Sunday, December 30, 2018

Vaera and Bo

VaEra and Bo will be combined because they both deal with the plagues in Egypt. Within the sequence of these plagues, as well as their character and the specific language used to describe them, we find numbers and groups of numbers, or ratios, that have a direct correlation with the mathematics and geometry of the star tetrahedron, which of course casts a shadow known as a magen david. In this portion of the Torah we see Moshe use the staff that he pulled from the stone behind Yitro’s house to perform all kinds of wonders. But before we get into that, we should review some of the background information.

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 connect and to reveal where the 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.



In the meantime, for those already familiar with the constructs mentioned above, we'll continue with a small dissertation on Pharaoh, a man who could be easily described as "the great pretender." People thought he was “a god” and he did nothing to dispel that belief. In fact he exacerbated it. The problem (for Pharaoh) was that "a real god" would have no need of a bathroom. Only humans need such things! And so to perpetuate the illusion that he was a god, Pharaoh had a secret toilet installed in the palace.
That way he could relieve himself without anyone being the wiser. The problem with Pharaoh's ingenious solution, was that a rather nasty odor would drift through the palace from time-to-time. To make matters even worse, a rat managed to make its way through the pipes one day and bit him on the lower posterior.  All facts considered, Pharaoh decided to have the toilet removed. However, this created an even bigger problem, because now he would have to take morning walks along the Nile in order to relieve himself. There among the reeds he could do his business in private, without anyone noticing. Or so he thought! What he didn’t count on was Moshe interrupting this morning routine. God said to Moshe, “Go to Pharaoh in the morning. He goes out to the water. Stand where you will encounter him on the bank of the Nile.” How embarrassing! You can imagine Pharaoh’s surprise when Moshe interrupted his morning routine, caught him with his proverbial pants down, and exposed him as a mere man. Oh the shame!

The important thing to note, is that the warning for the first plague (where the water was turned to blood) was given outside Pharaoh's palace near the river.



The second plague was somewhat different. Moses went straight to Pharaoh's house in order to deliver the warning. In the case of the frogs, God said to Moshe “Come to Pharaoh [in his house]."




The third plague was also different. There was no warning given at all. In the case of this plague, God  said to Moshe “Say to Aaron, ‘stretch forth your staff and strike the dust of the earth.’ It will turn to lice in all the land of Egypt,” so Pharaoh had no warning that this plague was coming!


The circumstances for each plague create a pattern. In the case of the first plague, the warning was given outside the palace; for the second plague the warning was given inside the palace; and for the third plague there was no warning given at all.




The fourth, fifth and sixth plagues repeat this pattern. For the fourth plague (which consisted of wild beasts) God said to Moshe “Get up early in the morning and place yourself before Pharaoh. He will go out to the water.” So Moshe met him outside the palace. In the case of the fifth plague (the death of domestic animals which provided the food supply for the Egyptians) God said to Moshe “Come to Pharaoh [in his house]” and in the case of the sixth plague (boils) God instructed Moshe to “Take a handful of ashes from the furnace and throw it up into the air... it will settle like dust on... Egypt... when it falls on man or beast... it will cause... boils.” As in the case of the lice, Pharaoh had no warning that this plague was coming. In other words, these three plagues follow the same pattern as the first three plagues where the first warning was given outside the palace; the second warning was given inside the palace; and for the third plague there was no warning given at all. 




As you might suspect, this pattern continues through the next set of three plagues. The seventh plague was hail. God said to Moshe, “Get up early in the morning and stand before Pharaoh.” By this time Pharaoh was starting to see a pattern too, and so he had no intention of going to his regular place along the river. Instead he chose a new place. It was a secret place. He told no one of its location. Still, Moshe was there waiting for him. You can imagine Pharaoh’s surprise. The point of all this is to show that the pattern continues to repeat itself. The eighth plague was locusts. God said to Moshe “Come to Pharaoh [in his house].” The ninth plague was the same as the third in the previous two sets, in that no warning was given at all. God said to Moshe “Stretch out your hand on the sky and there was total darkness.” In this third set of three plagues, we see the same as in the previous two sets. First, Pharaoh is approached outside the palace. Second, he is confronted inside the palace, and then finally, for the third plague in the set, no warning is given.




We have 3 sets of 3 plagues that follow a pattern:

  1. a warning outside the palace;
  2. a warning inside the palace; and 
  3. no warning at all. 



                                    The Triangular Matrix of the First 9 Plagues



                               1st Plague              2nd plague            3rd plague

                                   Blood                       Frogs                     Lice

                          Warning Outside      Warning Inside         No warning


                               4th Plague              5th Plague             6th Plague


                              Wild Beasts       Domestic Animals            Boils

                         Warning Outside        Warning Inside         No warning



                               7th Plague              8th Plague             9th Plague

                                    Hail                       Locusts                Darkness

                        Warning Outside         Warning Inside        No warning


 


Interesting pattern! It was interesting enough for Rabbi Yehuda Lowe of Prague (the Maharal) to make note. The Maharal was one of the most mathematically inclined rabbis in history (although we won’t go into specifics).


You might think this pattern would end after the ninth plague, but in fact it continues. We read in the text pertaining to this plague (which is at the end of these 3 sets of 3 plagues) that it would “last for 3 days,” and that “no one left his place for 3 days” (notice these references are specific references to "time and space" (3 are related to time and 3 are related to space). The mention of the number 3 twice in succession is like the geometry of a two-dimensional magen david, however the 3 sets of 3 plagues expand this similarity. There are a total of four 3s found in these particular events, which are characteristic of a star tetrahedron (which has four pairs of opposing faces with 3 corners). The geometry is ultimately characterized by 72 triangular corners that correspond
to the ratio of the triplets, as they relate to the 216 letters in the Explicit Name (explained in posts like: Ki Thetze, and Mattoth). However, the plagues did not end with the ninth, and the worst was yet to come. Interestingly enough, the tenth plague had 3 components involving the death of the first born. This plague was completely unique. Not only was it the most severe (because it resulted in the death of all the first born in Egypt) but it was also the straw that broke the camel’s back. It was this tenth plague that  caused Pharaoh to let Israel go! It was by this plague that: "Every first-born in Egypt [would] die..."



