After looking more closely at this geometry, we will then examine how it correlates with at least three great secrets that include:
- The miraculous history of Israel;
- The qetoret or incense; and
- The 42-letter (abbreviation of the) 'Name' and its place in the tetrahedral scheme of things within the luchot.
All of of these mysteries are connected to, and ultimately flow from the geometry associated with the 216 letters of the One Explicit Name. Having said that, be advised that the substance of this post is a bit steep and it would not be advisable to read the material or explore the geometry, without first having read the essential constructs listed in the column to the right (at a bare minimum) and the posts on Emor and Metzora as well; and especially the page on the Primordial Torah. The substance of this post might otherwise be to complex. It is nevertheless extremely profound, and worth taking the time to wade through the numbers, and to read and contemplate the implications.
Those who have met the prerequisites mentioned, should have no problem in understanding the subject matter. The first model (below) has already been presented in applicable posts. A quick review is nevertheless appropriate. The numbers are derived from the elemental properties of two opposing (three-dimensional) tetrahedrons, and the two opposing (two-dimensional) triangular halves of the magen david. Please note that each three-dimensional tetrahedron has 14 elements and each of its two-dimensional counterparts has 7 elements. The three-dimensional metric geometry of the tetrahedron can thus be expressed (one-dimensionally) in terms of its male components, opposite the number of its female components, or simply 14 opposite 14. Similarly, it's two-dimensional metric geometry can be expressed in terms of its male~female halves as 7 opposite 7. For now, our focus will be on the actual three-dimensional construct within the cube [of creation] rather than its well-known two-dimensional shadow (the magen david).
7 Opposite 7
14 Opposite 14
(this mathematic pattern is
hidden in nearly every parashah)
These two patterns (7 opposite 7 and 14 opposite 14) are found in the linguistic structure of the Torah as well as the character of various events throughout Israel's remarkable history, and they ultimately define certain fundamental elements of just about everything else in the physical universe, including certain aspects of quantum mechanics, the physics of time and space, celestial dynamics, human physiology, and even cycles of time, although specifically those pertaining to Israel. They also account for the fascinating linguistic structure in certain parts of the Torah, like the examples advanced in the post on Miketz. We would only add that these patterns are just some of the numbers and sets of numbers examined throughout this blog.
Keep in mind as you read this post, that this is the single most unique construct in the geometry of the luchot, and were these tablets not the template by which all things are measured, it would not be so important. It is unique because it is the only construct (among five) that is its own polar opposite. That is, it is distinguished from its opposite by virtue of its orientation (to its opposite) within the three dimensions of the cube. The remaining four constructs have different polar opposites, making their orientation (for purpose of distinguishing them from their counterparts) essentially irrelevant. These four remaining constructs and the part they play in the cosmic scheme of things are beyond the scope of this particular post. However, in the near future we hope to post an explanation on how these constructs give rise to the five string theories reconciled by dual resonance.
If you’ve read the essential constructs listed in the column to the right (and they should be read in sequence) then you are at least partially aware of the role that the geometry and the measure of the luchot play in the creation process and the physics of the universe. With that in mind, let’s examine the nature of this primary construct, its edges, faces and points, its internal metrics, and all of the elements or components that ultimately create the star tetrahedron. Since the human mind is not wired for complex three-dimensional geometry, this post will offer a comic-book explanation complete with pictures for those who are otherwise perceptually challenged.
When the geometry of the male tetrahedron is merged (or married with) the geometry of its female counterpart: the male ‘enters’ the female—the geometry of the opposing half is divided—and the number of original elements ‘multiply’ ...the end result being that the original number of elements (the 14 shown in the above model) increases to 70 (which will be shown in a subsequent model in the paragraphs ahead). The merging of the geometry is very much like human physiology when pregnancy occurs.
