Pekudey

In this parashah “[Betzalel] made the robe [for the Cohen HaGadol]” and they “made pomegranates [and] ...pure gold bells, and placed the bells in the pomegranates.” It then says “there was a bell and a pomegranate, a bell and a pomegranate all around the bottom of the robe.” 

Before getting into specifics, we need to mention that some of the information you are about to read requires a broader understanding of certain concepts. The reader is therefore strongly advised to read each of the essential constructs listed in the column to the right before proceeding. These constructs reveal the basic connection between the letters of the Name and the multidimensional mathematic and geometric characteristics of the luchot, both in their divided state and combined cubic form, along with an explanation of its 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 connect with one another, and to reveal where this unique 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.



With respect Parashah Pekudey, there is considerable debate in the Talmud and other sources as to specifics regarding the robe, the bells and the pomegranates. Was there a bell next to each pomegranate, or was there a bell inside each pomegranate? Some say there were 36, some say 70 and some say there were 72 pomegranates (and an equal number of bells).



Given the numbers, sets of numbers and ratios between the sets found in the last parashah, relative to the ratio of phi; relative to the measurements of the tabernacle; and their connection to God’s Name, the description of the robe with its components might be altogether different, and so I offer the following for consideration:



The number 70 has an indirect relation to God’s Name through a segment of the geometry in the royal seal (three-dimensional magen david). The number 36 is a facet of the Name as well, in that it is half of 72, but half of 72 is just that (only half) and so its connection is also somewhat indirect. The number 72 has a direct connection to God’s Name, in that it is the number of triplets, but still just one third the number of letters. We pointed out in (last week's) post on Parashah VaYakhel, that without the third wall of the Mishkan, there were only 2 x 72 (handbreadths) or 144 elements, and so the reflection fell 72 short of the mark. However, when we added the third wall, which was 72 handbreadths, the total measurement became 3 x 72 or 216 handbreadths, which properly reflected not only the number of letters in the Explicit Name, but also the triad nature of that Name.

In the case of the robe, if there were 72 pomegranates then there would be 72 bells and we would have the same problem we had in the last parashah concerning the walls (2 x 72 = 144). Each opinion thus far offered, is inadequate to reflect both the number and nature of the letters in His Name. In the case of 72, the total number is missing a vector or component of the triad form (see the post on VaYakhel for a more complete explanation). Briefly, in the case of the walls of the mishkan, there was a third wall (a third set of 72 handbreadths) that came along to reconcile the discrepancy. Is it possible there is yet another number to consider in describing the bells and pomegranates? 


The Torah describes the hem of the robe as having a “bell and pomegranate,” as though they were married together in pairs at the bottom of the robe. In other words, they could have been side-by-side; or maybe one inside the other (as suggested in some sources) but whatever their position, they were somehow juxtaposed one against the other like twins. Notice also that the phrase "bells and pomegranates," is repeated twice! Two sets of twins should sound familiar.

Since there were two items, bells and pomegranates (and the phrase is repeated twice) then by logical inference the form would seem to mirror that of the luchot, which were two mathematic and geometric twins brought down twice in the form of 108 opposite 108 (cubic handbreadths). Might there have been 108 pomegranates opposite 108 bells? 

If this were the case, from a purely hypothetically standpoint of course, how might they be distributed around the bottom of the robe. Consider what happens, for example, if they were not merely adjacent to one another, but also over top of one another, and staggered triangularly as pictured below.

This would mean first, that the 108 bells opposite 108 pomegranates, were like the 108 handbreadths opposite 108 handbreadths in the measure of the north and south walls of the tabernacle when portioned according to the ratio of phi. It would also mean they were like the 108 cubic handbreadths of the luchot, opposite its remaining 108 cubic handbreadths, all of which reflect the number of letters in the Explicit Name. It would however, mean something else. 

If the bells and pomegranates were juxtaposed as a series of triangles they would still be a string of alternating bells and pomegranates, but the staggered juxtaposition would form a series of triangles. 

Question: If there were 108 of each, then how many triangles would there be?
Answer: There would be exactly 72 triangles (72 x 3 = 216) as shown in the model below.

