The Story of the Cubit-Cubed

The story of the cubit-cubed and the physics of creation are found in the letters of God's Name. That is a profound statement. However, it is possible to illustrate the basic concept using simple geometry. The purpose of this post is to give the reader a glimpse of this reality, and more importantly to help the reader understand the assertion. It requires only that the reader have a basic understanding of the letters in God's Name, which have hopefully been acquired by reading the essential constructs listed to the right. A knowledge of these constructs is a prerequisite to this post.

Our sages said that God created the world with his Name: yud-hey ~ yud-hey-vav-hey (sefer yetzirah 1:1) generally translated as "Lord of Hosts." In english, these letters would take the form of YH ~ YHWH). Sometimes a picture is worth a thousand words and so, in this post, we’ll provide the reader with a third grade elementary school explanation, along with a couple of 'pictures' for the benefit of those who are mathematically challenged.

God is said to be a perfect unity. In fact, you might say He is the ultimate quantum singularity. We can represent this singularity with a simple dot (like the one pictured below). There is no multiplicity in this reality. At this primal level, there are no letters to represent God (no Name). He is "ein sof" ("without end" and incomprehensible).

.

In order to create our present universe, it was necessary to engrave a space "within" that singularity (referred to in kabbalah as the “tzimtzum”). The word “tzimtzum” (tzim-tzum) is a linguistic expression for the binary nature of “one” after it is divided in two, each 'half' being the polar opposite of the other (the primal concept of male and female). Hence, we have tzim and tzum as a portent of things to come, even before the existence of any form of 'duality.' The concept equates to yin and yang in the “eastern tradition,” knowledge of which was ‘preserved’ (presumably) by the descendants of Avraham’s other family (with considerably less substance).

At any rate, we are told that a 'line' then entered this “space” (i.e. the first dimension was ‘created'). A line is a geometric expression of a continuum consisting of two directions.

____________

 
At this level, these two directions (the primal equivalent of male and female within one-dimensional space) can be represented by the “primal energy” of two letters (because they are in fact 'created' by those very letters). These two letters were “essential” to this stage of creation. In other words, the two "directions" of the first dimension were "created" with the yud and the hey. This dimension is modeled with labels below.

י     ____________   ה 

       

(hey)                                                                  (yud)

 
The measure of this dimension was limited by the Holy One, Blessed be He, and the unit of measure for purpose of establishing the measurement of this newly created (but limited) cosmos1 was one cubit, and more specifically, it was one royal cubit, since the “measure” was dependent on the “letters” of the King’s Name (keep in mind we are talking about the mathematic system on which the universe is based).

These letters, and specifically their gematria (hey = 5 and yud = 10) include the 'potential' for additional dimensions in this cosmos, that are an intrinsic geometric aspect of the cosmic blueprint (luchot) that in 'combined cubic form' have "5" internal mathematic constructs. Together with their polar opposites, these constructs reflect the nature of all 10 directions within the space to be engraved, but we're getting ahead of ourselves. 

The nature of this one-dimensional continuum was limited simply because the process of creation had just begun. One dimension (with a continuum of two directions) was about to become two dimensions (with a total of four directions). 

This would require two more letters (because two dimensions have four directions). The “measure” of the second dimension was just like the measure of the first dimension (one royal cubit) so it’s total “area” was now 1 cubit x 1 cubit (or 1 square cubit). Even though this cosmos has now been expanded, it's “measure” still reflects the nature of its Creator, Who is One, since 1 x 1 = 1. 

We can represent this step in the process of creation (the second dimension) as follows:

( heh - vav - hey - yud )
(right to left) 

English YHWH (left to right)

God thus created the world with His Name: yud-hey ~ yud-hey-vav-heh. The yud-hey created the two directions of the first dimension and the yud-hey-vav-hey  then expanded on the first dimension, to create the second dimension, along with its additional 'directions.' The four letters were “essential” to this stage of creation (and it's subsequent expansion) hence the term 'havaya' or essential. 

One can now begin to understand why sefer yetzirah says that God  'engraved' the universe (created it’s multi-dimensional existence) with the letters of His Name, and specifically yud-hey (the first dimension) followed by yud-hey-vav-heh (the second dimension) and why God's Name is written this way in sefer yetzirah 1:1.
 