It is worth emphasizing at this point, that the purpose of this blog is not to extol the aesthetic virtue of geometry, or polyhedrons, or any other geometric object including the star tetrahedron. Its purpose is to allow the individual to peer ever so slightly past the superficial  curtain that masks reality, to see the inner workings of the Architect. It is His Name, behind the scenes, that is the focus of this blog. This Name has a unique Signature that is easy to see in the mathematic properties of each set of plagues, and those properties reflect the geometry of the royal seal (star tetrahedron) where the numbers and sets of numbers conform to the nature of the letters in God’s Explicit Name. The magen david or shield of David, was not some image that looks like a six pointed star. David's shield was God, and God has a Name. The image of the six pointed star is merely a mathematic representation of this Name. We find the Signature, not only throughout Torah, but all of creation as well, including such things as physics; celestial dynamics; biology; quantum mechanics; and the history of man, both secular and biblical, to name a few.

Fractal geometry is now being used by scientists to define nature (as mentioned in the last post) and that geometry finds a natural partner in triad structures like 'the monsters,' or the Koch curve for those familiar with such things. In the last post we talked about one face of the star tetrahedron and how, when it was paired with its opposite, it created a two-dimensional magen david, that could be seen as an orthographic projection of its geometry. We also pointed out, that one of the iterations in a fractal transform was 2.16 (1/100 of the number of letters in the Triad Name). Now, in this parashah, we see this same pattern appearing in the characteristics of the plagues.
 

Sets of Numbers that Define Plague “Geometry”
Matching the Two-Dimensional Face of a Star Tetrahedron and the
Three-Dimensional Opposing Stellation (in the Middle)
 
Click to Enlarge


The sets of numbers corresponding to the plague are a clear reflection of a two-dimensional face (shown above) and its 3-dimensional counterpart (shown below) on a tetrahedron that has been merged with an opposing tetrahedron (and again this is one of 5 constructs having dual form found within the combined cubic form of the luchot). 


Notice that each face is divided into thirds, leaving a center portion. The center is actually the stellation of the opposing tetrahedron. The picture above is a two-dimensional representation of one face on one tetrahedron. When two tetrahedrons merge together in opposing directions, each triangular face ends up with 3 quadrants having 3 corners (small faces) that are themselves triangles. The sides are each half the length of the original and the triangle in the middle is pointed in the opposing direction. This upside-down triangle is not a face. It is the stellation (or point) on the opposing tetrahedron, that also happens to have 3 faces. However, these 3 faces are in the third dimension. The 3 triangular faces each have 3 corners and the direction of any given corner on any given faces corresponds to its counterpart on the other faces. In other words we see a distinct pattern, one that is identical to the pattern of numbers and sets of numbers describing the plagues. The ratio and geometry of each face corresponds to the 3 sets of 3 plagues with 3 types of warnings. The magen david (in three dimensions) is like a mathematic map of the events that were destined to occur at a specific moment in time. But surely all of this must be a coincidence, and what about the point/stellation in the middle, the center triangle?

We've already mentioned the tenth plague. It was unique, more severe than all the others. The Torah says of this plague, that it effected every firstborn in Egypt, but then specifies who the word “every” pertains to: 1) “the first born of Pharaoh;” 2) "the first born of the slave-girl;" and 3) “the first born of the animals.” There are 3 specific references as to which of the firstborn were affected, just as there are 3 corners to the center triangle (which define the 3 faces on the stellation of the opposing tetrahedron in the 3rd dimension). As the tenth plague was unique, so too is the center “triangle” or stellation. Not only is it in the center of the other three, and not only does it point in the opposite direction, but its 3 faces are in the third dimension.
 

Three-Dimensional Projection of Plague Geometry
Conforming to the Geometry of a Star Tetrahedron
(Demonstrating the Connection to the Signature of the Architect)

Click to Enlarge



The parallel between the numbers, or really groups of numbers and their ratio, one to another, on the face of the star tetrahedron as it relates to the characteristics of the events in Egypt is remarkable. The odds of this being a coincidence is beyond calculation. We say that since all things come from God, there is no such thing as a coincidence. Even the word for coincidence in Hebrew means essentially that it was written from the beginning.




This is just one example. When we peer through the prism of the magen david (looking at these events in terms of the mathematics of the star tetrahedron, or reflection of the 216 letters of the Triad Name) we can see, that much more clearly, the Signature of the Architect who planned it all from the very beginning. All of this is found in the internal geometry of the combined cubic form of the luchot, the “cosmic blueprint” from which the 304,805 letters (of Torah) emerge.

If the nations learn anything from these events concerning the plagues, it should be that it is impossible to thwart God’s plan. Any attempt to do so will only exacerbate the pre-existing problems facing any given nation. The perfect example is found in Pharaoh’s attempt to kill the first born, by which he ended up raising Moshe (his nemesis) in his own household.

Another prime example (that has yet to occur) is that God has already defined the future borders of Israel. They are specified in great detail. These borders are an inheritance. Any nation or group of nations that tries to deny Israel this inheritance should be as concerned for their well being as Pharaoh should have been for his own, and perhaps more so, since the track record of those who oppose God's plan is so clearly fraught with destruction.

There is another interesting connection to the luchot (and thus to God’s Name in their volumetric measure) in this portion of the Torah. The measure of the luchot was 6 x 6 x 6. This measure is also found in the description of Egypt’s borders, where the plagues actually had an ancillary benefit. For many years prior to the plagues, there were border skirmishes between Egypt and its neighbors. That is, there were arguments as to the exact location of the borders. The plagues settled the issue once and for all. Whenever there was a plague, like the blood in the rivers, lice in the ground, or hail from the sky, that plague would end abruptly at the border. The border therefore became officially defined in the minds of everyone concerned, and arguments as to its location were settled once and for all. 


In addition, the Egyptians were consummate mathematicians and surveyors as evidenced by the amazing math and geometry built into the pyramids. In their construction we see the measure of a Royal (or Antediluvian) Cubit; the Meter; Pi; Phi; and any number of other standards. Interestingly enough, even the speed of light is found in the measurements of one of these monoliths (an explanation is beyond the scope of this post). The word for Egypt in Hebrew is Mitzrayim. The root letters of the first two syllables are the Hebrew letters “Mem" and “Resh" (M and R). The Egyptians used a triangle known as MR in their measurements and this is why they were so accurate in their surveys of the land. The name for Egypt (MitzRayim) is not only the name of a specific individual who gave rise to the kingdom, but this individual's knowledge of mathematics and geometry and the triangle MR was derived from his name.




The point of all this is to say that when the Egyptians surveyed the borders after the plagues, and the land of Egypt was said to be 600 miles square (6 hundred x 6 hundred)1. We cannot be absolutely sure what this means, but even on modern maps it is roughly square. The southern and western borders are almost perfectly straight and the corner between the two is essentially a perfect 90 degree angle.