In this configuration, the hem of the robe with its bells and pomegranates would be a mathematic and geometric mirror image, not only of the walls of the mishkan that were both 3 x 72 and 2 x 108 at the same time, as shown in the picture posted last week in VaYakhel, but it would also reflect the mathematic and geometric characteristics of the luchot. The requirement in the written Torah would be met, as there would be "a bell and a pomegranate, a bell and a pomegranate all around the bottom" as specified. More importantly, the characteristics of the bells and pomegranates would mirror the number and the triad nature of the letters in the Explicit Name.
 

108 Bells and Pomegranates Triangularly Arranged (Total: 216)
w/ 72 (Virtual) Triangular Forms

(There are 72 Triangular Corners in a Three-Dimensional Magen David)

3 of the 72 'Virtual' Triangles are Depicted Above
(the blue lines are used merely to show the triangular pattern)

(click on the diagram to enlarge it)

Contemplate this for a moment and remember also that when the top angle of an equilateral triangle equals 108 degrees, the sum of the 2 bottom angles equals 72 degrees, and when the top angle equals 72 degrees, the sum of the 2 bottom angles equals 108 degrees. If you wanted the robe to reflect the number and nature of the letters in God's Name, how many bells and pomegranates would you include, and where, and how, would you place them?

 

VaYakhel

Throughout this blog it has been asserted and repeatedly demonstrated that the mathematic Signature of the Architect is deeply embedded throughout the entire Torah in a variety of ways, and that it is hidden from the casual observer. In this week's parashah we see once again how this unique signature in the form of certain mathematic and geometric characteristics, is hidden just beneath the surface.

Before we get into the specifics of this parashah, we should mention that some of the information presented in this post requires a broader understanding of certain concepts. The reader is therefore strongly advised to read each of the essential constructs listed in the column to the right before proceeding. These constructs reveal the correlation between the letters of the Name and the multidimensional mathematic and geometric characteristics of the luchot, both in their divided state and combined cubic form, along with an explanation of its 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 and to reveal where the signature geometry is hidden in each week’s parashah. Once you are familiar with these constructs you can more fully appreciate each post. 



In VaYakhel, Betzalel and his assistant carry out the instructions that God gave to Moshe in Terumah. We are told that these men were gifted with the ability to know and understand how to build what God had directed. So with that in mind let's look at what they built and how it correlates with God’s Name.



The Torah says: “...they made 20 [golden] beams for the southern wall [and for] the north[ern wall] they made 20 [golden] beams.” At first glance this appears to be a simple, straight forward description. However, it also contains a great deal more. Embedded within these measurements is an accurate reflection of God’s Explicit Name. How on earth could the signature geometry be hidden within these simple measurements? 


We have shown on numerous occasions that the luchot were each 6 x 6 x 3 handbreadths and that a matrix with this ratio consists of 108 'elements' (cubic handbreadths). There were two luchot, so the measure of the combined form was 6 x 6 x 6, and the total number of elements in the twin matrix was 2 x 108 or 216. The number of cubic handbreadths in the luchot is of course equal to the number of letters in God's Explicit Name. In fact, when the tablets are placed side-by-side, the ratio becomes 6 x 12 (72) x 3 which reflects the number and nature of the 72 "triplets" (72 x 3 or 216 letters). This should be carefully contemplated.

As we proceed through this post you will see how these numbers, sets of numbers, and the ratio between the sets define the architecture of the mishkan (tabernacle).  First, consider the ratio known as 'phi' represented by the symbol φ, also called the golden ratio. Why is it called a "golden" ratio? What's so "golden" about it? You'll see why in a moment. The ratio is well known and can be found throughout the entire universe. Phi is essentially 'nature in action.' The curvature of a seashell is in this proportion. The number of male bees in a hive, to the number of female bees is in this proportion. The growth of leaves and stems in most plants conforms to this proportion, as does the shape of pine cones and sunflowers to name just a few. The hull of a modern ship when shaped to this proportion has the most efficient hydrodynamic advantage. The human body has many examples of this proportion as represented by Da Vinci's Vitruvian Man. The ratio of phi is the "logic of nature," juxtaposed against the chaos that opposes it. In many ancient cultures, it was proof of intelligent design so it should come as no surprise that this ratio is found within a certain mathematic relationship between the walls of the Mishkan and two specific aspects of God's Explicit Name! We see this quite clearly in the specifications of the tabernacle.