To put it another way, once this “marriage” or unification takes place, the original tetrahedrons are subdivided by each others geometry. The subdivisions result in additional geometry. We've modeled this below with a transparent artificially-exploded view of the star tetrahedron, so that the resulting subdivisions are easier to see, and the individual geometry of these subdivisions more easily discerned. Just keep in mind that a star tetrahedron does not really "explode" during the process of unifying two tetrahedrons. Rather, the individual geometric segments remain attached to one another. It is shown here 'exploded,' merely for the purpose of visual comprehension.
For the sake of distinguishing the internal subdivided geometry of one tetrahedron from that of its polar opposite, we'll now pull the male out of the female, so that their individual components can be perceived more clearly. The two opposing tetrahedrons (now separated) are shown below.
How many elements does a subdivided tetrahedron have? A tetrahedron by itself has 4 faces and 4 points, but as you can see from the above model, the division results in 4 tetrahedral 'stellations' surrounding an octahedral core. These 4 stellations are identical to the parent except for the fact that they are half the size. The 4 tetrahedral stellations have a total of 16 faces and 16 points. The octahedral core has 8 faces and 6 points. The lines of division are one-dimensional and should not be counted twice (you don't need two knives to slice a cake in half3). The number of lines is therefore 4 x 6 or 24, and together with the 22 points and 24 faces, the geometry (now) consists of 70 components or elements instead of just 14 (as demonstrated in the first few paragraphs). Since there are two tetrahedrons, the three-dimensional geometry of the resulting "marriage" can be expressed numerically as the ’sum’ of its 70 (elements) opposite the number of elements in its counterpart, depending on whether you consider it divided, or undivided. A transparent model of just one of these tetrahedrons (subdivided but not 'exploded') is shown below. Notice the lines of division that separate its various components from one another. Note also the orthographic projection of a two-dimensional magen david in the center of the model, hidden within the three-dimensionality of the octahedral core, a geometric reflection of both 'parents' (both the male and female tetrahedrons) or, to put it another way, a sort of geometric DNA.
You will see in a moment how the marriage or unification of this geometry (within the combined cubic form of the luchot) correlates with the written torah, but before we get into that, lets build a slightly more comprehensive foundation. The next picture (below) shows another “exploded view” of a star tetrahedron, except that the stellations of the female tetrahedron are depicted as being separate and apart from their male counterpart (the opposing tetrahedron that was “responsible” for its subdivision) and how the male counterpart relates to the female half of the equation.
There are several things you will notice about the above models, things that you can use to better understand the geometry:
- The stellations of the female tetrahedron are like 4 ‘wives’ surrounding the male tetrahedron.
- The male tetrahedron is like the geometric ‘husband’ to its female counterparts.
- When this “husband" ‘entered’ the female tetrahedron, it 'caused' the division of its structure and the subsequent multiplication of its elements.
- The grey triangular ‘bases’ (on the surface of the tetrahedron) are where the female stellations ‘mate’ with the male tetrahedron. The other triangular attachment points (or triangular bases) cannot be seen in this particular solid model because they are on the other three sides of the male tetrahedron. However, the other “wives” all attach to, or ‘mate’ with their husband in like manner, at their designated place and position.
- In this model we show the male tetrahedron remaining geometrically intact, and only the geometry of its surface is ‘disturbed’ (altered) by the marriage and subsequent attachment to its 'wives' (the stellations mentioned above). However, note that there are 4 quadrants on each of its 4 faces (16 in all) that result from its marriage to the female tetrahedron.
- The picture (above right) shows the male tetrahedron by itself, divorced from its 4 'wives’ (absent its 4 stellations). We’ve pictured it this way for two reasons. You can more easily see the full triangular base on one side (in grey) where one of the wives was attached to the husband (where the stellation attached to the male tetrahedron). However we’ve also done this to show that the male tetrahedron itself (also) has stellations (its 4 blue tetrahedral corners). These 4 blue corners (themselves stellations) are also tetrahedrons that are half the size of the parent, but in this case they are the geometric 'sons' of their 'father' (the male tetrahedron) whose 'birth' resulted from the marriage to the female tetrahedron. The "sons" of the male tetrahedron are pictured below in an 'exploded view," although once again, this 'explosion' does not really ocur in the actual geometry. It is depicted this way merely to clarify the geometry.
