized by n!. Six times the tetrahedrals (which alternate parity with step pyramids) complete a cube with the nexus formation of the octagonal numbers.

What all this means is that octagonal numbers are really very cool and hyper-mathematical. But octagonal numbers have even more up their sleeves. Like the centered polygonal numbers which tessellate around the basic structure of a hexagram (Star of David) octagonal star numbers tessellate in 2D but in a different way.

Tessellation is not particularly rare. Lots of shapes tessellate, lots of shapes fit together perfectly.

What is rare, however, is when something tessellates (in a well defined grid space) and these two tessellating bodies are important number series in their own right. That requires a mathematical rarity in the formulas and perfect agreement in a two dimensional grid. It's even rarer if every iteration tessellates. And if every iteration not only tessellates but also para-tessellates in three dimensions that is incredibly rare. And if all these factors come together and this really rare 2D&3D tessellation revolves around the very templates of all polygonal bodies, then this kind of tessellation is truly unique.

By maximally compressing octagonal numbers in two dimensions we create 'pinwheel' stars. The pinwheel configuration will tessellate with a four-fold tile, which paratessellates in three dimension by endless iterations of alternating cube-frames.

Each 'Tile' (the functional unit of tessellation) has a five-fold connection to the 12-gonal star numbers.

The numbers in blue are 2nd pentagonal numbers (x4) which are 'helper' numbers, numbers that often compliment other numbers in configurations:

Star Cubes

Star numbers, (centered 12-gonal numbers) are best known for their ability to form a 2D hexagram Star of David. This has been written about extensively by the online biblical mathematics community. HOWEVER, star numbers have a greater degree of geometry in that of the cube. This cubicity (if you will) of 8 and 12-gonal numbers is due to the number of vertices and edges of a cube (8 and 12 respectively).

Below is just a sampling of the cubic configuration that star numbers contribute to.

In the 'Checker-box' configuration of Stars*2, the remainder of the cube always equal the even numbers times

the odd squares plus one: 5^3 - 74 = 2*5^2+1, 7^3 - 146 = 4*7^2+1, 9^3 - 242 = 6*9^2+1... .

Fundamental to the mathematics of the star is of course the hexagram (six pointed star) made possible because star numbers are actually centered dodecagonal numbers, 12-based numbers (6 x 2). This allows then the stars to accommodate all the squares, perhaps the most important unit in all of mathematics.

To avoid ambiguity this configuration of the star numbers should be called Diamondstars.

The diamondstar configuration is a infinite deconstruction of the cube, seen below as nesting cubes:

This arrangement also follows the division of hexagrams into 1st and 2nd octagonal numbers (see below), the top and lower halves respectively in another new cubic nesting cube configuration.

These geometries display the eminent flexibility of the star numbers, and the likely reason for their selection.

The entire star series can be average out in this fashion, where every three stars are the central polygonal numbers times: 6*(n^2 - n + 1) - 1 ; every four stars: 6*(n^2+1) - 1; every five stars: 12*(n(n+1)/2 + 1) - 1, the other central polygonal numbers or the Lazy Caterer's Sequence.

By following this dissection of the stars into perpetuity, i.e. partial sums or continued addition, we find that all cubes (both odd and even) can be deconstructed into a series of star numbers.

Because stars isolate the squares both in 2 and 3 dimensions, they can be said to have an important relationship to the summation of six-fold squares.

That's important.

Because six-fold squares are essential to the understanding of the geometry of the right triangle, the basis of trigonometry, the practice of surveying, and the world's oldest theorem, at around 3,000 years old, most likely known by the Babylonians, but known collectively and commonly as the Pythagorean theorem.

The order and arrangement of right triangles can be seen by the square number adjacent to the vertical side-length of the triangle.

The above graphic (an original discovery of 37x73.com) shows how (possibly) all right triangles (and Pythagorean triples) can ordered according to the diamondstar-configuration of the star numbers. The ordering is arranged however, by AREA and NOT side length, a fascinating new development in the age old study of right triangle geometry.

Because of this, all cubes can be decomposed into stars numbers and their related geometry. This also includes all even cubes because all even cubes are the product of odd cubes (1^3 x 8 = 2^2, 2^2 x 8 = 4^3, 3^3 x 8 = 6^3). Indeed, 99%+ of the infinity of cubes can be decomposed into real star numbers, with a very minor nominal set of the powers of two cubes, which are all composed of the number 1 reiterated. The number 8, here, is also related being the top half of 13 (8 + 5 = 13) the 1st & 2nd octagonal numbers that combine to produce star numbers, themselves.

