Book of Formation - Sefer Yetzirah

Chapter II

Mishnah 2, part 1

By the grace of G-d 
Copyright © 2014-16 Nathaniel Segal 

Mishnah * II.2.1.  Twenty-two fundamental letters affixed to a wheel of 231 gates, and this wheel turns forward and back.

AH leff

BAYT (like 'bait', 'bate')

GIM mell

DAH let

HEH (lke hay)

VAHV

ZAH yin

HET / KHET

TET

YOOD (rhymes with 'hood')

KAHF

LAH med

MEM

NOON

SAH mekh

AH yin

PEH (almost like 'pay')

TSAH deek

KOOF

RAYSH

SHIN

TAHV

 
  1. Alef
  2. Beit
  3. Gimmel
  4. Dalet
  5. Heh
  6. Vav
  7. Zayin
  8. Het (Chet)
  9. Tet
  10. Yod
  11. Kaf
  12. Lammed
  13. Mem
  14. Nun
  15. Samekh
  16. Ayin
  17. Peh
  18. Tsadik
  19. Koof
  20. Reish
  21. Shin
  22. Tav

Companion - Consciousness / Speech

Mishnah II.2.1.  [II.4]  Twenty-two fundamental letters - consonants.

This mishnah addresses how the many different items were created through the simple Ten Statements of Genesis. For example, all species birds come from a general statement of creation.

In this mishnah, the Book of Formation demonstrates a substitution cipher for the twenty-two letters. I've listed the names of the letters in the sidebar.

Each of the letters in the Statements can be exchanged for another to reveal a message. We could say that the Statements of Genesis are in the form of a code, and substitution letters decode the Ten Statements to reveal many hidden details.

It seems awkward to say that the understandable narrative of Genesis is in a coded form. In truth, G-d presents the letters as a narrative that we believe that we understand. However, letter substitutions decrypt the surface text to reveal words and sentences containing details of creation that correspond to the multitude of created items.

This is not to say that we are able to understand words and sentences with letter substitutions. Let's keep in mind that this is a book that describes formation, but it presupposes an awareness of creation of something from nothing – ex nihilo.

Formation is like the work of a potter at the wheel who turns raw clay into an item, an artifact. Intention is in the mind of the potter. Someone watching may ask, "What are you making?" or "What have you made?" "Of what use is this?" "Does it represent art, a piece for decoration?"

Similarly, G-d's intentions for Creation lie hidden behind straightforward formation. The Ten Statements are words of both creation and formation. This mishnah's system of letter substitution serves to let us know how varied and abundant G-d's formations really are. The lessons from the Book of Formation are short on revealing G-d's intentions. The lessons concern the laws of nature, including human nature. These are the physical sciences and the metaphysical realms.

a wheel -The Hebrew word is galgal (gahl GAHL). It supports a family of meanings such as circle, cycle, rotation, and orbit. Here in the Book of Formation, it refers to both the wheel and its rim. This mishnah proposes attaching marks on a wheel for each of the twenty-two letters of the Hebrew alphabet. Do the same for the rim. In both cases the gaps between letters are all equal. We've spaced the letters around the circles so that the last letter of the alphabet is alongside the first letter. The gap between these two letters is equal to all the gaps. The letter Tav, the twenty-second letter, is as close to Alef, the first letter, as Alef is to Beit, the second letter.

We can then attach the rim to the wheel systematically so that we associate every pair of letters for substitutions. Otherwise, I don't understand this mishnah unless I note how a wheel has a rim. With the same twenty-two letters affixed around the wheel and around it's rim, the mishnah presents all sets of letter substitution pairs when the rim is put in place.

The lesson from this mishnah does not expect us to actually build a wheel with a rim and attach letters. We can and will visualize the wheel, its rim, and the attached letters to identify the 231 gates.

Two letters are called a gate. We visualize each letter as a post. Together, they form a gate. We can leave a gated enclosure with, let's say, an "Alef post" on our right and a "Tav post" on the left. Coming back in, the "Tav post" is on our right while the "Alef post" is on the left. This mishnah wants us to visualize how two letters form only one gate. We are visualizing pairs of letters for substitutions. With this Alef-Tav pair, we can replace a Tav with an Alef or replace an Alef with a Tav. This mishnah assures us that there are 231 pairs of letters but lets us ignore the order for how we identify the pairs.