There are several things in this second model that will help further your understanding of the geometry:
- The model on the left (above) shows the 4 'sons' (4 stellations) as a part of their father (the male tetrahedron).
- The model on the right (above) merely shows them separate and apart from their father, so that you can more easily see the individual geometry of each internal component.
- Notice (again) that the grey triangle in the middle of the surface (on any given face of the male tetrahedron) is the 'place' where a female stellation connects with its “husband,” to cause this geometric effect (giving birth to these 4 'sons').
Of course the geometry does not end with just these 4 children, because even the children have descendants. The model below shows how one of the stellations (as any given child of a parent) follows in the footsteps of that parent, having children of its own. The construct within the cube [of creation] thus goes from father to son (tetrahedron to stellation). Then the son (which is itself a tetrahedron) becomes a father to its geometric components, which again go from father to son and so on. Thus the sons possess the same geometric traits as their father and their grandfather. The model below shows how one stellation (or son) has children just like the father.
Ultimately, the father to son and/or mother to daughter iterations continue to infinity (the subject of fractal geometry). The geometric origin however, is in this case the star tetrahedron, and we see the first three iterations in the above model: the father, son, grandchildren etc.) The complex transparent model below will give you some idea of how the character traits (geometric properties) of the parents are passed to the children and even grandchildren.
This brings us to the additional “descendants” (of this geometry) and some of their specific properties. If you count the faces on all 4 of the wives (the female half of the geometric equation) you'll find a total of 12 small faces. These of course are a result of the marriage between the male and female tetrahedrons. You'll see a detailed model of these faces in the paragraphs ahead, along with a more comprehensive explanation of their significance. For now, it is sufficient to know that there are in fact 12 small faces on these 4 stellations (the wives) and that the total number of elements (children) coming from the female (once it is subdivided by the male) is 70.
Before we begin looking even deeper into the dual tetrahedral matrix (star tetrahedron) within the shnei luchot, please review once again the model of a subdivided tetrahedron.
Note again, that the 12 ‘children’ (visible faces) of a tetrahedron, can also be subdivided and that the number of elements in their geometry can also multiply. However, in each case, the resulting geometry mirrors that of its parents, the original mother and father (since they are all tetrahedrons, regardless of size) and remember that if you count the total number of elements in a subdivided tetrahedron (as we explained in the first few paragraphs at the beginning of this post) you’ll find there are a total of 70.
These family members (speaking of the geometric family) are defined by the triangular corners on the surfaces of both the mother and the father (male and female tetrahedrons married together) which happens to be 72. The set of numbers and the ratio of course reflect certain aspects of the real father (the Architect of the cosmic blueprint) behind the scenes, whose One Unique Explicit Name just happens to consist of 72 triplets (216 letters).
The geometric parent (tetrahedron) has even more specific attributes that correspond to the Name of the real father. For example, the grey octahedral core that is hidden in the middle of the star, has 8 triangular faces (or surfaces). These 'faces' are specifically oriented in male~female fashion, with 4 faces on one half opposite the 4 faces on the other half (hidden within the 216 corners of the outer geometry). This geometry is the geometry at the heart of the ‘marriage’ between the two tetrahedrons. The octahedral core is thus at the center of the blueprint, which is an actual reflection of the Name that is at the center of creation—a Name abbreviated by the 4 letters of the Tetragrammaton, specifically opposite the 4 letters of the opposing abbreviation, the Havaya Adnoot, as seen on the shviti in many synagogues. The two abbreviations together have a total of 8 letters (that are 4 opposite 4) as explained in the post on the marriage of the letters like the 8 faces of the octahedron, (that are 4 opposite 4) at the heart of the cosmic blueprint. When the letters that define this geometry are unified, they become one in the Havaya of Ekeyeh. The same is true with the geometry when it is flattened from three dimensions, down into two dimensions. The 8 corners of a cube in three-dimensions (that are 4 opposite 4) become the 4 corners of a square in a two-dimensional reference frame. The blueprint (luchot) is the geometric mirror image of its Creator's Name and all of its abbreviations, along with all of their various attributes1.