This math is entirely reversible, where both the formulas convert from n^2 to n*(n-1) and the concentric star perimeter converts from even to odd.

And these patterns can be demonstrated in a variety of ways.

Stella Octangula are the 3D versions of Star Numbers and are Star(n) + (n+1)

The stars also have a preponderance of three dimensional configurations in the world of the prisms. This too will paratessellate according to the 2D tessellations seen above.

Click to ENLARGE

This connection that star numbers have to all the polygonal and centered polygonal numbers, continues into the prism numbers the 3D renditions of the polygonals. All polygonal numbers create prisms, (the cube is the perfect or ideal prism) but all other shapes are created in the same manner.

Like the way the 2D polygonals have a template that unites all of them, in that of the Star Numbers (old and new, 8-gonal and 12-gonal) the template of the prisms is that of the cube (naturally enough) plus the 'Cross Numbers' (or four times the 2nd pentagonal numbers, see next and below) referred to above as the non-trivial cube-frames. Every fourth polygonal prism is a cube plus partial sums of cross numbers.

Octagonal Prisms (8-gonal) 8 x 2 = 16, 21 x 3 = 63, 40 x 4 = 160...

8 + 8 = 16

27 + 8 + 28 = 63

64 + 8 + 28 + 60 = 160

Star Prisms (12-gonal) 12 x 2 = 24, 33 x 3 = 99, 64 x 4 = 256...

8 + (8 + 8) = 24

27 + (8 + 8) + (28 + 28) = 99

64 + (8 + 8) + (28 + 28) + (60 + 60) = 256

Off-Centered Polygonal Numbers A, B & C (Red, Purple, Blue)

POLYGONAL TESSELLATION (PART III)

There were two types of polygonal numbers: the normal non-centered variety and the centered kind, but it appears we now have a third, which I am calling the Off-Centered Polygonal Numbers. This new figurative number system comes in three flavors and each flavor is determined by the divisions of the centered hexagon and the 1st & 2nd pentagonal numbers which produce it.

Off-Centered A (OfA) = Multiples of 2nd Pentagonal Numbers

Off-Centered B (OfB) = Multiples of Centered Hexagonal Numbers

Off-Centered C (OfC) = Multiples of 1st Pentagonal Numbers

The arrangement of figurative geometry is based on the distribution of objects in space. Non-centered polygonal numbers can be thought of as straightly arranged squares. However, centered polygons employ an alternating distribution, where one row is staggered from another. This staggered alternation is found first and foremost in the triangular numbers, when we center them:

A Straight Aligned Triangle

OOO

A Centered Aligned Triangle

OOO

But there is a third kind of distribution that employs different ratios than either of these. This pattern is well established in the series called the centered triangular numbers.

BUT WAIT! Aren't triangles already centered?

They can be centrally aligned but do not necessarily always have a center unit, but these centered triangles always do. This pattern however is actually a new form of figurative distribution, i.e. a third type of figurative (polygonal) geometry. It employs a totally different math and logic than either straight or centered aligned figurative geometry does. You can see this in the pattern of its rows. Each new row of a normal triangle will always be greater than the former row. But this is not true in centered triangles, where subsequent rows vary and this variation is the key to off-centered polygonal numbers.

By applying this new pattern to the old we find that it fits perfectly overtop it AND we find an exact replica of the hexagram star patterning all throughout this new polygonal system. This of course means, that all the puzzles and infinite families of tessellations (seen above) can be repeated in yet another family of mathematical shapes and formulas.

It also means this:

These sums of subsequent squares are unique to the Off-Centered A,B & C Star Numbers (this determination is based off of a search through the OEIS database). The square function is probably the most important and common feature to mathematics (and often physics E=mc^2). However, this specific type of formulation may be of particular interest in the study of Pythagorean Primes which are always the sum of two squares.

By counting the dots row by row we find three different sets of sums. When a row fills out to the outermost perimeter it is always a centered triangular number. But the subsequent, following-two-rows will always be 1st and 2nd pentagonal numbers respectively.