There were 20 beams on one side of the Mishkan and 20 beams on the other. If the 20 beams of each wall are divided to the nearest beam according to the ratio of phi (1/φ x 20) then the two portions of the wall are 12 beams and 8 beams respectively. Each beam was 1 1/2 cubits in width, which is the same as 9 handbreadths. The portion consisting of 12 beams would thus be 108 handbreadths and the remaining portion (which was the holy of holies) consisted of 8 beams, which would be 72 handbreadths. The sum of the two portions (108 + 72) equaled the length of the wall. The point dividing the two segments was also the point dividing the 'Holy of Holies' from the remaining area. To put it another way, the 20 beams of each wall were 180 handbreadths. The holy of holies, or inner portion, comprised 72 of these handbreadths, and the outer portion comprised the remaining 108 handbreadths.
 
Do those numbers sound familiar?  Of course they do! In terms of these proportions, the 108 handbreadths on the southern wall, opposite the 108 handbreadths on the northern wall were just like the luchot with a volumetric measure of 108 cubic handbreadths opposite 108 cubic handbreadths. These portions and their respective number of elements correspond to the number of letters in God’s Explicit Name. So too does the remaining portion, in that there are 72 handbreadths on the southern wall opposite 72 handbreadths on the northern wall. However, since there are only 2 x 72 (handbreadths) the remaining measurements would seem to be deficient!

The number of letters in God's Explicit Name is a function of 3 times 72 (the Triad Havayah of 72 consisting of triplets). The above model has only 2 'measures' of 72, so it appears to be missing a component or vector of the triad form of God’s Name.

Where is the third "set" of 72? Have we forgotten something? What about the third wall? So far, we’ve only discussed two walls. The tabernacle was closed at the end as pictured below. It also had a western wall in addition to the northern and southern walls. Care to guess what the measurement of the western wall was?



That's right! It turns out that this wall was made of 8 beams as well. These 8 beams were also each 9 handbreadths in width (8 x 9 = 72 handbreadths). There's our missing set! The total measure of the tabernacle where the ("golden" ark was present) was 3 x 72 or 216 handbreadths. The total measurement (of 216 handbreadths) is clearly divided into 3 groups (in the form of 3 walls) precisely conforming to, and mathematically reflecting the triad nature of the Name.



In summary, the walls have multiple characteristics that reflect the characteristics of the Explicit Name and the measure of the luchot, either in terms of the 108 x 2 (216) handbreadths associated with the twin matrix, or in terms of its triad form, 72 x 3 (216) handbreadths. The ratio of one to the other (phi / φ) as it relates to the Name and the measurement of His tabernacle, thus "defines" what He has created in the properties of nature which adheres to this ratio.

(Click to Enlarge)

The Section with the Golden Ark of the Covenant
is a reflection of the letters in God's Explicit Name


Once again we see the secret, subtle, sublime Signature of the Architect, hidden just beneath the surface of what appears to be ordinary circumstances (in this case the measurements of the tabernacle) in order to confound the casual observer. The significance of all this does not end with the examples above, and so, for the beggars among us who are not satisfied with a crust of bread, there is a bit more information to follow.



In addition to what has been mentioned thus far, the sum total of the opposing halves (180 handbreadths opposite 180 handbreadths) is the like the sum total for the two halves of a circle which is 360 degrees (180 + 180 = 360). This is like the two-dimensional shadow of a specific sphere within a specific cube (the combined cubic form of the luchot) with a specific measure, and thus a geometric absolute within the cube of creation defined by its 216 cubic handbreadths. That sphere is .5236 the volume of the cube that surrounds it, which happens to be the ratio between the royal cubit (the King's system of measurement) and the metric system (the royal cubit is .5236 of a meter).   

We have already mentioned that when the 20 beams of the wall are divided to the nearest beam according to the ratio of phi, the two resulting portions consisted of 12 beams and 8 beams respectively. What we have not yet mentioned is that the star tetrahedron with its 72 triangular corners (as a geometric reflection of the Triad Havayah) coincidentally has 12 edges and 8 vertices. The signature numbers of 72, 108 or 216, are repeatedly found as multiples in the total degrees of its various internal and external angular properties (like 2,160 in the 24 x 90 degree angles of the cube). 