In ancient times, the northern and Eastern Borders were apparently much the same. The description is interesting, both in terms of the numbers and the shape. In Isaiah 19:19 reference is made to a monument that is “on the border of Egypt” but also “in the midst thereof.” How can something be on the border and yet in the middle at the same time? Ancient Egypt was actually split into two kingdoms. There was a northern kingdom and there was a southern kingdom. The monument lay in the middle between the two, thus it was both on the border (of Egypt) “and in the midst thereof” at the same time. When you divide a square in half, like a square of 6 hundred miles, you get two rectangular polygons that are each 6 hundred by 3 hundred (the flattened two-dimensional ratio for each of the two luchot). The 6 x 6 (hundred) “measure” for Egypt (with a border in the middle) dividing between the two kingdoms (halves) is like the first two dimensions of the luchot (6 x 6 handbreadths) where the measure of each half in the third dimension is 3 handbreadths. It’s as if the numbers and sets of numbers are alluding to the fact that Israel would need to leave Egypt before they could receive the luchot and return to the Land of Israel where their knowledge of God, and their connection to God could rise to the next level (a higher dimension).


The ratio of 6 hundred x 3 hundred times 2 kingdoms, is two-dimensionally identical to the ratio of the shnei luchot (2 tablets) that Israel would shortly be receiving at Sinai. Whether there is a connection, or not, it is difficult to say. We only note the similarity because when the ratio of the two-dimensional version is extruded into the third dimension, it has the same ratio as the luchot that were 6 handbreadths x 6 handbreadths x 3 handbreadths, times 2, and where the total volumetric measure corresponds to the number of letters in God’s Explicit Name.

  
Mathematic Relationship Between the
Land of Egypt and the Shnei Luchot
(Two (Twin) Tablets)

Click to Enlarge


The Signature of the Architect is indeed everywhere. Inside of Israel it is much easier to see, just as something that is three-dimensional has more facets than its two-dimensional counterpart, a mere shadow of the original.

The names of the 3 portions of Torah leading up to the point in time when Moshe and company see the undivided form of the luchot (in Mishpatim) and its subsequent division when Moshe brings the two tablets down, suggest that the luchot are about to be given, as well as an understanding (revealing) of God’s Name. Last week's parashah Shemoth (Exodus in English) means “Names” in Hebrew. VaEra means “I revealed myself” and “Bo” means “come,” as if to say “when you leave Egypt and come to Mt. Sinai I [will] reveal my Name to you.” In fact, that's exactly what happens in the next post on Parashah BeShalach. It is in that parashah that we find the 216 letters of His Name as a contiguous stream of letters in the text (not permuted into triplets) but one-dimensionally separated into 3 segments of 72 letters, corresponding to the side-by-side form of the luchot, that is 6 in the first dimension, 12 in the second and 3 in the third (6 x 12 = 72 times 3 = 216). In other words, even though it is not (yet) permuted as is known (still one-dimensional in form) its numbers or sets of numbers, and the ratio between sets, preserves the triad structure in the 3 segments (the triad nature is more accurately perceived in two or three dimensions) as a function of the luchot's three dimensions.


Stay tuned for the next post which will be on BeShalach, where we will examine the implications in more detail.


________
Footnote 1 - It is interesting to note that the area of a 6 hundred mile square (360,000 square miles) is a multiple of 72 and in square feet, it is a multiple of 216 which would suggest this measurement has a connection not only to the Triad Havaya of 72 with 216 letters, but also to the imperial system of measurement. When you divide the 360,000 square miles by 72, the result is 5 hundred, which reflects a primary characteristic (in fact the primary characteristic) for the geometry in the combined cubic form of the luchot with its 5 internal mathematic constructs, one of which is the tetrahedron.   

Saturday, December 22, 2018

Shemoth

In Parashah Shemoth, the mathematic characteristics of the letters in the Explicit Name are concealed in the marriage, when Yitro: “...gave his daughter Tzipporah to Moshe." The circumstances surrounding this particular marriage are among the most fascinating, however the details are not found in the written Torah. We find mention of them in other sources, some of which are cited below. As you might suspect, the connection with God’s Name is in the details, so with that in mind we will reconstruct some of these details in their historical context, starting with Yitro, Moshe and Pharaoh.

Yitro and Moshe had much in common. Just as Moshe was close to Pharaoh, so too was Yitro. Yitro was one of Pharaoh’s closest advisors. Moshe was an advisor in that he was a member of the family. Just as Moshe was eventually forced to flee Pharaoh’s wrath to avoid retribution, so too was Yitro. But we’re getting ahead of the story!




Before Yitro fled from Egypt, he was actually a friend of Pharaoh (or at least as much of a friend as anyone could be, to the ruler of Egypt). Pharaoh would often walk with Yitro and they would discuss things of importance. One day they were walking through the marketplace, and Yitro saw two young girls who were for sale. The girls were twin sisters and remarkably beautiful, so beautiful in fact that Yitro suggested they purchase them. Pharaoh would take one of the girls, and Yitro the other (Midrash Talpios, Bithia / Zohar). The girls however, were not purchased as slaves. Rather, they were adopted and made daughters.


One of these girls was named Batya (Bithia) and she became the “daughter” of Pharaoh. The other girl was Tzipporah who became the "daughter” of Yitro. You may remember that it was Batya who would latter retrieve the reed boat from the river (with Moshe) after it was set adrift to save him from Pharaoh’s decree concerning the death of the first-born males. Batya would later marry Calev, and Tzipporah would later marry Moshe. Keep in mind that Moshe was with Batya in Pharaoh’s house. Since Batya and Tzipporah were twins they looked  alike. So naturally, when Moshe was forced to flee from Egypt and gets to Midian, he is surprised when he stumbles upon Yitro’s “daughter,” the young woman who is the twin sister of the “mother” who raised him. This is why the Midrash says that “Moshe married his mother.” It is not referring to Yocheved (his biological mother) or even Batya (his foster mother) but to Tzipporah who was Batya’s twin sister. But again we’re getting ahead of the story!




Since Tzipporah was not Yitro’s biological daughter, and because of her beauty, Yitro was evidently not anxious to see her given away in marriage. He therefore placed a condition upon anyone wishing to marry her. That condition is a story in-and-of itself, and it is here that we find a connection to God’s Name. You see, when Yitro fled the wrath of Pharaoh, Tzipporah was not the only thing that he took. Pharaoh had an amazing staff (sceptre) that he “inherited” after the death of Yoseph (a sceptre being one of the symbols of kingship). As the story goes, this ScePteR (of SaPhiRe) was the very same one that God had given to Adam. This scepter and a certain mantle were given to Adam in the 216th year of the first millennium as explained in the post on Bereishit and Noach, and so we see our first connection to the 216 letters of the Explicit Name. These items were the symbols of kingship (of the shepherding kind) that would be passed down to certain descendants (a facet of the birthright) until it finally reached Avraham, Yitzhak, Yaacov and eventually Yoseph.
The status of royalty (and scepter) went to Yehuda. The sceptre is mentioned in Genesis 49:10, which says that “the sceptre shall not depart from Yehuda, nor the ruler’s staff from between his legs.” The birthright however went to Yoseph (I Chronicles 5:1) and "the garments of Adam were [also] given to Yoseph" i.e. the mantle (coat) given to him by his father Yaacov (Bereishit Rabbah, Targum Yerushalami).   

