Twin Camps

This post is the beginning for a series of more advanced constructs. Before proceeding, the reader must have a basic understand of the "essential" constructs listed to the right. Here we begin with a meditation on a single word of Torah: "machanaim" (מחנים). The word refers to the camps mentioned at the end of Parashah VaYetze, when Yaacov encounters a group of angels. He names the place "twin camps." Genesis/Bereishit 32:3 "When Yaacov saw them he said, 'this is God's camp.' He named the place Twin Camps (machanaim) (R. Aryeh Kaplan's translation of Me'am Lo'ez on VaYetze pg. 108). You will see in a moment, how a single word can be mathematically unfolded to reveal many mysteries. In this case, it provides additional details of the historical account conveyed in certain commentaries. However, here, we will reveal by way of numerous examples, the hidden connection between the letters of the word and the root geometry of the primordial torah, which are both a function of the letters in God's Name.  

The twin camps, like many other references to twins, alludes to the primordial twins (shnei luchot / two tablets) and the mathematic nature of the created universe. Similar allusions are hidden in portions like Toldot and VaYeshev. Before getting into specifics pertaining to the meditation, lets review what you already know about the geometry of the foundation stone and the shnei luchot.

Prior to the twins becoming twins (before the luchot were divided and brought down) they were unified in one single stone in the form of a specifically measured cube. The geometry of this stone and its unique measurements can be expressed in several ways. At the most basic level, it is simply 1 cubit x 1 cubit x 1 cubit.  

The three dimensions of this cube are measured by the nature of the cube which has 6 faces. Since the cube is defined by its 6 faces, each of its dimensions must be defined by the same "measure" (6 units of measure). We call these units "handbreadths" because they literally correspond to "the Hand of God" in the creation process which is "defined" by the measured geometry of the evan shetiyah. Thus, the physical properties of the universe reflect these measurements (see the post on the primordial torah for a multitude of examples)

When each dimension of this cube is divided into 6 units of measure, it becomes 6 handbreadths x 6 handbreadths x 6 handbreadths, which of course defines its volumetric measure, or 216 of these units, like the 216 letters of the Explicit Name. These dimensions pertain to the three dimensions of physical space, but what about the dimension of time? 

The fourth dimension (time) must also conform to this measure of 6. Thus, creation will occur over (or last for) six days, or six thousand years (the name of the unit being irrelevant). Either way, these "six days" or "six thousand years" are discussed at length in many commentaries. The unit of measure for the fourth dimension  must be "6" because that is the requirement established by the geometry of the foundation stone. 

The former unit (the 1 cubit) is the most primal mathematic reflection of God, in that the Holy One Blessed be He and His Name are ONE. Just as 216 letters represent the Name of the One God, so too, 216 cubic handbreadths is the volumetric measure of the stone that is 1 cubit-cubed. 

1 Cubic-Cubit
216 Cubic-Handbreadths

Divide the original stone in half as it was when it was brought down by Moshe, and you get the primordial twins. The twins can be mathematically expressed as 108 cubic handbreadths opposite 108 cubic handbreadths. The letters of the Hebrew alphabet are a function of this geometry. Thus the gematria of the Hebrew letters in the word for "half" (chetzei") is also 108. The letters are bound to the geometry of the stone, or stones, as the case may be.

108 Opposite 108 

Before we get into the meditation, understand that in this "twin" form, there are 72 square handbreadths for each of its 3 layers. By understanding this form, we can more clearly perceive God's Name in the context of a three-dimensional universe, but only after its division. This has to do with the nature of human consciousness. However it also relates to the nature of the physical universe, as seen in its "duality" (well known in physics). It also explains why the Triad Havayah of 72 figures so prominently in such things as the nature of string theory and the properties of the E8 lattice (explained elsewhere in this blog).

At any rate, before we unfold the properties of the letters in the word machanaim (מחנים) to reveal the larger reality that is connected to this particular word, consider that: the geometry of the foundation stone is characterized by five internal mathematic constructs that exist in "dual" form.  

These constructs translate into five geometric polygons referred to as the platonic solids. It is important to note that one of these constructs (the tetrahedron) is different from the other four, because it is its own polar opposite. You will see why this is important insofar as the Hebrew letters, and particularly the letters of the word machanaim are concerned in a moment. The five duals are pictured below, along with their polar opposites.

                                            Tetrahedron     Cube       Octahedron      Dodecahedron       Icosahedron

                                    Tetrahedron    Octahedron      Cube     Icosahedron        Dodecahedron

Having explained the five duals in the geometry of the foundation stone, we will now proceed to examine the nature of the five dual1 Hebrew letters that flow from this geometry (especially since one of these letters precedes and then ends the word we are contemplating). 

There are, as you probably know, five Hebrew letters that exist in dual form1. These letters are unique in that one form is used when the letter appears at the end of a word, and the other form appears when it is positioned elsewhere in the word. The final form of any given letter is referred to by its name, with the addition of the suffix "sofit" ('sof' meaning 'end') for reasons which should be obvious. 