We also mentioned that the beams were covered with gold. You may remember in the portion of Tetzaveh, we showed how metallic gold had a crystal lattice that was face-centered cubic, with a measured hardness of 216 MPa; and that the latticework of gold had 24 triangular (or three-sided) facets (for a total of 72 corners).

While the external measure of the Holy of Holies consisted of 3 walls, each 72 handbreadths (mentioned previously) its internal measure (less the thickness of the wall) was 60 handbreadths by 60 handbreadths by 60 handbreadths (6 x 6 x 6 times 10-cubed) or 216,000 (cubic handbreadths). All of these measurements taken together reflect the nature of the cosmic blueprint (cube [of creation] or shnei luchot) that in turn reflect the nature and number of letters in the Explicit Name.

The portion of the 3 walls above (wherein each of the 72 handbreadths is found) is comprised of 8 beams on each of the 3 walls (thus a total of 24 beams just like the 24 facets in the latticework of gold). The golden section of the tabernacle (Holy of Holies) with its 3 walls of 8 beams, is a mirror image of the crystalline structure in metallic gold on the periodic table of elements. Although it might be more accurate to say that the gold; the tabernacle; the mathematic and geometric characteristics of the magen david within the sapphire cube, and indeed everything (witness the ratio of phi in nature) all bear witness to the One Who created the universe, and that the Signature of this Architect is hidden within the nature of each component.

(from left to right)

Three-Dimensional Magen David in the Cube of Creation

Face-Centered-Cubic Lattice of Gold

Holy of Holies in the Tabernacle

(Click to Enlarge)
 

More Detailed Model of the Holy of Holies

(showing the finishing beams that result in the mathematic connection between

the Name and the internal and external measurements)

With the above in mind, lets take a look at some of the more advanced geometric parallels between God's Name, the Tabernacle and the magen david in the third dimension. Each of these measures should be contemplated as they apply to the letters of God's Name.

Three-Dimensional Magen David
External Ratio ~ Cubic
Number of Triangular corners 3 x 72 = 216
8 Primary (Large) Faces
24 secondary (small) faces (w/ three-corners) = 72 triangular corners

~

Gold
External Ratio of its Crystalline Structure ~ Cubic
Hardness = 216 MPa
Each side/face = 8 lines
24 tri-angular (3-sided) facets = 72 corners

~

Golden Section (Holy of Holies) of the Tabernacle
Internal Ratio ~ Cubic
Product of Internal Ratio 60x60x60 = 216 (ten-cubed handbreadths)
Each side/face = 8 beams
24 beams = 3 sides/walls of 72 handbreadths
 

~

• Every face on a star tetrahedron or triangle of a magen david has 3 angles.
• Each of the 3 angles is 60 degrees (3 x 60 = 180).
• The north and south walls of the Mishkan were 180 handbreadths in length.
• The internal measure of the Holy of Holies was in the form of 3 measures of 60.
• The angles of the (equilateral) triangular faces on the star tetrahedron (or magen david) are 60 degrees, 60 degrees, and 60 degrees. 
• The internal measurements of the Holy of Holies is 60 x 60 x 60 (216,000 cubic handbreadths)
• The 3 measures of 6, or 3 measures of 60 mentioned above, reflect the measure and ratio of the combined cubic form of the luchot defined by 3 measures of 6 (6 x 6 x 6)
• When the face of any triangle (on the magen) is "tilted back" into the third dimension as would be the case on a star tetrahedron (which is three-dimensional) the ratio of the angles of one to another changes in terms of its 2-D orthographic projection.
• When the top angle is 108 degrees, the sum of the 2 bottom angles is 72 degrees.
• When the top angle is 72 degrees, the sum of the 2 bottom angles equals 108 degrees (72 + 108 = 180 or the measurement of the walls).
• When the walls of the Mishkan are proportionally divided according to the golden ratio, there are 8 beams and 12 beams respectively. The 8 beams are 72 handbreadths and the 12 beams are 108 handbreadths (72 + 108 = 180 handbreadths).
• The magen david (in the third dimension) has 8 vertices and 12 edges and the sum total of its various related angles are multiples of 72 and/or 108 degrees.
• The star tetrahedron or magen david (in 3 dimensions) has 8 primary (large) faces with 72 corners.
• The tetrahedrons of the "star" have 12 secondary (small) faces with a total of 108 corners (72 + 108 = 180 corners).