As mentioned above, Adam received this mantle and scepter in the 216th year of the first millennium. He passed it to Seth (the gilgul of Hevel) after he had walked the earth as a man for 216 years, as his father before him (again see the post on Bereishit and Noach for details). Its counterpart (the royal mantle) was given to Yoseph in the 216th year of the third millennium). Each event corresponds to the number of letters of the Explicit Name, and both demonstrate the significance of its possession.



It was therefore perfectly natural for Pharaoh to covet this sceptre and the title that it implied (assuming he understood its significance). So too did Yitro, but apparently for different reasons (since it was made of sapphire) which is why he stole it from Pharaoh before fleeing Egypt. 


When Yitro arrives in Midian and settles there, he uses his powers of occult to plunge the staff into the bedrock behind his house. There it would remain until one was found worthy to pull it from the stone, someone who would unite his people and lead them to freedom; and someone worthy of marrying his daughter Tzipporah. If that sounds familiar, it should. It is the premise of the Arthurian legend (King Arthur of mythic renown). Indeed the story of Moshe was taken into the diaspora by Jews who were fleeing roman persecution. Many of these people ended up in the British Isles (the origin of the term brit-ish, is itself found in the Hebrew words for covenant (brit) and man (ish). The story of how Moshe came to remove the real “sword in the stone” is rather interesting. There were many men who desired Tzipporah, but none were able to pull the staff from the ground. One day after Moshe arrived, he was walking through the garden and saw this unusual staff. There were strange marks on it which tradition says were the letters of a mystical Name of God. The Maharal says that it consisted of 10 letters corresponding to the 10 sayings of creation and that the [polar] opposite of each letter was a letter pertaining to one of the 10 plagues that Moshe would see when he returned to Egypt to free the people from bondage. But once again we’re getting ahead of the story!



When Moshe grasped the scepter, the power preventing it from being removed from the ground by anyone but God’s intended, immediately yielded to the one who was destined to possess it. Tzipporah married Moshe and the rest is history (except of course the real identity of the mythical King Arthur in British mythology). The above stories are loaded with allusions that one could contemplate without end. If you remember anything from this post, remember that the scepter—which was the symbol of kingship not only for Israel, but of all mankind—the scepter, with it's inherent connection to the 216 letters (as detailed in the post on Bereishit and Noach) —the scepter, that was passed from king to king before and after the flood—was miraculously pulled from the stone only by Moshe, who was the intended leader of Israel. Also remember that the star tetrahedron is a mathematic or geometric representation of the merging of male and female, which is a form of marriage like the marriage of Tzipporah and Moshe.

Now, about that sceptre....

 It is said that the sceptre (or ruler’s staff) was created on Friday evening (the 6th day) on the twilight of creation, like the measure for any given dimension of the combined cubic form of the luchot (6 handbreadths). Adam received the sceptre while he was in a garden (the Garden of Eden) just as Moshe retrieved it from a garden (at Yitro’s). When Adam died it was given to Enoch and after Enoch, it was given to Noah’s son Shem (who brought it aboard the ark). Shem gave it to Avraham and Avraham to Yitzhak. It was eventually given to Yaacov who brought it to Egypt during the famine (in the Hebrew year 2238). Yaacov bequeathed it to Yoseph but when Yoseph died it became the property of Pharaoh, who acquired all of Yoseph’s belongings (Shemoth 3, 2:21 Me’Am Lo’Ez pg. 67).

 


It is said the Yitro desired it because it was made of sapphire. Sapphire? In Hebrew, the word is “sanphirinon” (Shemoth Rabbah 8:3) spelled סנפירינון (samech-nun-peh-yud-resh-yud-nun-vav-nun sophit) which comes from the greek word σαπφειρινον meaning “sapphire-like.” 



Sound familiar? It should. The substance is the same as that of the block that Moshe saw under the throne of God (in Mishpatim) which was later divided and brought down as the luchot (6 handbreadths x 6 handbreadths x 3 handbreadths times 2 tablets = 216 or the same number of elements in the matrix as there are letters in the Triad Name). It is the substance with a clarity of “72” having a trigonal (three-sided) crystalline structure that alludes to the Signature of the Architect (see the post on the nature of sapphire).




There are people who would like to know where this scepter is today. However, I very much suspect that it will not be found at all. Rather, it will seek out the individual to whom it belongs at the proper time, much like the anointing oil that fled from Ishai’s sons, but flowed toward David HaMelech.




In the meantime, since this blog is about the mathematic and geometric Signature of God’s Name, it is at this point worth mentioning something that is referred to as “the emblem of fractal geometry,” which some mathematician’s have referred to as “the Thumbprint of God.” It is the "end result" (assuming it actually ends, which it does not) of an applied formula pertaining to a form of geometry that seems to exist somewhere between dimensions. What is unique about this form of geometry is that its most basic component is a "triplet," best represented by a broken line and fixed by two identical additional lines. However, it is also represented in triangular form, for example, in the Sierpinski fractal mentioned in last week’s post. All of this might sound cryptic, but what it means is that no matter how infinitely large or small (the number of iterations) may be, they can be represented by the same two-dimensional elements. Those elements are the components of a magen david, which is a shadow of the star tetrahedron (the three-dimensional construct to which we’ve referred so many times, because of its connection to the letters of the Triad Name.) One of the best examples of fractal geometry in action is shown below. This one is referred to as the Koch Snowflake and the example is animated to the 7th iteration.




What this amounts to is the fact that mathematicians all over the world have been staring at this concept for many years without ever realizing its underlying significance, or that the geometry has a connection to the letters of God’s Name. Certain curves in fractal geometry are a prime example. The 20 segment curve where DS = 2.160 (to the 2nd decimal place) and the step in the generator is minimal (k = 2) is shown below. This equates to a 2-dimensional orthographic projection of 1 side (of 8 sides) on the star tetrahedron, that subdivides into 72 triangular corners. No one seems to have noticed the connection to the royal seal (magen david) or the 216 letters of God's Explicit Name.