The only thing we would add at this point, to bring all of this into focus for the reader, is that the total number of degrees in the octahedral core is 1,440 and the total number degrees in the surrounding stellations is 2,880. The combination of these internal and external measurements (those that are hidden/nistar within the star, and those that are revealed/nigleh on the outside of the star) is 4,320 degrees (1,440 degrees + 2880 degrees = 4,320 degrees). These 'measures' are a function of the Architect's Signature, whose Name consists of the 216 letters (that define the measure of the luchot in cubic handbreadths) times 2, which is the number of times the luchot were brought down by Moshe (2 x 216 = 432) of which one set was hidden/nistar (withheld from Israel when shattered) while the other set was revealed/nigleh (when received) times the 10 mathematic constructs and polar opposites that are integral to the nature of the cube [of creation] that give rise to the 10 sephirot (that are 5 opposite 5 as referenced in sefer yetzirah). Everything in the blueprint is intricately connected with the physical universe that it defines. Take time to contemplate this.
Needless to say, the terms we have used in preceding paragraphs, like parent, child, husband, father, wife, pregnancy, sons, birth, descendants (etc) are merely anthropomorphic representations of the geometry we've been examining. However, in another sense, they are much more than that. If we assume that the nature of the cosmic blueprint (shnei luchot) has a role to play in establishing our reality in the physical universe, then we should expect this geometry to manifest itself elsewhere, in the form of some key event, or maybe a series of multiple events, and especially those pertaining to one nation in particular (Israel) whose ‘child’ is God’s first born ‘son’ (Exodus 4:22). With that in mind we have a story to tell.
Once upon a time, there was a man who wished to follow in the footsteps of his father (to do things, in essence, after the manner of his heavenly father). For the most part, this ‘heavenly father’ remained hidden from the man's peers who weren't concerned with such things. The man went a very long time before he acquired a wife. However, his heavenly father had prearranged this man's marriage, and he eventually ended up with a total of 4 wives. His first wife gave him 4 sons, during which time the other wives had no children at all. Then, when the first wife stopped conceiving, the other wives began bearing children, until the man had a total of 12 sons. Of course these sons had descendants as well, and eventually the number of people in his family multiplied, until one very special moment in time, when they numbered exactly 70 in all, once you included all the children and grandchildren.
The numbers should sound familiar and you should already know who these people were in real life. Who was the man? What were the names of his 4 wives? Which wife gave him 4 sons before any of the others conceived? Who were these first 4 sons? How many sons did this man eventually have? 12? And who were the 70? What you did not know was the connection between the history of this man and the geometry of the star tetrahedron within the cube [of creation] or shnei luchot. The father was of course Yaacov whose name was changed to Israel. He just happened to be ‘husband’ to 4 wives. His first wife (Leah) just happened to give Yaacov 4 sons before any of the other wives conceived. These 4 sons were Reuven, Shimon, Levi and Yehuda, and of course ultimately, Yaacov just happened to have 12 sons; but by the time they had children and grandchildren (descendants) and reached the border (edge) of Egypt, there were 70 in all. The question(s) you have to ask yourself are:
- Is the correlation between the geometry and the history of this man and his family a coincidence?; or
- Is ‘the story’ (in the Torah) somehow just a fabrication written by a mathematically knowledgeable author who had nothing better to do, than to figure out how to weave a tale based on complex three-dimensional geometry, at a time (over three thousand years ago) when computer modeling was unavailable?; or
- Are these actual historic events a reflection of a larger reality?