When we examine these duals, we find that one is different from the other four, just like the geometry of the stone where one of its internal constructs (the tetrahedron) is different from the other four. In the case of the letters, you need to know which one is different and what makes it different.

The letter that is different is the mem, which physically closes the loop, while the other four letters are physically open and extend downward (the "ם" as you can see is like a circle or a loop and distinctly different from the other four: "ך ף ץ ן" which all extend downward). When the mem is at the end of the word, it is a closed loop ( ם ) although when it is elsewhere in the word, it is slightly open ( מ ). This by the way is just like string theory, in physics, where the first type consists of both opened and closed strings. So the mem in the word machanaim is very special. It corresponds to the one construct that is different from the other four constructs in the geometry of the combined cubic form of the luchot (the tetrahedron and its opposite) that together cast a shadow we call the magen david. Now lets go camping!

The dual form of mem in MachanaiM is especially significant. Both forms have a gematria of 40. One is obviously at the beginning of the word, taking one form, and the other is at the end of the word, taking the other form (מחנים). Since the gematria of the letter mem is 40, we see 40 at the beginning of the word (the male component) opposite 40 at the end of the word (the female component). What does this mean?  

Look into the geometry of the foundation stone (the "stone" that would eventually be divided and brought down by Moshe). It is characterized by four points in any given two-dimensional plane. Thus, from any given position in time and space (within the context of our present universe) it can be expressed as four points within one plane, opposite four points in another (4 opposite 4). This is like the four letters of the Tetragrammaton opposite its four-letter counterpart (Adnoot). 

The dual tetrahedron within the cube can be expressed in like manner. Its points are ultimately the same points as that of the cube that surrounds it (modeled below). So there are four on one tetrahedron opposite four on its counterpart (again 4 opposite 4). Both the cube and the tetrahedron within, are the counterpart of the four-letter Name(s) within the geometry of the luchot (tablets). Each is a twin, but what does this have to do with the letter mem which has a gematria of 40?

In sefer yetzirah (1:5) the author describes creation in terms of five-dimensions. In a five-dimensional continuum, any given point has ten vectors (10). So the four points on one side of the cube or tetrahedron have a total of 40 vectors (4 x 10 = 40). These stand opposite the 40 vectors on its counterpart (the other side of the cube or the opposing tetrahedron that is within the cube). Whether it is 4 opposite 4, or 40 opposite 40, both are intrinsic functions of, and originate within the specifically measured geometry of the evan shetiyah. The mems' at opposing ends of the word machanim, that allude to the primordial twins, should now begin to make sense. Just as the twins (luchot) can be mathematically expressed as 40 opposite 40, so too, the twin mems in the word referring to these camps, are 40 opposite 40.
4 Opposite 4

40 Opposite 40

Understand that any division of the cube or its internal tetrahedral counterpart creates a set of identical twins (identical in size, shape, substance and clarity). The division of the cube results in two identical tablets (the luchot) and the division (or separation) of one tetrahedron from its counterpart, results in an identical tetrahedron facing the opposite direction. One is distinguished from the other, merely by its place or position with respect to the other in three dimensional space. The left half, or male half (108) stands opposite the right half, or female half (another 108) in both geometric sets. Twins.

With the above in mind, we will now begin to meditate on the word for these camps: MachanaiM. Meditate on the dual nature of the letter MeM in MachanaiM (מחנים) and the fact that it is one of the five twin letters. Meditate on its position(s) within the sequence of letters and see that it appears twice in the word, at opposite ends (40 opposite 40). Twins.

When Moshe went up to divide "the stone" and bring the resulting twins down, how many days was he there? Meditate on the 40 days and 40 nights he spent acquiring the twins. Contemplate the events as a reflection of their source in the geometry of the letters, which are in turn a reflection of their source in the geometry of the stone (40 opposite 40). Contemplate the geometry of the letters in God's Name, its root being the 4 letters of one Name (the Tetragrammaton) opposite the 4 letters of the other (Adnoot).

That being the case, we now meditate on dividing the word "machanaim / מחנים" into the first 4 letters, and juxtapose them against the last 4 letters: (מחני - the first 4) opposite (חנים the last 4). Meditate on the first four letters, and understand that they correspond to the first four points on the foundation stone, then meditate on the last four letters and understand that they correspond to the four opposing points on the foundation stone. Then contemplate the 40 vectors opposite their 40 counterparts, just like the gematria of the two opposing MeMs in the word describing the angels who were camping, which has a gematria that is 40 opposite 40. Contemplate carefully, then carefully contemplate, that each is a function of the four letter Name and its four letter counterpart as mentioned above, but here also contemplate that they are the root of the 216

Are these twin camps really a reflection of a larger reality defined by the twin stones?  Meditate on the gematria for the first four letters, then the last four letters:

The gematria of the first four letters (mem, chet, nun, yud) is 108, and the gematria of the last four letters (chet, nun, yud, mem-sofit) is 108. What is the volumetric measure for each of the twins? The male half is 108 cubic handbreadths and the female half is 108 cubic handbreadths. Twins, but not just any twins.