                                                                           k=2
The large upside-down triangle in the middle
is the two-dimensional shadow of
the stellation of the opposing tetrahedron

A mathematician by the name of Benoit Mandelbrot (Yale University) (who revealed the M-set) is really the father of fractal geometry. It was he who coined the term fractal. Mandelbrot was a Jew who grew up in Nazi occupied France. He spent 4 years eluding capture and deportation to the camps. When he was young he was fascinated by visual geometry. It was from this fascination that fractal geometry emerged.

Fractal geometry is actually a whole new kind of mathematics and it makes some traditional mathematicians a bit nervous. Unlike traditional math that defines smooth curves and comfortable shapes, fractal geometry can define everything in the universe from the craggy projections of mountains below to the movement of the clouds above, in fact literally everything. If you think the mathematic or scientific community is unaware of this, think again. It has created a firestorm of controversy! For more information on fractal geometry and its significance, watch Nova Season 36, Episode 4, entitled "Hunting the Hidden Dimension."

Meanwhile, in consideration of all the above (and in the interest of curiosity) let’s advance a hypothetical description of the sapphire sceptre. It is for the most part a story or fable; a parable of sorts (for purpose of instruction) the kind of tale that one might tell to children (and mathematicians) about what the sceptre might look like, and those who were destined to inherit it. 

Like all stories, this one starts at the beginning and so we’ll start with the usual: “Once upon a time,” and add that there was a king who wished to give his son the keys to the kingdom. Along with that birthright, he would provide his son (and his son’s descendants) with the symbols of kingship which included a sceptre and a royal mantle. The king wished to hide his signature within these things so that if the identity of the rightful heir was ever challenged, the signature could be revealed and remove all doubt. And so he created a scepter, a staff by which his descendants would be known, a staff that a shepherd might use to protect his flock. 

The king in this fairytale was much more than just a king. He was also the mathematician of mathematicians, and so he decided to place his signature upon the staff in these terms, so they would match the numbers that not only reflect  the letters of his name, but the geometry of the royal seal and the official documents and records of the kingdom.

To that end, he made a staff with 3 sides, like a triangle (a 2-dimensional shadow of a tetrahedron) where the length of each side was 3 “fingers” (3/5th’s of a hand of 5, and fractionally the same as 6/10ths, or ‘.6’ of one hand, which happens to also be a static measurement of 5.236 centimeters, if one bothers to put his fingers to a calculator).

He would make the length of each side equal to 24 handbreadths. That way, the combined length of the 3 sides would be 72 handbreadths (alluding to his name). However, 72 handbreadths also just happens to be 2 x π (Pi) in meters which is one of the fundamental aspects of mathematics.

The 24 handbreadths of each side (that are each 3 fingers in length) therefore end up being .666 x π (Pi) in meters (a little more than the height of a man). The 3 fingers and the .666, together would also bring to mind the 3sixes” describing the measure and the ratio of the king's blueprint (6 x 6 x 6) that would be used to define the kingdom.
 

6 x 6 x 6 Handbreadths


A handbreadth is 1/6 of a royal or sacred cubit. It is 1/6th of 1/6th of π (Pi) meters. That's 8.72666 centimeters (there are 6 of these handbreadths in a cubit). Please note the sequence of integers in the fractional portion: 8.72666. Thus, the length of 3 “fingers,” would be 5.236 centimeters which coincidentally happens to be 10 times the ratio of a cubit to a meter (which is .5236) as defined within the primordial torah (luchot) as the ratio between the volume of the sphere within, and the cube without (that ratio being well known in mathematics). Seriously? Must be a coincidence!

You may already understand the meaning of this fairytale. The king is God and the kingdom is the physical universe (creation). The blueprint for the kingdom is the combined cubic form of the luchot (the principle having been adequately demonstrated in the post on the primordial torah).
 
The combined cubic form of the luchot was coincidentally .5236 meters per side (6 handbreadths = .5326 meters). When it was divided and given to Israel it would have 10 commandments (just as the measurement of any given side of the sceptre would be a factor of 10 in relation to the aforementioned ratio). All of the numbers for our hypothetical sceptre allude to the King’s system of measurement, by which all things are calculated and compared, both physical and non-physical.


The 3 original or primary sides that are each 3 fingers in length (and 24 handbreadths in length) further allude to the King’s Signature (in the geometry of the star tetrahedron) since it has 24 small triangular faces that ultimately define (3 x 24 or) 72 triangular corners (72 x 3 = 216) as depicted in one of the figures above dealing with fractal geometry. The Signature would thus be hidden from the casual observer.

Click to Enlarge

 
 
You may  remember that in the narrative of our story above, we said that when “the king” designed the scepter, he started with a triangle of 3 sides. In the picture above, you can see what happens when the fractal set previously mentioned is applied to just the third iteration. There are many possibilities. I like this one because it so eloquently reveals the mathematic and geometric connection between the Name, the royal seal (magen david) and the luchot.

Needless to say, the above story is just a story, except for the mathematic and geometric correlations that do in fact connect to God’s Name; and the substance of the sceptre actually being like-sapphire, which it is; and the fact that the length of 3 “fingers” is exactly 10 x .5236 centimeters; or that the ratio of those 3 fingers to a hand (with its 10 fingers) is 6/10ths (.6) and also 1/6th of the ratio of a cubit to a meter (.5236); and that the sapphire cube that is 6 x 6 x 6 handbreadths really is a cube of 1 cubit, or .5236 meters; or that the 10 commandments that were part of the divided cube (luchot) were “a measure” by which man is judged; all of which is factual. 


Nevertheless, the description in our story is just hypothetical. It is full of coincidences pertaining to actual facts and thus hypothetical. Whether it is in anyway true, is anyone’s guess. It is posited here, because it is perfectly acceptable to draw logical inferences from known facts. And besides, it made for a good story and served as the vehicle for revealing how certain fundamental numbers and ratios really do correspond to God’s Name and the many things mentioned in the Torah!

_______

Author's Note: The scepter that is modeled above (shown without the head so that the facets of the staff can be examined) is 6 handbreadths in length (like each dimension of the evan shetiyah). It has 48 external facets with a total length to the 288 handbreadths, like the "288 sparks" (explained by the ARI) that equate to the shattering of the vessels. The total length of these facets is equal to the number of square handbreadths on the surface of the luchot (288 square handbreadths). The number 288 is the sum of 216 (the number of letters in God's Name) and 72 (which is the number of triplets pertaining to this Triad Name). The head of the scepter (not shown) is a sapphire star tetrahedron.