First, you may want to re-read the post on the primordial torah that shows just how precisely the celestial dynamics of the sun, moon, earth and stars; the rest of the physical universe; and even the history of mankind, mirrors the mathematic parameters that are specified in the cosmic blueprint (shnei luchot) because if the physical universe does indeed possess the characteristics specified, then the history of mankind, as revealed in these events, can be no less a formal “specification,” within the same blueprint2.
Second, if these events are a specification in some sort of 'cosmic blueprint' (more specifically the luchot) we would expect to find additional witness to that effect, and certainly in Jewish sources, advanced by Torah scholars. We do. Consider for example, the first reading in Parashah Yithro, where Moshe tells his father-law the story of the exodus from Egypt. The verse is generally translated as: “Moshe told his father-law all that God had done to Pharaoh and Egypt for the sake of Israel, as well as the frustrations they had encountered on the way, and how God had rescued them.” The words that refer to the telling of this story are “V’Yi SiPuR Moshe” (‘SiPuR’ referring to the “story”). Rabbi Morgenstern of Slovenka points out that the root (letters) of the word ‘story’ (SiPuR) are the same as the root for the word ‘sapphire’ (SaPphiRe) alluding to the sapphire tablets that were received along the way. Because of the double meaning, the sentence could be rendered as: “...in the Sapphire [were the events of the exodus that] "Moses told Yithro," which would allude as much to the nature of the sapphire as to the receiving of the tablets themselves. This would certainly seem to be the case, as demonstrated by the tetrahedral ratios described by our sages explained in such posts as Parashah BeShalach. All of this has profound implications.
Ultimately, an understanding of the geometry associated with the letters of God's Explicit Name, and how they define the luchot and its internal properties, tends to shine a new light on the stories which comprise the literal history of Israel found in the Torah. This 'light,' or knowledge and understanding of "the story" (in terms of this geometry) reveals clearly that the letters of Torah describing various events are an actual function of the 216 letters that define the measure and the nature of the SaPphiRe blueprint (216 cubic handbreadths ~ 216 degrees and various multiples etc). Take time to contemplate this larger reality, and you can begin to see why these particular letters in the text quoted above, would indicate that the events in question (pertaining to the exodus) were in the sapphire. The history of Yaacov and family, up to, and including, their descent into Egypt, as previously described, is just one more example.
With this in mind, we will now look even deeper into the geometry of the Primordial Torah, and specifically those parts that define the geometry of the star tetrahedron. You may remember that the male tetrahedron was shown geometrically intact, and only its surface was ‘disturbed’ (or appearance altered) by the marriage and subsequent ‘pregnancy’ of the wife. How was the surface ‘disturbed’? Look at the picture of the 4 stellations on the male tetrahedron again, in the model below, and you will note that each of its 4 (original) faces is now divided into 4 quadrants on each of the 4 stellations. It thus has a total of 16 quadrants on each of its 4 stellations (each being just like the father).
The next few paragraphs might appear mathematically daunting at first, but once you finally grasp the implications that we'll be discussing in a few moments, the significance of this as it pertains to the letters and resulting narrative of the written Torah will become apparent.
To lay the foundation, we need to zoom-in on one of the original sides of the female tetrahedron (modeled below in light blue). We see it as three light blue triangles (in the foreground) because the original side was 'subdivided' by the penetration of the 'alpha stellation' of the male tetrahedron (shown in dark blue). There are of course 12 small faces on each tetrahedron, with 3 on any given side of the original. These faces (resulting from the 'penetration') have a total of 9 corners.
We should also point out that while there are 12 small triangular faces on the female tetrahedron, we can only see 3 of these triangular faces (depicted in light blue) because the remaining 9 triangular faces are hidden behind the opposing (male) tetrahedron. The description of the opposing tetrahedron is identical. While there are 12 small faces on the male tetrahedron, 3 of these faces are hidden, and so we see only the 9 that are visible (depicted in dark blue). From this place and position our perspective is incomplete because of the limited nature of three-dimensional space.