The 'strangeness' of the math here, is due to the odd nature by which centered triangles add their rows. What this does, is create certain inequalities in the symmetry of the star, where the internal hexagon is of a different measure than its triangular rays. This also explains why there is both a difference in the formulas and the sums of the squares.

1st and 2nd Pentagonal numbers are highly important to mathematics itself (in combinatorics & number theory) and Biblical mathematics too, as they reside in the center of every Star Number. Compared to the triangular and square numbers, the pentagonal numbers seem like strange and unfriendly fellows. The angles of their geometry is all wrong and the way they add up numbers seems almost suspicious.

Because the pentagonals follow the two most important polygonals (triangles & squares) and are themselves followed by the similarly important hexagonals, it means that they fall in between the cracks. This inbetweener status is how the pentagonals act as a mathematical facilitator, connecting and complimenting other numbers systems. This graphic best displays this type of behavior:

It is exactly this strange connector behavior that informs and constructs the off-centered polygonal number system, starting with (OfA) the patterns produced by the 2nd pentagonal numbers.

Off-Centered Polygonal Numbers A. (OfA) above.

A careful bit of calculating reveals that the partial sums of ALL Off-Centered-A polygonal numbers produce centered polygonal Prisms. The Off-Centered-C polygonal numbers do something similar and create semi-centered-prisms, i.e. c.hex-prisms = 2x7, 3x19, 4x37, etc. semi--centered-prisms = 2x19, 3x37, 4x61, etc. And ALL (OfC) polygons do this as well.

The above is just one example. All (OfA) polygonal numbers repeat this pattern.

What this means, from a puzzle-perspective, is that every single one of the infinity of the polygonal puzzles (tessellations) mentioned in the first section, can be infinitely reiterated employing the full array of 3D prisms as pieces instead and each one of these puzzles can translated into a layered series of puzzles of the off-centered-variety (OfA).

That's a lot of puzzles...

The middle Off-Centered Polygonals B, however, can only be purely formulated with certain symmetries.

Off-Centered Polygonal Numbers B. (OfB)

Off-Centered Polygonal Numbers C. (OfC)

Strangely, of the three new off-centered polygonal categories the first, (OfA) produces an inordinate amount of mathematical connections across the board on the OEIS database, whereas the third (OfC) produces only one that I could find (A304163). This is strange because it means that the 2nd pentagonal numbers are more mathematically interconnected than the first!

Here again we see that the numbers can only coalesce around hexagons and their stellated cousins, the hexagram or six-pointed star.

To sum it all up, it means that the Star Numbers are the perfect or ideal pattern. The hexagonal pattern is perfect too, but not ideal because it does not include the stellating rays. The 12-gonal star numbers finish the pattern of the rays and all polygonal patterns afterwards are simply repetitions of that ideal pattern of the star.

Although many of these number series were already known to mathematics, it was not known that they were all connected or how they were connected. The first off-centered polygons connect to the five platonic solids in a newly discovered and yet unexplored way:

(OfA)

Off-Centered Polygonals A

Centered Triangular Numbers (A005448) = Natural Numbers + 2nd Pentagonals,

i.e. 1+0=1, 2+2=4, 3+7=10, 4+15=19, etc. & creates Tetrahedrons, i.e. a(n) = A000292(n) - A000292(n-3)= centered triangular numbers

Centered Square Numbers = Triangular Numbers + 2nd Pentagonals (A001844) & creates Octahedrons, and square pyramids, i.e. a(n) = A000330(n) - A000330(n-2)

2nd C. Pentagonal Numbers = 5*(2nd Pentagonals) + 1 (A093500) & creates Icosahedrons

2nd C. Hexagonal Numbers = 6*(2nd Pentagonals) + 1 (A082040)

2nd C. Heptagonal Numbers = 7*(2nd Pentagonals) + 1

2nd C. Octagonal Numbers = 8*(2nd Pentagonals) + 1

2nd C. Nonagonal Numbers = 9*(2nd Pentagonals) + 1 (A093485) & creates Dodecahedrons

2nd C. Decagonal Numbers = 10*(2nd Pentagonals) + 1

2nd C. Hendecagonal Numbers = 11*(2nd Pentagonals) + 1

2nd C. Dodecagonal Numbers = 12*(2nd Pentagonals) + 1 (A081272)

All five of the platonic solids are created in connection to the 2nd pentagonals, the first and foremost being the cubes, where 1st & 2nd pentagonals sum and become centered hexagonals which sum and become cubes.