VaYechi

Once upon a time there was a diamond cutter who was an expert in the cutting of various gemstones. He was asked to cut a certain unique stone that was made of sapphire, into two halves. It happened that this particular stone was a cube and the measure of each side was 6 handbreadths. Before cutting the stone, he studied it to ensure that the division was made according to his employer’s request. The cutter made a mental note of the cube’s 6 faces, its 12 edges and 8 corners, and realized that its geometry was defined by these 26 elements, a number that just happened to be the gematria of his employer’s name.



The cutter was told that when the division was made, each half of this stone should be identical in size and shape (6 x 6 x 3 handbreadths). The time had not yet come to make the division, but still, he contemplated how to make the cut by envisioning a dotted line down the middle. The cutter noticed that there were 5 faces on each half of what he would cut1. He then counted the edges and noticed that there were 8 on each half, of what he would be dividing. And finally, he noted the 4 corners on each half (before division). When he added them together together, there were 17 elements defining the characteristics of each half.



This particular stone was like no other the cutter had ever seen and so he was very careful in dividing it, so as to ensure the proper result. The twin stones were to be given to the employer’s family. 

An explanation of the story will continue in a moment. In the meantime, if you are new to this blog then you should read each of the essential constructs listed in the column to the right before proceeding. These pages explain some of the basic connections between the letters of God's Explicit Name and the multidimensional mathematic and geometric characteristics of the luchot (tablets of the covenant) 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 that creates a unique signature. The purpose of this blog is to show how they connect with one another and to reveal where this Signature of the Architect is hidden, not only in each week’s parashah, but the physics of time and space, quantum mechanics and celestial dynamics to name a few. Without the background provided by these essential constructs, the posts will make little sense.


Combined Cubic Form of the Luchot
(its geometry before division)

Click to Enlarge


17 Elements                 -               17 Elements
(5 Faces, 8 Lines, 4 corners)         -       (5 Faces, 8 Lines, 4 corners)

The actual story is of course well known. The sapphire block (in Mishpatim) was divided and brought down by Moshe as the luchot. Moshe spent 6 days in the cloud, and for 6 days he brought [the letters of] Torah to the people. It was the 6th of Sivan. The description of the event alludes to the measure of each side and the ratio of the stone that was to be divided (6 x 6 x 6 handbreadths). 



The measurements of a cube are all equal. Its edges, points and faces are defined by the characteristics of it 26 elements. However, after you divide a cube in half, each half still has 26 elements (although not all of the edges or faces remain equal). 

The 26 elements of a cube (as in the combined cubic form of the luchot) are like the gematria of the Tetragrammaton which is 26. The Tetragrammaton is the root of the Triad Havayah of 72. God is One and Indivisible, and so the division of a cube, and the combined cubic form of the luchot, specifically, into two halves, does not in any way effect the number of elements (the essence) that defines it. It merely establishes "duality," as in the dual (male ~ female) nature of reality, that is well known in physics.




From this 'blueprint' came forth 10 sephirot and 22 Hebrew letters, which were combined and permuted until all 304,805 letters [of the written Torah] had been received. 


In addition to the 26 basic geometric elements that define the outer measure of the luchot (6 x 6 x 6 handbreadths) the volumetric measure of the combined luchot was 216 cubic handbreadths. After division, there were 108 cubic handbreadths (elements) in each half of the matrix, each half being 6 x 6 x 3 handbreadths (6 x 6 x 3 times 2 = 216).

 Keep in mind that the gematria for the word "half" in Hebrew is 108.

These sapphire twins were given to Israel, but hidden within their mathematic and geometric characteristics, is the Signature of the Architect (the "employer" who ordered the cutting of the stone). The combined cubic form of the sapphire block was/is the cosmic blueprint for the created universe. It should be noted that at this moment in time (as the historical account in VaYechi takes place) the luchot had not yet been divided but were “soon” to be given (along with many secrets) and entrusted to Israel who would be charged with the duty of preserving its specifications (fulfillment of halacha).


Thus...

When it came time to give birth...”
“...there were twins in her womb.”



Some reflections are of course brighter than others. There are those that are buried so deeply, that they could easily be considered coincidences were it not for their place and position within the scheme of things. The date of an event, is a good example. Sometimes they are very bright and the significance is rather obvious. At other times they are not.

The blueprint does cast its shadow into the physical universe below, that is undergoing the process of creation. One of these shadows appears in the year 2199, when “Rachel [gave] birth to Yoseph” (VaYetze). The number 21, for example, has special significance throughout the Torah. It is factored by three 7s. The number 99 is also significant and can be factored as three 33s. Ordinarily, something like this would just be a coincidence. However, we've seen these same numbers many times whenever key events are concerned (the three 7s for example were a prominent mathematic feature in Parashah Miketz where three 7s were repeated four times, and the number 7 itself, a total of 28 times). These three 7s are a function of the 7 points of intersection in each of three dimensions, that are hidden between each measured handbreadths of the evan shetiyah (foundation stone). However, it is also found in its internal tetrahedral geometry, or what we call "the royal seal," (the star tetrahedron (three-dimensional magen david) that is mathematically at the heart of the foundation stone. This tetrahedral geometry has three small faces on any given side, with 3 triangular (3-cornered) facets. The three 7s (associated with the 28 elements in the tetrahedral geometry of the luchot) define each edge of a triangular matrix of 7 and 'create' the face of a tetrahedron defined by 72 three-cornered elements (as shown in the post on Miketz) all alluding to the signature geometry of the Architect (with 3 x 72 or 216 letters). The three 7s also allude to the movement of time (measured in cycles of 7) as it relates to the three dimensions of space. This can be seen in the cubic matrix of 7 that is defined by our calendar related halachic observance (7 days, 7 x 7 days, 7 weeks, 7 years, 7 x 7 years etc). The points of intersection between the measured handbreaths (7 x 7 x 7) are a function of the 42 letter Name, and each measured handbreadths is a function of the 216 letter Name. For more on this geometry and these unusual sets of numbers see the post on BeHar and Bechuko-thai).

The diagram below shows any one of the two-dimensional surfaces on a star tetrahedron. The grey triangle in the middle is the stellation of the opposing tetrahedron which is "three-dimensional" with respect to the smaller, two-dimensional faces surrounding it. Each of those small faces in turn has 3 triangular corners, so the end result is 3 within 3, within 3 or three 3s (3,3,3). The final 3 is inevitably hidden within the fourth dimension and can only be perceived over time (the three white triangles in the middle are a fractal representation of the fourth dimension with respect to its two and three-dimensional counterparts1

This pattern of 3 things within 3 things with 3 more things is prevalent throughout Torah and in many mitzvoth.  On a practical level, the current model of string theory posits a hologram with nine dimensions of space and one dimension of time (three specific 3s plus one). In the Torah, we see this aspect of the blueprint or combined cubic form of the luchot (three specific 3s) in such things as the plagues of Egypt, where there were 3 types of warnings in 3 places (nine plagues) along with 1 additional plague, corresponding to the 10th plague that was totally unique (as is known). Each is a reflection of the geometry in the blueprint, which is in turn a reflection of the letters of the Triad Name of 72, which has a total of 72 (nine x 8) 3s.