Before performing certain mitzvot, we say “for the sake of the unification of the Holy One Blessed be He, to unify the Name yud-hey with vav-heh” lest we err and assume the nature of the Holy One (represented, or perceived here, by more than one letter, which simply means more than one form of primal energy) is somehow less than a perfect unity.


The gematria of the additional letters, by the way, is six for the vav, and five for the second heh. These provide the 'potential' for yet another dimension, the third dimension, where the measure of the primordial 'cosmos' is a function of the number six (as in six handbreadths for all three dimensions) and where the three-dimensional character is in the form of a cube, which will now have five polar opposites (to the first five above). Note that with the additional letters there are now two letter hehs. 
 
And so as the process of creation continued and a third dimensions was created. It too was “limited” by the same measure. It too was one royal cubit. The total measure was now 1 x 1 x 1 cubit (and still “One” like its Creator) but more specifically, one royal-cubit-cubed (a three-dimensional representation of One). 

The cosmos that was created by these letters might seem rather small to the 'observer' (being only one cubit-cubed) but it is important to remember that at this phase of creation it was still in its primordial form as a simple construct in the mathematics of the 'combined cubic form of the luchot' (cube [of creation]) where "size" was merely a primordial reality directly proportional to its potential. The potential was, in essence, limited only by a divine thought, framed within the mind of the Creator. Its properties would thus conform to the mathematic characteristics of His foundation stone (primordial torah) with a geometric reality that pre-existed the universe. The 'creation' specified by the stone would ultimately be a continuum consisting of five dimensions with 10 'directions' (sefer yetzirah 1:5)

           ....like the 5 'dual' mathematic constructs of the cube 
                        ...like the 5 string theories reconciled by 'dual' resonance
                                    in quantum chromodynamics). 

Contemplate this. You will understand why it is important as we proceed. When God looked into this 'primordial torah,' He perceived everything that would come into existence, and once it was fully 'expanded' it would evolve into our present universe (which by the way is still expanding). The facts presented in our post on the primordial torah (that revolve around its internal and external geometry as well as the measure of the universe as a multiple of its various internal and external aspects) prove this assertion beyond a reasonable doubt, and you should now begin to understand what was meant by the sages of Israel when they described the 'engraving' of space in terms of these letters. If not, it will become increasingly clear as we proceed.
 
To that end, you may have noticed that a third dimension consists of two more 'directions,' and that it requires four more letters to sustain its duality or male/female nature (a simple mathematic fact pictured below with the associated letters). 

The additional four letters were those of Adnoot (aleph-dalet-nun-yud or ADNY) which is the counterpart of the four above that comprise the letters of the Tetragrammaton. They are pictured below as a geometric function of the primordial torah.

(click to enlarge)

Yud-Hey-Vav-Hey

'Opposite'

Aleph-Dalet-Nun-Yud

4 opposite 4
These 4 'pairs' are the template for a 'reality' with:
4 'couples' on an ark
4 'couples' in the Cave of Machpela
4 'pairs' of rings on the aaron
4 righteous men opposite 4 unrighteous (Kiriyat Arba)

...each is a reflection of a larger reality

These letters are depicted on the shvitis in most shules (synagogue / beit kenesset) without reference to the associated geometry. However, the connection that these Names (and specifically their letters) have to both the measure and geometry of the combined cubic form of the luchot (primordial torah or 'cube' [of creation]) and the multi-dimensional nature of the universe should start to make sense.

You may have also noted that the geometry of a cube (the three dimensions of its physical space) are technically neither male nor female, although as two Names, this duality exists in potential. The only way to turn it into a “binary” (male~female) reality, is to divide it in half (like the sapphire brick under the throne that was divided and brought down by Moshe). The subsequent “multiplicity” (in the universe) thus became a reality, but not by multiplication (as there cannot be more than 'one'). Rather, it was the result of division, as just explained, and which we will model (according the letters of the Name) in a moment.

