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 multi-dimensional 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 hand-breadths and that a matrix with this ratio consists of 108 'elements' (cubic hand-breadths). 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 hand-breadths 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 hand-breadths. The portion consisting of 12 beams would thus be 108 hand-breadths and the remaining portion (which was the holy of holies) consisted of 8 beams, which would be 72 hand-breadths. 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 hand-breadths. The holy of holies, or inner portion, comprised 72 of these hand-breadths, and the outer portion comprised the remaining 108 hand-breadths.
Do
those numbers sound familiar? Of course they do! In terms of these
proportions, the 108 hand-breadths on the southern wall, opposite the 108
hand-breadths on the northern wall were just like the luchot with a volumetric measure of 108 cubic hand-breadths opposite 108 cubic hand-breadths. 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
hand-breadths on the southern wall opposite 72 hand-breadths on the
northern wall. However, since there are only 2 x 72 (hand-breadths) 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 hand-breadths in width (8 x 9 = 72 hand-breadths). There's our missing set! The total measure of the tabernacle where the ("golden" ark was present) was 3 x 72 or 216 hand-breadths. The total measurement (of 216 hand-breadths) 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) hand-breadths associated with the twin matrix, or in terms of its triad form, 72 x 3 (216) hand-breadths. 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.
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 hand-breadths opposite 180 hand-breadths) 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 hand-breadths. 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 hand-breadths (mentioned previously) its internal measure (less the thickness of the wall) was 60 hand-breadths by 60 hand-breadths by 60 hand-breadths (6 x 6 x 6 times 10-cubed) or 216,000 (cubic hand-breadths). 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 hand-breadths 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.
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.
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 hand-breadths)
Each side/face = 8 beams
24 beams = 3 sides/walls of 72 hand-breadths
- 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 hand-breadths 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 hand-breadths)
- 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 hand-breadths and the 12 beams are 108 hand-breadths (72 + 108 = 180 hand-breadths).
- 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 hand-breadths): the two luchot that were 2 x 108 cubic hand-breadths (216) or in their ratio of a single side-by-side matrix of 72 x 3 cubic hand-breadths (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.”
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