All of the above are mathematic or geometric reflections of characteristics within and without the original sapphire cube that was 6 x 6 x 6 (216 cubic handbreadths): the two luchot that were 2 x 108 cubic handbreadths (216) or in their ratio of a single side-by-side matrix of 72 x 3 cubic handbreadths (216); and the geometric characteristics of the magen david in the third dimension. All have a direct correlation with God's Name. Is this a coincidence or is it the result of careful design, and if it is of design, then who is the designer? The answer should be somewhat obvious!


This week's post will end with one last assertion. As hard as it may be to believe, it can be shown in the geometry of the 3 walls consisting of 8 beams covered in gold (with its 24 facets) together with the resulting internal measurements, that these numbers, sets of numbers and ratios between sets, provide an accurate reflection of the number of letters in the written Torah. However, an in-depth explanation of this particular mathematic relationship is beyond the scope of a weekly post because it involves aspects of the twins that are far more complex. However, a glimpse of this relationship can be seen in the post on  “The Eight Dimensions.”

Ki Thisa

In this parashah the sapphire twins are 'born' when they are divided and brought down, as it is written: “Moshe... began going down the mountain with the shnei luchot” in hand (two tablets of the covenant) (Ex 32:15).



The event is the culmination/fulfillment of the prophetic allusions that were seen in both Toldot and VaYeshev: “When it came time to give birth... there were twins in Her womb,” alluding to the Shechina and the giving of the luchot. The exact same phrase is repeated twice, in two entirely different portions of the Torah, involving two different women and two entirely different sets of twins. The unusual language that references the twins (becoming twins when it was time) is a reflection of the primordial division (duality) of the cosmic blueprint (two tablets) and that which it defined (the physical universe) and the two times the phrase/events occur, is a reflection of the two times that Moshe would actually bring down two different sets of twins. The numbers and sets of numbers attributed to the luchot in its volumetric measure and the number of degrees in its various angles equate to the number of letters in God's Explicit Name (216).

We will look more closely at this allusion in a moment. In the meantime, 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 shnei luchot (two tablets) 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 and to reveal where the signature geometry is hidden in each week’s parashah. Once you are familiar with these constructs you can more fully appreciate each post.

The prophetic reference(s) to the giving of the luchot were hidden in the events pertaining to Rivka and Tamar (in Toldot and VaYeshev). The important point to remember is that the prophecy appears in the form of two sets of “twins” as well, like the luchot that are mathematic, geometric and substantive twins, a template for the "duality” of a universe with male and female aspects.

6 x 6 x 3  (108) Plus  6 x 6 x 3  (108) Equals 216


The people were not prepared to receive the first set and the two halves were destroyed. A second set was then brought down.

These two events involving the 10 commandments and the luchot, parallel the descent of the 10 sephirot. The tablets with the 10 commandments are shattered, but then brought back. Likewise, the 10 sephirot are shattered, but then brought back (as is known and discussed in numerous commentaries). The blueprint defines reality, and reality reflects the geometry of the luchot, all of which reflect the nature of the letters in God's Name and its abbreviated forms.

Each event is a reflection of it’s source above, which is the cosmic blueprint or cube [of creation] consisting of 5 opposing constructs (10: that are 5 male ~ 5 female) as explained and modeled in the post on string theory and the luchot. The primordial (Torah) luchot before division, was not just 'a cube,' but the cube [of creation] or what we generally refer to as the combined cubic form of the luchot. The tablets are mathematically and geometrically defined by the 216 letters of God's Name (as seen in its volumetric measure of 216 cubic handbreadths; 216 elements/angles; 2,160 degrees etc) regardless of whether they are perceived in their combined cubic form (as they were above) or their divided state (as they are below). In their combined cubic form they are also referred to as the cube [of metatron] the mem-tet angel. The word 'MeTRon' means to measure (or 'to judge'). It is the origin of the word 'MeTRic,' a system of measurement as well as a unit of measurement that is also found in words like MeTRonome, which is a device that signifies the passage of time (used by musicians for keeping time). In English, the word MeasuRe has an 'S,' but the origin of the word and its letters are the same: MeTRon. 