Three (Corners) within Three (Triangles) within Three (More Triangles)
Plus One Different from the Others


The Geometry of any Two-Dimensional Face on the Royal Seal2
(...is One "Face" of a Three-Dimensional Magen David)
(The Grey Triangle is the Stellation of the Opposing Tetrahedron in the 3rd Dimension)
(The 'Blueprint' or Luchot Defines 'the 5' Reconciled by 'Dual Resonance' and the 9 Plus 1 of String Theory)

 
In addition to what is mentioned above, the number of polygonal elements in the first 6 layers of a tetrahedral matrix of 7 is 91 (shown/diagrammed in the post on Miketz). Yaacov just happened to be 91 years old when “Rachel [gave] birth to Yoseph” (one of 'two children' and a geometric reflection of the original moment of creation). It also happens to be the sum of the gematria of the two primary abbreviated forms of the Explicit Name: the Tetragrammaton (Yud-Key-Vav-Key 26) and the Havaya of Adnoot (Aleph-Dalet-Nun-Yud 65) where 26 + 65 = 91. Are these numbers and patterns a coincidence? Or, are they evidence of a sophisticated design (seen here in the historical narrative) that conforms to the internal geometry of a cosmic blueprint of divine origin (the shnei luchot)? We should look for additional witnesses.

Yoseph grows to be a man and in the year 2216 another shadow appears when... Yoseph “reaches the age of 17,” so Yaacov “assigns to Yoseph [the royal] garment” (VaYishlach). Within days, he is “sold” and becomes a slave. The years of happiness in which Yaacov was together with Yoseph (17 years) come to an abrupt end, but another shadow is cast. A period of 22 years begins, years in which they are separated. During this time, Yoseph becomes ruler of Egypt and a famine engulfs the land. Yoseph’s brothers come to Egypt and are eventually reunited with Yoseph. In the year 2238 they bring their father Yaacov to Egypt and at long last, Yaacov and Yoseph are once again together. Yaacov says to Yoseph “I have seen your face and you are still alive,” revealing yet a third shadow, a period of time that will last for another period
(an opposing period) of 17 years (years in which they are together again as opposed to being apart from one another). Yaacov says to Pharaoh, “My journey through life has lasted 130 years” (VaYigash) and then later in this weeks parashah Yaacov.... breath[s] his last having lived to be 147 (147 - 130 =  the 17 years they were together again).

With Yaacov’s death, the stage is set for the eventual exodus from Egypt and the events at Mt. Sinai where Moshe goes up, sees the sapphire block, it is divided in 2 and given to Israel as the luchot. These 2 luchot contain the plans and specifications for the events herein described (and quite literally everything else). The 17 elements on the 2 halves of the cube (before division) cast their shadows below in the events that transpire, and so we see the 2 periods of 17 years in which Yaacov and Yoseph are together (as when the luchot were in combined cubic form) separated by 22 years (like the 22 letters that ultimately emerge from the primordial form of the Torah). The 22 years are a reflection of the 22 letters (22 primordial forms of energy) that were combined and permuted to create the universe (a limited subset take form in the 304,805 letters of the written Torah). These geometric aspects of the primordial torah cast their shadows and so we see them manifested as Yaacov and Yoseph separated for 22 years, like the letters that came forth when the two halves of the luchot were separated from one another (divided and brought down). 
The lower 'events' are the mirror image of the upper/higher reality.

Are these numbers, sets of numbers and patterns merely a coincidence? Taken individually there is certainly that possibility, but considering that opposing periods of time (that conform to very specific 'opposing' aspects of the internal or external elements of cubic geometry and the measured geometry of one stone in particular) continue to repeat themselves throughout the Torah, the interconnected nature of the various elements suggest otherwise. The history of mankind reflects the mathematic characteristics of the blueprint either way, and the mathematic and geometric characteristics of the Name hidden beneath the surface, that we see in so many mathematic and geometric forms, that all bear witness to the Signature of the Architect who so carefully designed everything from the beginning. 


There are additional shadows in this parashah that allude to the Triad Name, for example: “Yoseph died at the age of 110” having had 10 years subtracted from his life for the 5 things said against his father Yaacov and translated (5 times) without rebuttal (5 opposite 5). This has already been explained in other posts and will not be repeated here.

_____

Footnote 1 - With 5 faces on each 'half,' there are 10 halves that are '5 opposite 5'. This is an external reflection of the internal geometry within the combined cubic form of the luchot (5 mathematic constructs opposite 5 counterparts, or more specifically the geometry giving rise to the platonic solids, and well known in mathematics).

Footnote 2 - The concept being integral to the third iteration of what is called the Sierpinski Fractal (or Sierpinski Triangle) a two-dimensional representation of higher dimensionality.

Sunday, December 9, 2018

Vayigash

This week’s parashah has a beautiful geometric reflection of the letters in God’s Explicit Name, hidden in at least three places, beginning with the text that tells us, ‘...these are the sons of Yaacov's wife Rachel."

If you are new to this blog then you should read each of the essential constructs listed in the column to the right before proceeding. These pages explain some of the basic connections between the letters of God's Explicit Name and the multidimensional mathematic and geometric characteristics of the luchot (tablets of the covenant) 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 that creates a unique signature. The purpose of this blog is to show how they connect with one another and to reveal where this Signature of the Architect is hidden, not only in each week’s parashah, but the physics of time and space, quantum mechanics and celestial dynamics to name a few. Without the background provided by these essential constructs, the posts will make little sense.

Regarding the first instance of the three mentioned above, this Name is hidden within the details of the phrase referring to: “the sons of Yaacov’s wife Rachel,” and its counterpart: “all the souls coming to Egypt... not counting the wives of Yaacov's sons.” 

In this segment of the text, Rachel is specifically identified as Yaacov’s wife. In the case of the other three matriarchs, Bilhah, Zilpah and Leah, the Torah merely says “these are the sons of...” without identifying them as Yaacov’s wives. Rashi finds this noteworthy, saying that Rachel is mentioned singularly as the wife because she was mistress of the household. However, there is another explanation. The ratio and the numbers are a faint reflection of the same math and geometry that appeared in last week's parashah. 