The only reason I ever looked into this type of geometry was due to Bill Downey's discovery of Jesus Christ 925 creating a centered star. It was this discovery that lead to the idea that this particular figurative distribution could produce its own polygonal geometry.

The star on which the Jesus Christ—925 blue star rests is the 22nd centered dodecagonal number (2273). This is a configuration created by the joining of 12 triangles, each triangle being the 21st (231) which itself houses the 2nd pentagonal number 77 (Christ). All this is of some great importance to the Three-Seven Code in that 21 is 3 x 7 and 231 is 3 x 77 & 7 x 33.

The name Jesus in Greek is a 'Cross-Number', numbers that lock together in three dimensions and marry a cubic star number.

Cross and Star

The star numbers (c. dodecagonals) have been explored probably more than any other math connected to the Bible, largely, because of their prominence in the first verse and rightly so. Mathematically, star numbers represent twelveness, since what they really are, are centered twelve-shaped numbers, i.e. all star numbers are the triangular numbers times twelve plus one. There should be something terribly familiar to these numbers to anyone at all familiar with the Bible and the life and ministry of Jesus.

Twelve Disciples plus an eternal ONE.

This simple math is made infinite with the star numbers and transcendentally corresponds to the theological numbers of the Bible in that of the 12 tribes and 12 disciples. One might call them infinite echoes. One might also call them prophetic ripples since they start in Genesis 1:1 and ripple throughout the rest of Scripture.

But what about the cross?

If the star (a mathematical hexagram) represents twelveness and moreover Judaism, as can still be seen today in both the flag of the state of Israel and as the universal symbol of Judaism as a religion, then what about the cross?

In the same fashion, the cross is the universal symbol of Christianity recognized world round. We have of course a great number of mathematical crosses in the text. Naturally enough, the most basic operation in mathematics is addition and all others are related: Subtraction = Addition in reverse. Multiplication = Advanced Addition and Division = Multiplication reversed. So let us call it the positive operation and it is of course the positive operation is nothing other than the sign of the cross.

What makes pentagonal crosses so crossy, however, is how they are essentially mathematical and integrate seamlessly with the star numbers into a unified whole. Such a process is the perfect analogy of the progressing theology of the Bible.

Old Testament Covenantal Judaism = The Star (of David)

New Testament Christianity = The Cross

It is in the cubic configuration of the star numbers that we find the missing piece of the puzzle and we find it in the frames of all cubes. By saying we find it in all the cubes, what we are really saying is that we can find this pattern in all number. All of them. Because all of them, the whole infinity of number, create regular cubes whose odd frames marry infinitely with the star numbers in the most basic of Trinitarian math:

n x n x n = The CUBE

All odd cubes contain a hidden 3D puzzle that is astonishingly symbolic of the Old and New Testament...

Not long ago I discovered certain strange patterns in the mathematics of the biblical text that indicated that octagonal numbers were somehow important in ways I didn't then understand. The evidence found on this page goes a long way to rectifying my ignorance.

We find these octagonal number patterns in the first five words of the Bible (225), in the extended prime mathematics of the first verse (in the over 25 million references to the name and number of the Lord Jesus Christ (294)) and in the most important transcendental number in science (pi) in perfect connection to once again the name the Lord Jesus Christ (3168) and of course in the Lord's Prayer, the hidden riddle and pearl of the New Testament.

A quick glance at the first few iterations of this number show an intense connection to the name Jesus-888 and the full name and title the Lord Jesus Christ-3168.

What makes these numbers so much more cross-like than others is that their importance is found in 1/4 of the value or each arm of the cross. So it is not so much that they look like a cross (many number-patterns will) but that they are a cross from an important mathematical perspective, a thing far more rare and significant.

The next iteration in this sequence is 888, the sum of Jesus in Greek.

Each arm of each cross is a second pentagonal number, which may not sound significant but is indeed ridiculously significant in how all numbers are divided in the discipline of combinatorics.

Pentagonal Numbers are an important element in the mathematics that govern partitions in the field of combinatorics. The generalized pentagonal numbers (generalized polygonal numbers are numbers whose formulas employ both positive and negative numbers for the n-value) are fundamental to Euler's Pentagonal Number Theorem and the Euler Function.