During the four bows of Shemoneh Esrei (the 18 blessings) we "marry" the opposing letters of these 'Names' (again lest we err) thinking that God is less than a perfect unity, or somehow schizophrenic, for having "divided" the letters of His Name to 'create,' just as we did when we mentioned the unification above (with the letters of yud-hey with vav-hey) except here, in this three-dimensional case, it is yud with aleph; hey with dalet; vav with nun; and hey with yud (as shown in some siddurim). 

The exact nature of this marriage is explained more fully in the post on the marriage of the letters (listed among the advanced constructs). When this unification is performed during each bow, it has the potential to expand human consciousness. The practitioner inevitably becomes more keenly aware of his or her place in the cosmic scheme of things, acquires a spark of ruach hakodesh, and connects with God on a perceptibly higher level (having unified these letters in his or her mind). 

In the meantime, insofar as this post is concerned, it is sufficient to understand that when the letters of yud-hey-vav-hey (YHWH) are “married” to their polar opposites, which are the letters of aleph-dalet-nun-yud (ADNY), the result of the primal chemistry is the Name "Ehyeh" or alpeh-heh-yud-heh (AHYH). For now, just know that the primal energy associated with the physical combination of these letters does in fact result in the letters of Ehyeh, and this “result” is the 'catalyst' for the division of the cube [of creation](combined cubic form of the luchot) into two halves. The potential for 'duality' as a function of these two Names is now realized. At the same time, it is the glue that holds the male and female halves (that same duality) together. We received these two halves of the sapphire blueprint ("the keys to the kingdom") at Sinai, when the two “tablets” of the covenant (shnei luchot) were brought down.

Each “tablet” was said to be 6 x 6 x 3 (handbreadths) and the two, together, are one royal cubit-cubed (in combined cubic form) or 6 x 12 (72) times 3 in side-by-side form, like the letters of the Triad Name of 72 that define its fully expanded existence.

The divided cubit-cubed is shown below, along with the letters of these 'Names' that ‘engraved’ their 'duality' into the physical nature of our universe. This geometry and the letters of the Name are the heart of quantum mechanics with it's four forces; along with their polar opposites (+/-) and the mysterious force that binds them all together, so that the universe does not fly apart at the seams.

yud-hey-vav-hey     (YHWH)

aleph-dalet-nun-yud (ADNY)

alpeh-heh-yud-heh  (AHYH) 

(see footnote 1) 

So it was, that the Holy "One" looked into "One" (sapphire) blueprint or primordial torah, for purpose of creating a universe based on the letters of His Name, and a primordial reality even before the beginning. It was measured in terms of "One Royal Cubit-Cubed," which was of course the same as 6 handbreadths cubed; a measure resulting in a volume of 216 cubic handbreadths (like the 216 letters in the fully expanded Explicit Name of the Creator). It was of course divided and brought down by Moshe, and we received these halves (which can be seen in the geometry of the Names that define it above) as the shnei luchot. 

The zohar says (zohar II pg. 222) that "...when the Holy One, Blessed be He, wanted to create the world, He detached [the foundation] stone from underneath His throne of glory, and cast it into the abyss (the newly allotted 'space' that would become the physical universe). One end of it remained fastened therein, whilst the other end stood out above... out of which the world started spreading itself to the right and left [one dimension] and [then] into all [6] directions [of its three 'spatial' dimensions]. The "foundation stone" from under the throne (as a mathematic and geometric reality apart from creation) is depicted above. It is not a boulder on the temple mount, as espoused in legend, but rather, it was a "mathematic reality" that took physical form in a sapphire block brought down by Moshe (the shnei luchot) that were placed in the ark, which was itself "on the temple mount".

The early sages of Israel understood the significance of it's unique measurements, but they believed that the units of measure (like the 'royal' cubit and the 'royal' handbreadth) were too holy for everyday use, and so they ‘invented’ a slightly different measure for the cubit, in order to avoid using the “royal cubit” to measure ordinary things. That is why the Hebrew cubit(s) are different from the royal cubit (ante-diluvian or pre-flood cubit) that is generally attributed to the Egyptians (as though it were royal because of some Pharaoh).