RAMBAM (Rabbi Moshe Maimonides) in his “Guide for the Perplexed,” Ch. 5 on Verse 24:10 of Exodus, points out that (prior to the division of the luchot) those who had gone up with Moshe “...saw the God of Israel and  [something] ...under His Feet,” rather than simply saying that “they saw the God of Israel.” Why? According to Maimonides, this comes to explain that while they were able “to see” [the sapphire cube] as “something,” they failed to describe [to understand, perceive, comprehend or delve into its significance]. This opinion has a number of fascinating implications worth pondering in light of the information advanced in such posts as the primordial torah; the measured speed of light and string theory and the luchot (mentioned above).

Maimonides then asserts and admonishes the reader: “How much more is it incumbent on us...” (having the benefit of hindsight) that we “...persevere in perfecting our knowledge of the elements,” which is further clarified in: “the principle part [to which he was referring] which is what “...was under His feet [the] 'paved work' of a sapphire stone,” (Ex 24:10) the 'paved work' alluding to the plan or path, literally "written in stone" (the geometry of the stones) established by the letters of the Name that (again literally) define the geometry of the 'sapphire cube.' The gematria or mathematic sum of the letters in “HaEvanim” (heh-aleph-bet-nun-yud-mem) which is the Hebrew phrase for “the stones,” is 108. This is, once again, the volumetric measure for each of the stones/luchot. The phrase is invariably mentioned twice (as demonstrated in the post on Parashah Eikev) alluding to the two sets of tablets that were brought down twice. 

Each cubic handbreadth of the original undivided cube of 'stone' (with a total of 216) corresponds mathematically to a letter of the Name (216 letters) but each handbreadth further consists of 5 "elements” (a hand having 5 fingers). These 5 elements are like a geometric extension of each letter corresponding to each cubic handbreadth. There are “twenty-two basic forms of energy (represented by the letters) associated with any given 'hand' and we are told (in sefer yetzirah) that the divine law "engraved them in voice, carved them in breath and placed them in the mouth in five places: the throat, palate, tongue, teeth and lips.” This is how man creates (as in the words he speaks). How did God create? He spoke the world into existence. Speech is sound, and sound consists of vibration or oscillation as in vibrating strands of energy, in particular those associated with 22 specific letters. The 5 string theories as they relate to the 5 'elements of each 'Hand' (insofar as the 216 cubic handbreadths associated with each of the 216 letters are concerned).

In contemplating man's participation as co-creator, it should be remembered that 'man is created in the image of God.' When we speak, we make use of 4 places in the mouth, opposite 4 in the throat that combine to control the flow of air (4 opposite 4 like the letters of the Name yud-key-vav-key opposite the 4 letters of the Name aleph-dalet-nun-yud). Moreover, there are 4 specific letters of one type involved (the Gutterals: אחהע) opposite 4 of another (the Palatals: גיכק) as is known (the ARI). The gematria of the first set is 84 and the gematria of the second set is 133, the total being 217. The ARI points out that when you subtract one unit for the whole, the result is 216, adding that this is 3 x 72 (Hashem Hameforash / the Explicit Name) which gives rise to the remaining "3" phonetic groups (the Dentals, Labials and Sibilants). The primal (abbreviated) form of the Explicit Name thus expands to the full form, as creation progresses. There is of course much more to the allusion and its implications. 

The greater the awareness we have of God’s Name, the easier it is to properly create. The alternative is to improperly create (which on a certain level) is the same as using God’s Name in vain. In failing to comprehend the 'something' that was seen under the throne, or even trying to comprehend it, all of those who were with Moshe, could see only that “under His feet was ‘something’ like a sapphire stone,” never really describing it or recognizing its significance. Something. In the works of our sages like Maimonides, the Maharal and the Vilna Gaon, we can see this mistake was later rectified (as demonstrated in posts like the Four Species and God's Name and The Eight Dimensions. In the works of Maimonides, in particular, we see this in his admonishment to the people as well, where it is explained that it is incumbent on us to (essentially follow in his footsteps and) describe what we see around us in terms of God's Name.

Tetzaveh

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 'vickers 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.

The underlying mathematic signature is everywhere in nature. 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.


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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).