Last week, Pharaoh’s dream encompassed a 14 year period, one in which Yoseph was not only a ruler but the most powerful man in the known world. This 14 year period of time was directly mentioned in the text, while the preceding 14 year period (starting when he was sold a slave) was not directly mentioned. The numbers, the ratio and the nature of the elements all correlate with the male-female polarity of the royal seal or magen david (with 14 elements in one half, opposite 14 elements in the other half) and the corresponding male-female letters of the Name (associated with each tetrahedral point) that sustain a physical universe in like form.

The time in which Yoseph was a slave had to be calculated by subtracting the year in which the “years of plenty” started, from the time when Yoseph was sold. These 2 periods of 14 years stood opposite one another, and also in stark contrast to one another, alluding to the Name of the Architect found in the geometry of the royal seal (three-dimensional magen david) as the 14 elements of the male and 14 elements of the female halves of a star tetrahedron. This in turn is a reflection of the mathematic nature of the Triad Name (recalling once again a tradition that the luchot were “tetrahedrons” and knowing that their combination in the form of a star tetrahedron is one of 5 internal mathematic constructs found with the combined cubic form of the luchot). 

The marriage of the abbreviated forms of this Name can also be seen in the narrative where 1 wife (Rachel) was mentioned by name in the text. She was "different" from the other 3 (Bilhah, Zilpah and Leah) who were not mentioned. This is a reflection of the ratio between the letters of the 4 "couples" that are married together (in unification) or more specifically the Tetragrammaton and those in the Havaya of Adnoot, as explained in the post on Parashah Ki Thetze and also the Marriage of the Letters. It is also a characteristic in a primary geometric component of the royal seal, specifically the tetrahedron, which has 3 points in one (lower) dimension (corresponding to Leah, Bilhah and Zilpah) and 1 point in the dimension above (Rachel) specifically identified as Yaacov's "wife" in the text. This same geometry is observed throughout the Torah, one example being the 4 strands of tzitzith that are doubled over: 3 white and 1 techelet. Each is a facet that is a characteristic of the blueprint (luchot) which is in turn a facet of the Name and its abbreviations.

In this week's parashah we see the same tetrahedral geometry and the same reflection of the male and female halves, where Rachel was the matriarch with 14 male descendants, hence the specific reference to her as Yaacov's wife. As with the previous reflection, these 14 souls are also mentioned directly in the text “...these are the sons of Rachel... in all 14 souls.” The 14 daughters of Yaacov however, are not mentioned directly, but alluded to where it refers to the wives who were not counted (each son by tradition having a twin sister at birth). The combination of male-female “elements” (in the 14 sons mentioned, and the 14 daughters not mentioned) is a very bright reflection of the tetrahedral geometry in the royal seal. These were the daughters of Yaacov, 14 in all (the female) counting Dinah and the triplets of Rachel (2 of which were daughters). In other words 14 male opposite 14 female. So once again we have an allusion to the internal geometry of the luchot (tetrahedrons) in this particular form. For more information on this one-dimensional mathematic expression and its three-dimensional counterpart (which are both reflections of the royal seal and the letters in the Name) see last week's post on Miketz 
and also the post on Emor.

In the second instance, the Triad Name is seen as a mathematic reflection of the words Israel (Yaacov) spoke to Yoseph “Now I can die, I have already seen your face, for you are still alive.” An explanation of this reference will have to wait until next week’s parashah, when this particular geometric reflection culminates in the events and the correlation is complete (in VaYechi). 


The third and final instance that we will mention, is one of the more significant in the entire Torah where: “all the souls coming to Egypt with Yaacov... were 66,” together with “all the souls of Yaacov’s house coming to Egypt were 70.” This would seem to be a contradiction, except for the fact that 4 souls were already in Egypt (Yoseph, Aseneth, and their two sons  Ephraim and Menashe). The ratio of 4 to 70 is found numerous times in the Torah. It is found twice in this parashah alone. The first example is mentioned (directly) above. The second example is alluded too indirectly, where “Yoseph gathered up all the money [and] ...brought the money to Pharoah’s treasury.” It is well known that there were 4 treasuries of gold. What is not so well known is that each contained “70 quintles of the precious metal.” The ratio of 4 to 70 is also found in Mishpatim where, Moshe, Aaron, Nadav and Avihu (4) along with the 70 elders ascend before the throne (of God) and it is found in “11 kinds of spices were in it, as follows: stacte, onycha, galbanum, frankincense (4 spices) each weighing 70 maneh,” to provide the obvious examples.



These numbers or sets of numbers, and this ratio in particular (whether the 4 are included or excluded from the 70) holds a very deep secret that is examined at length in the post on "the Incense and the 216 Letters."

In-so-far as this post is concerned, it is sufficient to point out that when a tetrahedron (with its 14 elements) is subdivided by an opposing tetrahedron, the result is not only a magen david (in the geometric form of a star tetrahedron) but the number of elements in each (subdivided) tetrahedron increases (from the 14 mentioned above that were described in the post on Miketz) to 70 elements. There are 4 stellations (or points) on each of the tetrahedrons of the star after the subdivision (see the pictures below).

View of a Star Tetrahedron from the Top
Click to Enlarge


In the meantime, the number of elements resulting from the division is calculated as follows. The subdivision results in 4 tetrahedrons that are half the size of the original, and 1 octahedron in the middle. A single tetrahedron is made up of 4 faces, 4 points (8 elements). The 4 tetrahedrons together thus have 16 faces and 16 points (32 elements). The octahedron in the center has 8 faces and 6 points (14 elements). The lines that cause the division (from the intersection) are one-dimensional and should not be counted twice (as separate elements on each of the resulting geometrics). Therefore the number of lines equals 4 x 6 or 24. The 24 lines together with the 22 points and 24 faces equals 70 elements. The stellations, or points on the star (4 on each) can be further subdivided in like manner, the result being a star tetrahedron with 2 tetrahedrons, each with 4 stellations having 70 elements, with a total of 72 triangular corners that geometrically reflect the 216 letters of God's Explicit Name in their triad form. Note also (below) that a two-dimensional magen david can be seen in the orthographic projection of just one tetrahedron so divided.   

The 70 Elements of a Tetrahedron
(after being subdivided by its polar opposite)


The Orthographic Projection in the Three-Dimensional Magen David Above
(seen as a “shadow” in each 'half' below)


The important thing to remember is that God’s Name is imprinted (mathematically and geometrically) on every aspect of time and within every dimension of physical space in His creation, as well as the intrinsic characteristics of things Jewish.