The first and second pentagonal numbers which make up the series of the generalized pentagonals pair together to produce the centered hexagonal numbers, those found in the center of every star number.

Generalized Pentagonal Numbers: 2, 5, 7, 12, 15, 22, 26, 35, 40, 51...

2 + 5 = 7

7 + 12 = 19

15 + 22 = 37

26 + 35 = 61

40 + 51 = 91

This pairing process is not exclusive to the pentagonals, indeed it is universal to all polygonal numbers. In particular it unites the octagonal numbers with the dodecagonal numbers.

1st and 2nd octagonals also combine to produce the net of a cube or 6n^2 (six times the square numbers)

Generalized Octagonal Numbers

The two components of the hexagram (Star of David) that connect to biblical mathematics so strongly is the center(ed) hexagon and the surrounding rays which produce the star. This simple and elegant shape hides an enormous amount of complex connections throughout many of the bodies of mathematics. The all-important center (1st & 2nd pentagonals and the Euler Function) is the basic template for the greater template of the rest of the generalized polygonal numbers which is fulfilled in the completed stellation of the star numbers —the centered dodecagonals.

Both 1st and 2nd octagonals are differences in squares:

If we reverse the operation from subtraction to addition, we get the Octo-Numbers (5*n^2 - 6*n + 2) and (5*n^2 - 4*n + 1)

This 1st and 2nd octagonal split can also be envisioned as a rectangle and in so doing we find a new formula for the star numbers by subtracting the square numbers x2 from the odd-hexagonal numbers (of which T(37) and T(73) are both members).

The way in 1st and 2nd octagonal integrate with Star configuration split is mathematically important and diversified.

Where star numbers are figurate representations of a hexagram, octagonal numbers (1st & 2nd) are very similar with two opposed and interpenetrating triangles not yet fully equalized. These Hourglass Octagonal Configurations also serve as a template for other important number series.

The octagonal numbers are also perfectly modular with the centered octagonal numbers, aka the ODD SQUARES.

1st and 2nd octagonals also create a sixfold grid-variation pattern that can be seen on the hexagonal spiral of all number:

See above section Star Cubes for more on this integration. It also means that LCJ-294 is a hybrid of the 7th & 8th stars (1st&2nd Oct)

This relationship between 1st and 2nd polygonal numbers is universal for all polygons and follows the star template.

Generalizing the polygonal numbers (i.e. employing positive and negative values for n) produces an infinite array of centered polygonal offspring, whose pattern can be seen in the chart below. One could make a strong argument that the generalizing of a formula is what produces centered polygonals, via the moderating process of positive and negative values for n, which translate into centered geometries.

Pairing all the 1st and 2nd polygonal numbers together also produces centered polygonal offspring.

The progression here is the polygonal number minus two times two

So 8-gonals make centered 12-gonals, and 12-gonals make centered 20-gonals and these c. icosagonal are also starry:

These are the odd squares (centered octagonals) plus triangular numbers x12 and can be seen here in my addition to the OEIS: A069133

Perhaps the biggest trick of all for the generalized polygonal numbers is how they regulate the Multifactorials:

What we are seeing is how perfectly stellated two-dimensional bodies are perfectly enabled to do the same in higher dimensions. We can also see a profound connection then between the non-centered Star Numbers (octagonal numbers) and the Star-of-David-Star-Numbers (centered dodecagonal or C. 12-gonal). Adding cross-numbers to the mix creates a 2-dimensional and 3-dimensional life-cycle.

This 3D patterns can also be seen in the 2D version of the octagonal star. The opposite cross numbers (four times the first pentagonals) can be found likewise in all the even octagonal stars.

Both the 8-gonals and the 12-gonals are stellated polygonals, i.e. star configurated polygons. The 8-gonal 'stellate' off of the square and the 12-gonals stellate off of the hexagon. Both of these can be seen as twin triangles in the stages of merger (see animation below).

Where the odd octagonal numbers (see above) highlight the star and the cross, the even octagonal numbers highlight in the same way the first pentagonal numbers. Indeed, the even octagonals are the pentagonals numbers x8.

The two merging triangular configurations tessellates with centered hexagonal numbers,

each pair then becomes all the odd ordered second pentagonal numbers (A033568)

Octagonal numbers come in a variety of forms. The classic (for which they are named) are a series of concentric octagons. The next (for which they used to be named) is a four pointed star configuration.

