Sunday, February 18, 2018


Tetzaveh has a number of remarkable allusions to the signature geometry of God’s Explicit Name as well as the cube [of creation] in both its combined and divided state (shnei luchot) and their sapphire-like attributes.

 The first reference is the command to “Make the ephod... out of gold (filament) blue, purple and crimson (strands of) wool, together with twined linen."

How was this done? They would take...

6 strands of blue wool, and weave it together with one filament of gold
6 strands of purple wool and weave it together with one filament of gold
6 strands of crimson wool and weave it together with one filament of gold

In other words they would weave 3 threads with 6 strands of wool, just like the 3 measures of 6 in the combined cubic form of the luchot with a total volumetric measure of 216 cubic handbreadths. The element of gold, interestingly enough, has a measure of hardness equal to 216 MPa1 (a unique measure that will be further expanded upon in succeeding paragraphs). All of these elements are a reflection of a larger reality that revolves around the letters in the Explicit Name. The divine service was always performed with a knowledge of these letters.

If you are new to this blog and have not yet read the essential constructs listed in the column to the right, please do so before proceeding. They explain some of the basic connections between the letters of God's Explicit Name and the multidimensional mathematic and geometric characteristics of the luchot, both in the divided state and combined cubic form, along with the internal geometry that casts a shadow known as a magen david. There is a special connection between the Name, the luchot and the geometry of the magen. The purpose of this blog is to show how they all connect with one another, and to reveal where this Signature of the Architect is hidden in each week’s parashah. Once you are familiar with these constructs you can more fully appreciate each post.

You may recall other places in the Torah where the events were a reflection of these letters, specifically in the form of 3 measures of 6, like the 3 measures of flour that were each 6 ounces in VaYera; and the 3 periods of time measured as a function of ‘6’ in Mishpatim (6 days in the cloud; 6 days bringing the words of God to the people; and the events beginning or ending on the 6th of Sivan). All of these numbers allude to the ratio of the sapphire cube of creation measured in terms of three 6s (handbreadths) 6 x 6 x 6, where the number of elements in the matrix (216 cubic handbreadths) corresponds to the number of letters in the Explicit Name or  Triad Havayah of 72 (3 x 72 = 216).

There was of course another thread, but it was made of linen, not wool. It too was woven together with a filament of gold. These 4 threads together are a reflection of the two primary abbreviations of the Explicit Name: the Tetragrammaton and the Havayah of Adnoot that are married together to create the 4 letters of Ehyeh as is known. You’ll note that one thread (the linen) is different from the other three (of wool). This is like the 4 “couples,” in the marriage of the two abbreviations, where one couple (the vav and the nun) are different from the other three. This ratio is mentioned in numerous other posts like Parashah Ki Thetze, Massey and Devarim, and the Marriage of the Letters, so it will not be repeated here, except to say that when the 4 letters of one are married to the 4 letters of the other, one couple (the vav and the nun) is different from the other three couples, because the form or substance of the letter is “subtracted,” rather than "added."

When the threads of wool and the threads of linen are woven together in proper fashion, they properly reflect the nature and number of letters in the Explicit Name and its abbreviated forms. Were we to wear an ordinary garment containing both wool and linen, it would not necessarily reflect these characteristics and would or could be tantamount to what is called chillul Hashem (a desecration, diminishment, tarnishing or distortion) of the King's Name. Hence it is forbidden. There are recognized exceptions in halachically required garments such as a tallit, but the tallit is to be made with techelet, where one strand (the one with techelet) is actually different from the other three, and where the 4 are "doubled over" making them 4 opposite 4, like the letters mentioned above that are married to one another during unification. The reason why we should not wear an ordinary garment containing both wool and linen, should be somewhat obvious. 

One is different from the other three? That ratio should sound familiar for another reason. It also happens to be a dominant aspect in the geometric ratio of the royal seal, between the number of points in any one dimension (any one point of a tetrahedron ) in contrast to the number of points above that dimension (the other three points of that tetrahedron) as pictured below.

The Two Tetrahedrons as a Reflection
of the Marriage Between the Letters

The total number of points (which is 8) is equal to the total number of letters in the Tetragrammaton and the Havaya of Adnoot (also 8) and the geometric marriage of each pair of points (a total of 4 pairs) corresponds to the marriage of the 4 'couples' in the Name. Further, the marriage in the geometry of the two alpha points (one couple) is different from the marriage between the remaining six beta points (three couples). See the model above. The geometry of the royal seal (a subset of the geometry in the combined cubic form of the luchot) merely 'reflects' the characteristics of the letters of the Name and its abbreviated forms that define it's properties (as an aspect of the luchot).    