If you have trouble understanding why these great sages did this, then consider that they did the same thing with the original holy letters (the type of font/script) ‘received’ at Mt. Sinai. This script was considered too holy for everyday use, and so a different font/script was employed for ordinary writing. By the time the people of Israel returned from the Babylonian exile, they were so far from God that many did not even know His Name, and so Ezra built a scaffold and pronounced the Explicit Name so that the people would awake. He also resurrected the original style of letters in order to resurrect the people from their spiritual slumber (human consciousness being a function of letters, words and specifically the holy letters of Holy Name(s).
 
The same condition is prevalent today. People are asleep. They have lost their connection to things of a divine nature, and it is long past time to resurrect the original systems of measurement that were all based on the “one” royal cubit defined by the letters of His Name, that gave rise to the cosmic blueprint (sapphire luchot) and ultimately the entire universe as dictated by the physics explained in such posts as 'string theory and the luchot.'

The picture below is from r. Aryeh Kaplan's commentary on sefer yetzirah (pg. 26). It is a model of the sephirot depicted in the form of a natural mathematic array (two-dimensional). If you compare this model to the two-dimensional orthographic model pictured above, you will notice that it is geometrically identical, except for the addition of three diagonal lines that horizontally connect the opposing points. 

Sephirot Depicted as a Natural Mathematic Array

You may have noticed that the first model had 12 letters, or 12 points/vertices, whereas the model above has only 10 (representing the sephirot). This is because the more primal reality (the primal nature of the cosmic blueprint) was two dimensional (two dimensions preceded three-dimensions just as just as yud-hey preceded yud-hey-vav-hey).

At any rate, along with the above model of the sephirot, r. Kaplan provides a table on page 28 showing the connection between the 22 letters as they pertain to this geometry, explaining that there are 10 points in the (above) system corresponding to the 10 sephirot; 3 horizontal lines corresponding to what are called the 3 'mother letters' of the Hebrew alpeh-bet; 7 vertical lines corresponding to the 7 'double letters' in the alpeh-bet; and 12 diagonal lines corresponding to the 12 'elemental letters'; the total number of lines being 22, like the total number of letters.

It is easy to see the connection between the geometry of the shnei luchot or two tablets (cube [of creation] in divided form); the sephirot, and the 22 letters. Although, few realize the connection that these numbers, sets of numbers and ratios have to the measure of the luchot and it's 216 elements, or that these in turn originate with the 216 letters of God's Explicit Name.

We are told that God “looked into” this (primordial) torah before He ‘created’ and as you no doubt saw in the post on the “primordial torah,’ the physics of the universe and its quantum nature are quite literally “engraved’ (defined) by all 216 “letters” of His Name and its opposing male/female 'halves' (108 opposite 108) which can be perceived according to the volumetric measure for each of the luchot in handbreadths (each one being 108 cubic handbreadths) that do in fact dictate this quantum reality (as enumerated again more fully in the posts like "the primordial torah," "string theory and the luchot" and also the "E8 lattice and the Name").

The "216" cubic handbreadths of this primordial cosmos; along with the 216 square handbreadths of its external surface; the 2,160 degrees (in its twenty-four 90 degree angles) and the 72 triangular corners (216 such corners) of the star tetrahedron (three-dimensional magen david) within, are all mathematic/geometric reflections of the letters in the Name (and its various abbreviated forms) that 'engraved this space' destined to become the model or template for the physical universe that would follow.

You may remember that in the post on the primordial torah, we explained how the geometry of the luchot defined all systems of measurement, although many have admittedly been distorted over the millennia. The ratio of the volume of the sphere within the luchot to the cube that surrounds it is .5236. This is well known in mathematics. What is not so well known, is that the ratio defines the metric system, where one cubit is equal to .5236 of a meter (this will be modeled in the paragraphs ahead). Since this measure is derived from a static ratio, a ratio that does not vary, it was not lost over time. The royal or imperial system of measurement was not so lucky. It was not derived from a static ratio. Rather, it was a function of a linear measure that was unique to the cube [of creation] and derived from the nature of the letters in the Name. The units of measure were anthropomorphically 'named' after parts of the body so they could more easily be perceived by mankind. Hands and feet by the way, are both numerically 5 opposite 5, like the mathematic nature of the luchot with 5 internal constructs and their 5 polar opposites). The luchot were "brought down twice" and so we have two sets of 5 opposite 5, like the two sets of digits: fingers and toes (both being 5 opposite 5). Man was (quite literally) made in the image of the signature geometry originating with the letters of the Name (thus made in the image of God).