Notice that the same set of numbers (12, 3, and 9) describe the marriage of the tetrahedrons in different ways. So the set of numbers is actually rather unique. For example, from the same place and position (in three-dimensional space) we can in fact see 12 small faces at the same time. It's just that we can only see 3 faces of one tetrahedron and 9 faces of the opposing tetrahedron. Or, we could say the 12 faces each have 3 corners that together have a total of 9 corners. In each case, the same set of numbers invariably describes the tetrahedrons from a single point of view that is limited at any one time by the observer's one position in three-dimensional space.
3 in the Foreground
9 in the Background
Study the model above and you will see these numbers, sets of numbers and applicable ratios apply to every aspect of the star tetrahedron, and specifically note that you cannot see all 12 small faces of any one tetrahedron at the same time because a certain number of its faces are always hidden behind the stellations of the opposing tetrahedron, while others are revealed. That is the nature of our reality. From our 'place' (within time and space) we see only the 3 on one face (in the foreground) and the 9 on the other faces (in the background). In each case, the ratio is the same. The two models below show these 12 small faces. The one on the left emphasizes the 3 in the foreground (on the female) and the one to the right emphasizes the 9 in the background (on the male).
12 - 3 = 9
Up until now, our focus has been on the geometry of the two opposing tetrahedrons and the set of numbers that define their characteristics from one point in time and space. The set of numbers 12, 3, and 9, along with their respective ratio, again, result from the marriage of the two.
Now we'll shift our attention to the stellations of those tetrahedrons which differ from the parents in two specific ways: 1) a stellation has only three sides because the fourth is hidden between the stellation and the octahedral core; and 2) it has no stellations at all, merely an extra quadrant in the middle where the stellation would be, were it mated to a polar opposite (which it is not). The thing you should notice about each stellation is that, with this additional quadrant, it still has 12 faces, only the faces are now very small, and half the size of those that were half the size of the parent. These very small faces populate the 3 sides of any given stellation, which together have 9 corners. So it still has something in common with its parents, and even though the geometry has changed, the same set of numbers still apply.
However, the geometry has nevertheless changed, ever-so-slightly, because the middle quadrant is different (it lacks a stellation) and so this modification could be described more accurately by saying that there are now a total of only 9 triangular corners on each stellation found on its 3 visible triangular sides, which of course each have 3 triangular corners. You can see these geometric properties and this specific set of numbers in the blue triangular corners of the stellation that is emphasized in the model below. However, notice something else.
Notice that the difference between the stellation and its parent, is the absence of an actual stellation, and that this absence 'creates' an extra quadrant in the middle of each side. This quadrant is one fourth, or one quarter, of the total area (the word 'quadrant' actually comes from the word 'quarter' even though it is often used to describe any fractional portion of a given area). The quadrant on one side of one stellation is emphasized in green on the model below.
This new set of numbers (9, 3, and 3) and the quadrant they create (that is one quarter of the face of any given stellation) is also highly significant as you will see in a moment.
However, before getting into that, lets summarize these unique sets of numbers. We have the 4 stellations on the female tetrahedron that when subdivided by an opposing tetrahedron can each be described in terms of 70 elements. We also have the 4 stellations on the male tetrahedron that when subdivided by its polar opposite can each be described in terms of 16 quadrants. The unification of these tetrahedrons results in 12 small faces on each tetrahedron, with 3 faces on each original side, that have a total of 9 corners. Finally, the unification creates stellations that are best described as having 9 corners on their 3 visible sides with 3 specific corners. Although, the subdivision of the stellations does create a quadrant in the middle of any side, since it lacks a stellation. Do those sets of numbers by any chance sound familiar? They should.