Two other important configurations are either new or relatively unknown. The cubic configuration, is well explored on 37x73.com and corresponds to some of the most important passages and names in the Bible, including the Lord's Prayer and the 3168 decimals of pi, to name but two. The latest configuration (the third) represent two triangles beginning to merge, which makes these 'old' stars of the same category of the 'new' stars which are two triangles at equilibrium.

To see how this 'merger of triangles' connects to the 3168 decimals of pi and the most quoted passage in all the Bible check out pi.

Plato's Star of David?!

Figurative geometry, i.e. counting dots, is at the same time, strangely enough, one of the oldest of sciences and one of the least explored. It was of course the Greeks (Pythagoras, Plato and the rest of the gang) who established the science and importance of figurative geometry as an aid to understanding the mathematics that governed the universe. However, since then, figurative geometry has fallen on hard times and is now something of a mathematical backwater.

Nevertheless, it is science, real science, it is true and it may be terribly important.

Not long ago I began to realize that figurative geometry, after all this time, was still relatively unexplored. A savvy student can, if they so desire, find dozens of new connections between figurative bodies on a daily basis, for who knows how long. All that being said, it is still something of a shock to find unknown connections related to the Platonic Solids, the five most majestic geometric bodies, that are the crown jewels of geometry itself and among the first such bodies to ever be discovered.

If figurative geometry is a backwater of mathematics, then figurative frame geometry is the absolute hinterlands in the middle of nowhere. It should noted however, that is in such places that science often hides its most important secrets. This being said, there really should be no connection between star numbers and the Platonic Solids. Yet, looking closely we find a strong connection to every one of all five of the shapes. Here however, we must remind ourselves of a very important fact: yes, figurative geometry is not cool and yes figurative frame geometry is almost unheard of, BUT they are both just as true and just as scientific a concept as all other geometry, or the power of pi or even the Riemann Zeta Hypothesis.

Scientific Truth is scientific truth.

And figurative frame geometry is just as important as line geometry only it is more rigorous and more exacting. By examining the frame geometry of the hexagram (Star of David) we find it is highly integrated with all five of the Platonic frameworks.

New and true science:

The first two platonic frames doubly connect to the hexagram. The same n-valued tetrahedron and cube frames produce a hexagram while a triangle of tetrahedrons does the same.

To a scoffer, this may look like a lark, but the following cannot be so considered, for it is fundamental to the very construction of the hexagram:

So far that makes a tetrahedron, a cube and an octahedron, 3 out of 5.

The next platonic frame is then the dodecahedron, or 12 faced solid body. Here the hexagram frames helps out the cube frame in order to build the dodecahedron frame, which is the master frame of all five of the platonic bodies.

Finally we move onto the icosahedon (20-sided body) which has at least two strong connections to the stars. The hexagram (Star of David) is one of the very few 2D geometries that has both a perimeter and a distinct proper frame. In most cases the two are one and the same. It is this difference and distinction that produces yet another profound connection to the Platonic Forms.

The second connection is a triple concoction of starry wonder. By taking the star numbers proper and subtracting a n - 1 frame and then subtracting an n - 4 star one finds the framework of the last and perhaps most beautiful of the platonic solids.

Cave-Man Calculators: The Calculi Codices

To the mind of a historian, one of the most interesting things about biblical mathematics may be how it is largely informed by figurative geometry. This, it will be remembered, was a creation of the first Greek mathematicians, whose pebbles (in Greek calculi, which is where we get the word calculus) were the first prototypes of a calculating machine. In the absence of computers, handheld calculators or the ever handy abacus, these little patterns of stones helped the Greeks show the relationships between number, something we now call mathematics.

These 'rock-calculators' were the very beginnings of the disciplines of science and mathematics, which now hold court over all the world. So it is a great irony that the wise men of ancient Greece barely tapped the surface of the calculating potential of these rock calculators. Only 2,500 years later are we beginning to understand the infinite depth of possibility that these humble patterns made of pebbles can produce. The amount of information packed into these simple geometric formations is something akin to an ancient alien technology, whereby arranging rocks in a certain way one can demonstrate the entire array of polygonal numbers.

Somebody tell the producers of Ancient Aliens stat!