After the flood, the accumulated knowledge along with the reason for such things was lost: including and especially the knowledge of the 'cube' [of creation]; it's specific measure; and the origin of it's measurements as they pertain to the letters of the Name. Confusion was allowed to enter the minds of men who were more concerned with themselves, than with their Creator. Mankind became increasingly distant from God, until God finally restored such knowledge to those more trustworthy, specifically the descendants of Avraham, Yitzhak and Yaacov.

At any rate, you can now begin to see, more clearly, how the geometry of the luchot defines various systems of measurement, including the imperial system, where we find handbreadths, feet-lengths; yards and fathoms. One need only "do the math" to see that these units of measure are all part of the "royal" system because they are a function of the letters in the King's Name. There are 6 'royal' handbreadths per linear cubit defined by the luchot (as known) and in three dimensions this yields a volumetric measure of 216 cubic handbreadths like the 216 letters in the Name. There are also 6 'royal' feet in the circumference of the sphere within the luchot (of its combined cubic form). Just as the ratio mentioned above (.5236) defines the metric system, its various linear measurements (both square and circular) define the units of the imperial system. 

The two-dimensional model below shows the measure of the luchot (as one 'royal' cubit) divided into 6 'royal' hands (handbreadths) and also the measure of the circumference of the sphere within, that is likewise divided into 6 halaquim (parts) each being a 60 degree segment (6 'tens' of degrees) of a 360 degree circle. One segment, or one sixth of the circumference, defines one 'royal' foot. The 60 degree segments that equate to the measure of a foot, are the origin of the sexagesimal or base-60 numeric system, however it is important to emphasize that the measure of a foot as it pertains to this system, is only true when the size or measure of the cube is that of the shnei luchot in combined cubic form. Please take time to understand all of this.

The imperial system of measurement is derived from the cube, as well as the divided cube, but more importantly, from the specific "measure" of the luchot. The 1/6th ratio (or 60 degrees in case of the sphere within) are also a function of the three-dimensional magen david within. When a tetrahedron is married to its polar opposite, they are 60 degrees out of rotational phase with one another. The cube [of creation] or foundation stone is a sophisticated mathematic system that defines the multidimensional nature of our present reality.

Today, the measure of a handbreadth is for the most part a forgotten imperial unit of measure. A measure of a foot (as well as yards and fathoms) have not been forgotten. However, the exact measure of a (royal) foot was lost after the flood, and so today, the measure of 'feet' and 'yards,' varies among the nations who have forgotten the origin of the system (i.e. the geometry of the cube [of creation]). As of the writing of this post, there is no other source on the internet, or anywhere else for that matter, that explains the relationship between the imperial system of measurement and the signature geometry of God's Name.

A popular online encyclopedia named Wikipedia provides relevant examples quoting: William of Malmesbury's Deeds of the Kings of England [that] records that during the reign of Henry I "the measure of his arm was applied to correct the false ell of the traders and enjoined on all throughout England."[13] The folktale that the length was bounded by the king's nose was added some centuries later. Watson dismisses William's account as "childish" but William was among the most conscientious and trustworthy medieval historians, the French "king's foot" was supposed to have derived from Charlemagne, and the English kings subsequently repeatedly intervened to impose shorter units with the aim of increasing tax revenue. 

All of these tales and the resulting measurements were of course nonsense, though they did serve a purpose (like increasing tax revenue). The point being, that an honest system of measurement based on the measure of the real King's luchot (the geometry of the two tablets) was corrupted. So it was that over time, the origin of the units of measure and the actual measurements were either lost or distorted.

In summary, we would ask: How does the cube [of creation] (defined by the 216 letters of God's Name) define imperial measurements, like the measure of a foot? Just as the external linear measure divides into 6 handbreadths, the internal linear measure of the circumference (of its inner sphere) divides into 6 feet. 