"Eleven kinds of spices were in it: Stacte, Onchya, Galbanum, Frankincense (4 spices) each weighing 70 maneh; Myrrh, Cassia, Spikenard, Saffron (another 4 spices) each weighing 16 maneh; Costus 12 maneh; aromatic bark 3 maneh; Cinnamon 9 maneh. All of these measures (pertaining to the spices only) constitute one “set” of measurements (all are measured in terms of a unit known as a maneh) whereas all subsequent ingredients (those ingredients that are not spices) are measured in terms of different units, like "carshina lye in the amount of 9 kab (as opposed to maneh); Cyprus wine [in the amount of] 3 se’ah and 3 kab (as opposed to maneh); Sodom salt [in the amount of] a quarter kab (as opposed to maneh) and a minute amount of Maaleh Ashan (an unspecified amount existing outside the framework of any system of measurement). If he left out any of the spices he is liable to the death penalty." Why? Because the amount of spices reflects the tetrahedral geometry within the cube of creation that is defined by the 216 letters of God’s Explicit Name, which is outside the framework of human understanding. Change any measure, and it no longer reflects the specifications that mirror the geometry corresponding to the 216 letters of the Name. The end result would be a distortion of, or 'desecration' of that One Unique Name, and any adjustment to the mixture would be like an adjustment to the geometry associated with the letters that sustain the universe.
If you learn nothing else from this post, know that there is a one-to-one correspondence between the geometry of the qetoret and that of the star tetrahedron within the luchot. Even more specifically, there is a one-to-one correspondence between the 11 organic constituents that are spices, and the measure of the ‘parent’ geometry (in the tetrahedrons) that are all conveyed in terms of maneh. At the same time, as if to act as a witness, there is also a one-to-one correspondence between the geometry of the remaining ingredients (which is the ‘child’ geometry of the stellations) and the ingredients that are not spices, that are not measured in terms of maneh, but rather, other units of measure, with one special measure (that is not expressed as a whole number at all, but rather a fractional proportion of a single unit of measure in) a quarter kab of sodom salt) pertaining to the difference between the two. So, the first unique set of numbers applies to the parent geometry (the original tetrahedrons) whereas the second set of numbers applies to the child geometry (the stellations) and a third set applies to the difference between the first two sets. The connection between “the measure of the incense” and the tetrahedral geometry in the combined cubic form of the luchot (primordial torah) is unmistakable, as are the letters of the Name that define the measure of the luchot in the first place, along with various aspects of its internal geometry, or more specifically, the geometry arising from the letters that sustain the universe. This has many implications that must be contemplated.
We might ask ourselves, what exactly is incense to God? Sweet smelling fragrance? What is "sweet' to God? What could be 'fragrant' to the Creator of the universe? He peers at his children through the lattice (Song of Songs) wondering when His children will notice him. He is hidden from them. How then can they possibly see Him? But when a child suddenly becomes aware of his or her father's watchful eye, and his all-encompassing love, the father is pleased with the child's awareness (having reached this new level of consciousness required to perceive what was otherwise hidden). Then, when that moment comes, it is like a sweet smelling fragrance to the father. The sweet fragrance you see, is the ability of man to finally see his Creator, and this is a function of his consciousness, a consciousness that man does not yet (but will soon) possess (timed to coincide with the 6,000 years mentioned in the post on Metzora) when 'every knee will bend' (an anthropomorphic term that refers to man's ability to acknowledge or otherwise be aware of God and His handiwork) (Isaiah 45:23).
All of this brings us to the 42-letter (abbreviation) of the Name, which is in essence a set of married triplets (3 + 3) written as seven sequences of six letters by the Hai Gaon. These letters to a certain extent reflect the very moment in time when the consciousness of man allows for him to more clearly perceive his Creator; and they reflect the same (tetrahedral) geometry wherein the 216 are perceived, as pictured below!
In Parashah Massey, we read: "and these were the journey's of b'nei Yisrael (the people of Israel)." The gematria3 of "these are the journey's" is 216 (אֵלֶּה מַסְעֵי). How many journey's were there? There were 42 journey's. Both are reflections of the letters in the Name, and both are a function of the geometry in the luchot. The 216, you already know. These are found in such things as the 216 cubic handbreadths in its combined cubic form and the 2,160 degrees in its 24 external angles of 90 degrees, as well as the 72 triangular corners (216) of its internal tetrahedral geometry (etc). You are about to see the 42 letters / 42 journey's in that same geometry. This is where time, space and history all collide in the properties of the primordial torah.