Imagine, if you will, the great Greek minds of history, Plato, Pythagoras and the like, playing with pebbles in the dirt not knowing the incredible calculating potential that they had in their very hands. A simple example of this function can be found in the frames of triangles, which when combined, create centered hexagons whose frames then create centered 12-gonal numbers (star numbers) whose frames then combine on and on forever.

The opposite alternation (the even iterations of any polygonal series) will always produce an appropriate multiple of the squares: triangles x3, hexagons x6 and hexagrams, which are actually dodecagrams (12-geonumbers), and render x12 the square numbers.

But this is just the beginning of the rock-calculators hidden functions. Here we will use square numbers as an example. Square numbers and centered square numbers both share the same perimeter as seen below:

By comparing the square perimeters to their index numbers we find a coordination that runs the gamut of all centered polygonal numbers. But this is by no means a feature unique to squares. All figurative bodies are encoded in this perimeter to index relationship.

But the best way to understand this is backwards!

Looking at the centered square numbers, listed below, you can see that by subtracting from each number you get a square perimeter number. By cross-referencing that number with its index number (the top row) that index number reveals the monogon or central polygonal numbers. The next is digon (2-gonal) followed by the centered triangle and so on.

In other words the index numbers are the 'Equals' window.

Take the centered square numbers (1,5,13,25...) multiply (x1 - 1) and you will always get a square perimeter number; using (x2 - 2), (x3 - 3), (x4 - 4) will produce another series of square perimeter numbers. By comparing these numbers against their original square perimeter index value, you will find that those index values will produce the entire array of all centered polygonal numbers.

And here's the catch: ALL POLYGONAL NUMBERS WILL DO THIS!

That's a 'Caveman Calculator'.

What's really interesting, however, is when we try this trick with the 'super-number series' the Star Numbers. As can be seen on this page and almost everywhere else on this website, Star Numbers do things that no other number series can. They connect to other number series like no other number series can. So it is no surprise then that Star Numbers, once again, have a special function that stands out from the rest of the polygonal numbers.

What's really surprising is that Star Numbers do this with three of the three-dimensional frames of three of the five platonic solids. A simple example of this can be found in the matrix between Stars times n - m, which index with tetrahedron frames which connect to multiples of the triangular numbers.

Take any star number, multiply it times one and subtract nine: 1 x 13 - 9, 1 x 37 - 9, 1 x 73 - 9 = 4, 28 , 64. All of which are tetrahedron frame numbers. Take the index value of those tetrahedron frames and you will find two-times a triangle. Do it again but now star numbers times two minus ten and you get four times the triangles. And on and on it goes forever.

By playing around with the multiplier and the subtracted value, you will find these tetrahedron frames hold the pattern to all even central polygonal numbers.

If you muck around with the numbers again you will find another entire array of number series:

And of course the Star Numbers do this with tetrahedrons, hexahedrons (cubes) and octahedron frames. How many different way it does this, is unknown. But it does do this infinitely in probably an infinite number of different ways.

So what value do these caveman calculators have?

We have yet to find out!

What I mean by that, is that this is an entirely new form of scientific calculation. The basic polygonal perimeter calculator produces (as seen above with the square perimeters) Calculi Codex I, which is the infinite array of all central polygonal numbers. Calculi Codex II is the infinitely infinite series of arrays that the Star-Rock Calculator creates. Calculi Codex II is almost entirely unexplored, virgin mathematical territory. I played around with it and found the coefficients will still connect to known numbers series listed on the OEIS well up into the hundreds. After that the numbers become to large for most of the entries at the OEIS database.

So here is the thing though: Calculi Codex II or the Star-Rock Calculator will produce positive connections to series after series of well known and established mathematical formulas. But just because the Star-Rock Calculator blows out the OEIS database doesn't mean it's not connecting to important mathematics. It just means that these are unknown and yet to be discovered. And it does this infinitely, so it very well may be the key to an infinite number of mathematical formulas new to science.

In other words, the caveman calculator, the Star-Rock Calculator and the Calculi Codices could be sitting on an entire world of new and undiscovered mathematics. Literally one could use the most ancient calculative technology, the original information technology, a bunch of stupid rocks arrange this way and that in the dirt, to find the latest most modern cutting edge mathematics. Talk about ancient alien technology!

And all that is just from Calculi Codex I & II, we don't know if there is a Calculi Codex III.