The mishnah's instructions are such that "this wheel turns forward and back." I understand this to mean that the substitution pairs work both ways as I just wrote. When we visualize how "this wheel turns forward and back," the top of the wheel and its rim are rotating. The top moves to the bottom; the bottom arrives at the top. A stationary observer sees Alef above Beit and later Beit above Alef. Alef can substitute for Beit and Beit can substitute for Alef. To repeat, we are counting pairs. Otherwise, we cannot arrive at the count of 231.

a wheel - The Hebrew word galgal has several meanings according to context. It's used to refer to the 'orbit' of a planet, for instance. It can refer to a 'cycle' such as the cycle of the year.

The Hebrew word galgal is independent of technology, so we can also visualize a dial with a ring closely around it. This dial has a knob, and we will be turning the knob one letter at a time to match a letter on the fixed ring. This also fits the mishnah's lesson of "affixing the twenty-two letters to a wheel."

For the sake of simplicity, we will visualize both sets of letters in the same order, the traditional order of the Hebrew alphabet, one complete set around the ring and one complete set around the edge of the dial. See the sidebar for the names of the letters in the Latin alphabet and for their order. We will also affix the letters in the order of reading Hebrew – from right to left. We will be orderly and systematic by deciding on a configuration even though we could choose a different one. We could visualize the letters running from left to right just as we could use the Latin alphabet – what you're reading now – and order the letters alphabetically according to their name. What matters is that I am setting up this visualization to be systematic.

As a sensible beginning, let's align the Alef on the dial with the Alef on the ring. Since both letters are the same, we have no substitution. Then, let's turn the dial by one letter counter-clockwise. Alef on the dial is opposite Beit on the stationary ring. Immediately to the left on the dial, Beit is opposite Gimmel on the stationary ring. This dial setting creates one set of substitutions – Alef-Beit, Beit-Gimmel, . . . , and Tav-Alef. The substitutions are by adding on * one. This set consists of twenty-two substitution pairs. My name for this set is Alef-Beit based on the name of the first pair of letters. I'm not aware that actual names for substitution sets appear in the Book of Formation. Just as we have leeway to visualize a method to form all the pairs according to our own choosing, we can assign names to organize our system.

Now let's turn the dial in the same direction, counter-clockwise, by another single letter. The ring never moves. Now, Alef on the dial is opposite Gimmel on the ring. This creates another set of substitutions – Alef-Gimmel, Beit-Dalet, . . . , and Tav-Beit. The substitutions are by adding on two. This set consists of twenty-two substitution pairs, too. My name for this set is Alef-Gimmel.

@[We can use this first pair to substitute Alef for Gimmel or Gimmel for Alef, just as we can do with the second pair. We can use this second pair to substitute Beit for Dalet or Dalet for Beit, and so on.]

When we turn the dial again by one letter, we are creating a substitution by adding on three. We are "dialing up." Now, Alef on the dial is opposite Dalet on the ring. This creates a new set of substitutions – Alef-Dalet, Beit-Heh, . . . , and Tav-Gimmel. This set has a count of twenty-two substitution pairs, and my name for this set is Alef-Dalet.

With another turn of the dial, the substitution set is Alef-Heh, Beit-Vav, . . . , Tav-Dalet, another set of twenty-two substitution pairs. This set is a substitution by adding on four, called by me Alef-Heh.

We turn the dial again and create a substitution set by adding on five. The pairs are Alef-Vav, Beit-Zayin, . . . , Tav-Heh, another set of twenty-two substitution pairs, with my name Alef-Vav.

We'll continue doing for this for the next sets of twenty-two new pairs of letters with names for their substitution sets.

My names for the next three substitution sets are:

For a substitution by adding on nine, the pairs are Alef-Yod, Beit-Kaf, Gimmel-Lammed, Dalet-Mem, Heh-Nun, Vav-Samekh, Zayin-Ayin, Het-Peh, Tet-Tsadik, Yod-Koof, Kaf-Reish, Lammed-Shin, Mem-Tav, Nun-Alef, Samekh-Beit, Ayin-Gimmel, Peh-Dalet, Tsadik-Heh, Koof-Vav, Reish-Zayin, Shin-Het, Tav-Tet, another set of twenty-two substitution pairs. My name for this set is Alef-Yod.