Just as it was in the example pertaining to the incense and our perception of God's Name in the geometry of the star tetrahedron (which was severely limited by our place or position within time and space) so too is our perception of the 42-letter (abbreviation) as it pertains to the same geometry. However, when the stellations are subdivided like their parents, the geometry is easier to see. The view that we presented in the model above, becomes slightly more complex when 'the (geometric) children' are divided, as you can see from the model below, which shows each quadrant on each stellation, on both tetrahedrons. If you count the total number of quadrants visible to the 'observer' from his one position within the framework of 'time and space' (as determined by the measure of the incense) you will note that there are exactly 42 visible quadrants, no more, and no less.
The remaining geometric elements corresponding to the remaining letters, cannot be seen from this perspective, because they are hidden within the limitations of three-dimensional space. Each element is nevertheless a triangle or 'triplet' associated with the 72 triplets (that define the 72 triangular corners of the star tetrahedron within the combined cubic form of the luchot).
Since there are 42 letters in this (married subset) of the 216 letters, we again see that there is a one-to-one correspondence between the geometry of the cosmic blueprint and the One Unique Explicit Name that defines this geometry, not to mention actual events like the "journey's in the wilderness" (42) outside the "kingdom," as opposed to being inside that "kingdom," where we have a more intimate connection to the Creator (216) through knowledge (grace ~ torah study) with particular knowledge of the shnei luchot. This corresponds to being inside the borders of Israel. Please make aliyah as soon as possible, or at least make plans to return from your sojourn (the time of your soul's journey outside those borders).
We would only emphasize at this point, that the 216 letters and its various abbreviations, are what define the measure of the luchot and its internal geometry; that in turn measure everything else; all of which defines our reality in the physical universe, including but not limited to the letters of Torah themselves, that in turn clarify or define such things as halachic requirements, like those pertaining to the requirements for the measure of each ingredient in the incense, the history of Israel etc.,
Without the prism of the magen david (a dynamic within the combined cubic form of the luchot) it would be difficult to see the connection between the letters of the Name; the luchot; halacha; the physics of time and space; quantum mechanics and just about everything else that should be viewed as a function of these letters.
Click here to read the next advanced construct: The Ulam Spiral and the Name
Footnote 1 - If you study the geometry of the octahedral core you will see 4 faces on one side (the male half of the creation-equation) opposite the 4 faces on the other side (the female half of the creation-equation). Note that your view is limited by your position in time and space, so you can only see 2 of the 'sides' on one half the octahedron, and 2 of the sides on the other half. This is a geometric reflection of certain mitzvot which come to rectify our lack of awareness, when we say "for the sake of the unification of the Holy One, Blessed be He... to unify the Name Yud-Key with Vav-Key," since our perspective is so severely limited by our presence in this world. In like manner, we can only see two faces, of the octahedron's two halves, that geometrically correlate with the two letters of the first half of the abbreviated Name, and the two letters of its opposing half, that are visible at any one 'time,' from any one 'place' by any given observer. Man is not omniscient, or this would not be necessary. The geometry of the 4 opposite 4 gives rise to the full geometry reflecting half of all the letters (108) that are distinctly opposite the other half of the letters (another 108) where the total is 216.
Footnote 2 - This would mean that academics in various universities who reject what is referred to as "intelligent design" would be genuinely foolish to dedicate their lives to promoting the ongoing illusion, that the universe and the history of man, is nothing more than a series of random events, a proposition that simply does not bear up under scrutiny. Were this the end of it, then we would dismiss the effect as a coincidence, but in truth, its really just the beginning.
Footnote 3 - R. Aryeh Kaplan in his commentary to sefer yetzirah explains that “In general, a knife or cutting blade has one dimension less than the continuum it cuts. (Ch 1)
Footnote 4 - Thanks to Avraham Agassi for pointing out the relevant gematria.