The letters of the word "MachanaiM" is a lofty reflection of the original "twins," the shnei luchot (two tablets). The division of the camps, or the measured geometry of the stones, correlates with the division of the letters in God's Name for the sake of creating a universe with "duality" (the duality previously mentioned). Understand this well. Each tablet has a corresponding volumetric measure (108 opposite 108). Moshe spent 40 days and 40 nights on har Sinai to receive these twins. This however, is just the beginning of the story (SiPpuR) in the geometry of the SaPphiRe stone (understand that this "stone" is the mathematic foundation of the universe, i.e. "foundation stone").

In the case of the machanaim, it ends with Yaacov's encounter with Esav, when Yaacov divides his entourage into two camps. Two camps? His two camps also reflect the primal counterpart (the twin camps) which are a reflection of the original twins (the luchot). What happens during this encounter between Yaacov and Esav? Before we answer that question, meditate on the year that Yaacov and Esav were born (you should know this if you've studied the essential constructs listed to the right).

So what happened during the encounter between Yaacov and Esav? When Yaacov was  returning to eretz Israel (the land of Israel) Esav sends 400 men to intercept him (the nature of this interception and the outcome is more thoroughly examined in the post on "Every Second Counts" where we explain the significance in great detail). Here, we will remind the reader that Yaacov countered Esav's 400 men with the grace (chein) that was attributed to him for his study of torah (for 14 years) in the yeshiva of Shem and Eber. The Hebrew word for grace (chein) is spelled chet-nun (חן). The chet has a gematria of 8 and the nun has a gematria of 50. The mathematic product of this gematria is 400 (8 x 50 = 400). Coincidence?

Look at the word מחנים and you will see the letters yud and mem (10 x 40 = 400) surrounding the letters chet and nun (8 x 50 = 400). Esav's men surrounded Yaacov and his entourage. 

Machanaim is a word-level reflection of this particular story, but what about the details? Esav brought 8 "armies" (groups of men) to confront Yaacov. Each group consisting of 50 men (as is known) which is the same as the gematria of the relevant letters. The total number of men was thus 400 (8 x 50 = 400). These 400 men are specifically mentioned in the written Torah.  

In addition to the yud-mem that surrounds the chet-nun, there is another yud-mem adjacent to one another (the two final letters). Meditate on the mathematic product for the gematria of this yud (10) and that of the adjacent mem-sofit (40) which is also 400 (10 x 40 = 400) and understand that this constitutes a "doubling" of the numbers. We will see the significance of this in a moment.


In previous posts we demonstrated the degree to which the story of Israel is hidden in the geometry of the sapphire stone or stone(s). So it is in Parashah VaYishlach where we later see another set of twins, when Yaacov splits his entourage into two camps. In the previous example, the word for these camps was machanaim (a masculine word). In VaYishlach, the word used to describe Yaacov's Camp is machnaot, generally translated as "camping" (here a feminine word similar to machanaim). 

One set of twins is therefore decidedly different from the other set of twins. How many times were the twins brought down? Twice. The first set of twins were "received" only by Moshe, but then later, a second set of twins is brought down for all to receive. In the written Torah, the narrative thus refers to twin camps of angels seen only by Yaacov, but then later, Yaacov splits his group into two camps that everyone can see. Twins appear twice. Contemplate each of the above parallels carefully, then meditate on the doubling previously mentioned, as it applies to the 400.

A number of sources double the above numbers. It is said by some, that Esav actually brought 400 "generals" that each commanded 400 men, the total being 160,000 men and these stood against 160,000 angels who were with Yaacov (various commentaries). Contemplate the letters of the word machanaim again, but this time calculate the mathematic product for all of the first four letters, and then, the mathematic product for all of the last four letters.

When you meditate on the word machanaim, note that the product of the first four letters, the mem being 40, the chet being 8, the nun being 50, and the yud being 10, is 160,000 (40 x 8 x 50 x 10 = 160,000). These stand opposite the product of the last four letters which is ultimately the same, except that the sequence starts with the chet 8, followed by the nun 50, then the yud 10, and the mem-sofit 40. The product is again 160,000 (8 x 50 x 40 x 10 = 160,000).

Some say that it was only 400 men, but each was capable of fighting with the strength of 400 men, in which case, there were 400 angels who could each fight like 400. The opinions are a function of the gematria which was known to the sages. It's connection to the evan shetiyah is of course a different matter, but this too was known to a select few, like the ARI, Yoseph Tzeyach (his mentor) who wrote "sefer shnei luchot" (the book of/on "the two tablets") and Abraham Abulafia to name a few (we elaborate on these elsewhere).

There are many additional allusions that are beyond the scope of this post. For now it is sufficient that the reader understand the connection between the events; the gematria of the letters, be this the sum, or the product; the geometry of the stones, and the geometry of its division; and the letters of the Name and their division for the sake of creating a universe with duality. 

This post was a meditation on a single word of Torah. The next is a meditation on three spaces (between six letters) in the written Torah: "The Root of Torah"

Footnote 1 - There are three mother letters, Seven duals and twelve elementals as is known, for a total of 22 letters. The seven duals are letters that have two different pronunciations. However, in addition to these seven duals, there are also five duals that possess "duality" by virtue of the way they are physically written.

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