As we said before, the measure of the foot today, varies from country to country because knowledge of the original system (based on the measure of this particular "cube") was lost eons ago. However, according to this royal or imperial system, a foot is almost exactly 274mm (in metric) which is about 10% less than the measurement of what is called an "international foot" today (which is recognized only by the United States). Nevertheless, there are 'feet' in other countries, like some of the Belgian measures for a foot (specifically 273.8mm to 274.3mm) that are still close to the original measurement. 

You may have already noticed that 180 degrees of the sphere (half of the circle) within the luchot equates to one yard (3 feet) and the full circle (which is 6 feet) equates to the measure of one fathom. The difference between the half-circle and the full-circle, is the result of dividing a cube. A cube so-measured, when divided in half like the sapphire brick that was divided and "brought down" by Moshe has the specific circumference within that 'defines' these imperial measures.

The associated numbers are modeled below in the division of the luchot (two halves). It is easy to see how these numbers and sets of numbers, as well as the ratio between one measure and another, originate in the geometry of the luchot, once the basis of the system defined by the letters of the Creator's Name has been explained. Like the signature question that was made at the beginning of an old, once-famous television game show might ask: "...will the real Creator of the Universe please reveal His Royal Name." 

The two-dimensional model for each half of the cube (each of the luchot) is shown below. The linear measurement of the divided side is 3 handbreadths. The corresponding division of the circumference is 3 feet, which is 180 degrees, or half of the 360 degree circle within, and the origin of the unit of measure known as a "yard". 

At this point you may be asking yourself, "why are these lengths divided by 6?" Why 6 handbreadths. Why 6 feet? Why 6 (tens) of degrees? Why is 6 the magic number? Why not some other number? 

The simple answer is that the cube and the sphere (that is within the cube) define their relationship, by virtue of their intrinsic nature. The cube has 6 faces. When the sphere touches the surface on each side of the cube that surrounds it, there are 6 points of intersection (see the model below) where the surface of one kisses the 6 surfaces of the other (6 places where the surface of one is tangent to the surface of the other). While the sphere has just one "surface," the cube has 6 surfaces. The ratio between the number of surfaces on one, to the number of surfaces on the other is 6-to-one, where the "One" is in the nature of the Holy One. His cosmic blueprint (primordial torah) defines all things, and so the number 6 (as in 6 handbreadths or 6 feet) is not an arbitrary number. It is a function of the marriage between the geometry of the sphere and the geometry of the cube that surrounds it, and thus becomes the primary divisor that defines each "unit or measure" as it pertains to (one) royal cubit. Take time to fully understand this.

click to enlarge

When the cube is divided in half (as the luchot) the sphere is also divided in half. The division of the sphere results in the next two units in the imperial system of measurements, specifically the "measure above" and the "measure below" the specific unit of measure known as a "foot" that is defined by the sphere. The measure below is the royal "inch" that is defined by the 12 sides or surfaces on what are now two halves, and so each foot is divided into 12 parts (halaquim) which is of course is 1 (royal) inch in length. The measure above is defined by the number of "feet" in half of the circle, which is 1 "royal" yard.

In summary, the letters of this Name define the geometry and measure of the cosmic blueprint (the shnei luchot) which in turn defines the geometry and measure of the sephirot, the letters and indeed, the entire universe.

The purpose of this blog is to explore this geometry and its various measurements, and to establish its divine nature as it pertains to the Name of the Creator, to the end that this Name will be revealed to the nations; that the people of these Nations will awaken; recognize One God; and reject the idolatry inherent in the many forms of ancestor worship prevalent in the world today (the false religions currently recognized and accepted by so many) as explained in the post on Bereishit and Noah.  



___________________

Footnote 1 - When the Names are perceived as a function of the sephirot, AHYH is 'on top,' with YHWH in the middle and ADNY on the bottom. When perceived as a function of the foundation stone and its three-dimensional geometry, the "division" places the uppermost event in the center. When the luchot are side-by-side, Ehyeh-Asher-Ehyeh (Keter) is on top and (ADNY) and YHVH are paired below (Tiferet/Malchut) as depicted in Shaarey Orah. The former is a function of kabbalah and the sephirot. The latter is a function of physics and the mathematic system defined by the cosmic blueprint (luchot).