For purpose of clarification: a tetrahedron is half of a three-dimensional magen david, or royal seal of the King as explained in our post on Emor. When two tetrahedrons are married together in opposing directions (much as the letters are during unification) the result is a star tetrahedron. The “star” casts a two-dimensional shadow known as a magen david. The number and the nature of the letters in the King’s Name are hidden within the seal as a function of its geometry.

The Marriage in the Royal Seal
and its Shadow

The tetrahedron is one of the five mathematic constructs in the combined cubic form of the luchot. When it is juxtaposed against its polar opposite, the two form what we call the royal seal containing the geometric Signature of the Architect. The tetrahedron is unique in that it is only construct within the combined cubic form of the shnei luchot that is its own polar opposite (oriented in the opposing direction). This construct is shown below within the geometry of 'the cubit-cubed.' The 5 constructs and their polar opposites are modeled and explained in greater detail in the post on String Theory and the Luchot.

The Combined Cubic Form of the Luchot
and the Royal Seal of the King Hidden Within

Although we've said this before, it bears repeating. The star tetrahedron (three-dimensional magen david) or royal seal, is a geometric reflection of the 216 letters in the Explicit Name. When the two tetrahedrons are joined, the marriage results in 72 triangular corners. This triad nature is an aspect of the Explicit Name or the Triad Havayah of 72 as it is called. We see the geometric nature of this Name (revealed) in the measure of the luchot, but only when it is divided and in side-by-side form, where its volumetric measure is expressed as 72 x 3 handbreadths (the total being 216). Otherwise, it is (hidden) in the volumetric measure of the cube (before division) and its internal geometry in the star tetrahedron with
72 triangular corners, where the sum of various angles (in degrees) is a multiple of either 72 or 216 (like 720, 2,160 degrees etc).

In Tetzaveh we see these same reflections. The 3 threads with 6 strands of wool are a reflection of the volumetric measure in the combined cubic form of the luchot (6 x 6 x 6 = 216). The 4 types of thread consisting of one type (linen) opposite three others (wool) is a reflection of its eight dimensions (that are 4 opposite 4) both externally and internally, and thus the geometry is a perfect reflection of the primary abbreviated forms of the Name as well. However, the nature of these geometric reflections does not end with the 216 elements hidden in the combined cubic form of the luchot.
For example, when the gold is added to each thread, it adds a whole new dimension to the connection to the letters of the Name. What part does gold play and how does it reflect the nature of the letters?

The gold filament completes the geometric picture. It was of course real gold that had been beaten into fine sheets and then sliced into filaments. When these were added to the threads of wool and the thread of linen, there were a total of 28 strands or filaments. The number 28 is a prominent mathematic feature of Torah as demonstrated in posts like Emor and Miketz because it is also the number of elements in two tetrahedrons, and thus an integral aspect of the tetrahedral geometry in the royal seal

For the purpose of this post on Tetzaveh, we’ll merely point out that the number 28 is invariably embedded in the linguistic structure of the text, generally as a function of two opposing 14s (14 opposite 14). The two 14’s are essentially a one-dimensional mathematic expression for the 14 elements of a three-dimensional tetrahedron, opposite the 14 elements of its counterpart (28 as shown below) whose marriage creates a star tetrahedron.

Students of the ARI will recognize this aspect of the geometry as a reflection of what is known as the supernal stream (nachal) consisting of seven names with 4 letters each, or 28 letters total

All of the above helps us to further understand why we find this pattern subtly hidden in the linguistic structure of the text in so many places. More specifically, the letters of Torah are a reflection of the mathematic nature of the combined cubic form of the luchot, which are in turn a reflection of the letters in the Name. Our reality is literally orchestrated by those letters!

The 28: 14 Opposite 14

The gold thread has additional significance as well. As it turns out, gold has a crystalline structure known as “lattice face-centered cubic.” The curious thing about this structure is that it is like the outside of a cube but with an external latticework that is identical to the combined cubic form of the luchot with a star tetrahedron inside. If that sounds complicated, don't worry. We'll draw you a picture in a moment. 

The  crystalline structure of gold is pictured below/right. The picture to the left is the star tetrahedron inside the cube [of creation] or combined cubic form of the luchot. As you can see, the surface structure is an identical match to the face-centered cubic latticework of gold!

The Face-Centered Cubic Latticework of Gold
as a function of
The Star Tetrahedron within the cubic form of the Shnei Luchot

The mathematic characteristics of the sapphire cube, the star and the Explicit Name converge in the geometry of the “royal seal” found within “the cube of creation,” and in this case we see an aspect of the same geometry in the "Signature" embedded in the latticework of gold. In fact, if you count the number of “corners” in the lattice, you will find a total of 72 angles (corners) that once again reflect the Triad Name of 72. The numbers pertaining to the outer shell, its lines, faces, angles and so forth, reflect the internal characteristics of the magen david, both in the nature of sapphire, and the gold, where there are 3 times as many corners (3 x 72 = 216) along with an associated number of lines, faces and edges. An explanation of how these other associated elements connect is beyond the scope of this post. We'll only add at this point a small passage from the Song of Songs: "My beloved... is standing behind our wall, He is looking through the windows, He is peering [at us] through the lattice."

The Signature of the Architect is subtly hidden not only in this parashah, but throughout every aspect of creation and especially in things that have to do with whatever is instrumental to divine service, like the use of gold or requirement for its use. In this case we find it in the substance, the number, and the ratio between the various threads in the ephod, but as you can see, it goes far beyond that! 

Aside from the connection to the latticework of gold explained above, gold has another property that connects it to the letters of God's Explicit Name. We briefly mentioned this connection in the first paragraph of this post, but here we will expand upon it. Gold is commonly thought of as the most precious metal, so its use in the Temple (and many other things) is logical as they are intended to honor God, but they do so in a hidden manner as well. It turns out that gold  directly mirrors the characteristics of all 216 letters in the very nature of its atomic structure!

In addition to the nature of the latticework mentioned above where 72 (times 3) alludes to the letters, metals are often measured according to their "hardness.” The hardness of gold happens to be 216 on the 'vicker's scale.' 216? Isn't that the exact number of letters in God's Name? The vickers scale is not an arbitrary scale. Rather, like the 'abbe scale' used to measure the clarity of transparent substances like sapphire, the vickers scale was logically designed to account for the natural characteristics of the materials tested, and therefore a natural mathematic extension of those characteristics.

When you know God's Name, you can see His Signature everywhere. This is just one of many examples. It is found elsewhere in this week's parashah as well, albeit in yet another form. The text continues: “Make an altar of incense....” The altar was 1 cubit x 1 cubit x 2 cubits. The dimensions result in an object that amounts to 2 cubes, one on top of another. Moreover, each cube was 1 cubit-cubed. If you’ve not yet read our post about ccSpace216 on the Eight dimensions of the Name, and the measure of the cubit-cubed, it should be considered required reading. In this case, the altar was in the measure of 2 connected cubes that were each 6 x 6 x 6 handbreadths.

In mathematics or physics, if you want to visually portray a four-dimensional hypercube, also called a tesseract, you would use two 'connected' cubes. Nowadays, the hypercube is generally animated as a cube inside a cube. However, given that the depiction/picture is static, and that the first cube is solid and non-transparent, the depiction of the second cube would be lost. In such cases it may be depicted as simply two separate cubes, side-by-side, connected by lines as pictured below. The effect has the purpose of depicting the three dimensions of physical space and the passage of time (or fourth dimension).

The Hyper-Cube

When we rise above three-dimensions, we must find a way to express the passage of time. The connection of two cubes serves that purpose: a given space defined by the cube [of creation] as a function of time. In last week's parashah, we saw how the area beyond the curtain in the Ohel Moed was 216 thousand handbreadths (6 “cubed” times 10 "cubed") alluding to HaShem's (the Name's) rule over the three dimensions of space and the one of time. The measure of the altar of incense alludes to His rule over both time and space. It also alludes to something else. In fact, the measure of these 'two cubes" (like the shnei lucot that were brought down twice) is equal to the exact speed of light through a true vacuum as explained in the post on that subject.

One other thing of note. On the ephod, there were stones of Sardonyx, one for each shoulder. The names of Yaacov’s sons were engraved on these stones, 6 on one stone and 6 on the other stone. Hidden deep within the structure of Sardonyx is a trigonal crystalline structure (because it is actually a crypto-crystal) like the trigonal structure of sapphire. It could certainly be a coincidence that the product of these numbers, 6 x 6 x 3 = 108, and that there are 2 stones like the luchot. However, considering all of the above, it's likely more than just a coincidence. We’ll let you decide.


Footnote 1 -  This system of measurement is a modern method referred to as the Vickers Scale proposed and developed by Robert Smith and George Sandland. It is important to note however, that the measure of hardness was not based on an arbitrary scale. Rather, like the Abbe scale for clarity in transparent substances, it was derived from a logical ratio, which is in this case the ratio between a given force as it relates to a given surface area (F/A).