For a substitution by adding on ten, the pairs are Alef-Kaf, Beit-Lammed, Gimmel-Mem, Dalet-Nun, Heh-Samekh, Vav-Ayin, Zayin-Peh, Het-Tsadik, Tet-Koof, Yod-Reish, Kaf-Shin, Lammed-Tav, Mem-Alef, Nun-Beit, Samekh-Gimmel, Ayin-Dalet, Peh-Heh, Tsadik-Vav, Koof-Zayin, Reish-Het, Shin-Tet, Tav-Yod, twenty-two substitution pairs named by the first pair Alef-Kaf.

My names for the substitution sets so far are:

We've now visualized a system of substitutions at letter distances of one through ten, each set containing twenty-two letters. The subtotal is 220, ten times twenty-two.

For a substitution by adding on eleven, the pairs would be Alef-Lammed, Beit-Mem, Gimmel-Nun, Dalet-Samekh, Heh-Ayin, Vav-Peh, Zayin-Tsadik, Het-Koof, Tet-Reish, Yod-Shin, Kaf-Tav, Lammed-Alef, Mem-Beit, Nun-Gimmel, Samekh-Dalet, Ayin-Heh, Peh-Vav, Tsadik-Zayin, Koof-Het, Reish-Tet, Shin-Yod, Tav-Kaf. However, we are now seeing duplicate pairs for the first time. The last pair of Tav with Kaf is the same pairing as Kaf with Tav. Alef-Lammed and Lammed-Alef are the same, and so on. By adding on eleven, we only see eleven new pairs – Alef-Lammed through Kaf-Tav. My name for this substitution set is Alef-Lammed, with a set of only eleven pairs of letter substitutions to add to the subtotal.

Now add the eleven pairs of new substitutions that I've named Alef-Lammed to the subtotal. The total of 220 plus eleven is 231.

These 231 letter substitution pairs explain this mishnah. We have visualized what the mishnah proposes if I am justified in my methodology to form all unique pairs – the gates. However, let's proceed to turn the dial to check for new pairs.

For the next substitution by adding on twelve, the pairs would be Alef-Mem, Beit-Nun, Gimmel-Samekh, Dalet-Ayin, Heh-Peh, Vav-Tsadik, Zayin-Koof, Het-Reish, Tet-Shin, Yod-Tav, Kaf-Alef, Lammed-Beit, Mem-Gimmel, Nun-Dalet, Samekh-Heh, Ayin-Vav, Peh-Zayin, Tsadik-Het, Koof-Tet, Reish-Yod, Shin-Kaf, Tav-Lammed, a set named by me Alef-Mem.

All the pairs of Alef-Mem have appeared before at a letter distance of ten, in the group that I've named Alef-Kaf. This group Alef-Mem gives us no new letter pairs. For the purposes of this mishnah, this group Alef-Mem is equivalent to the group Alef-Kaf.

For the next substitution by adding on thirteen, the pairs would be Alef-Nun, Beit-Samekh, Gimmel-Ayin, Dalet-Peh, Heh-Tsadik, Vav-Koof, Zayin-Reish, Het-Shin, Tet-Tav, Yod-Alef, Kaf-Beit, Lammed-Gimmel, Mem-Dalet, Nun-Heh, Samekh-Vav, Ayin-Zayin, Peh-Het, Tsadik-Tet, Koof-Yod, Reish-Kaf, Shin-Lammed, Tav-Mem, a group named by me Alef-Nun.

All these pairs have appeared before at letter distance of nine, in the group that I've named Alef-Yod. This set that I'm calling Alef-Nun contributes no new pairs of letters. This group Alef-Nun is equivalent to the set Alef-Yod.

For the next substitution by adding on fourteen, the pairs would be Alef-Samekh, Beit-Ayin, Gimmel-Peh, Dalet-Tsadik, Heh-Koof, Vav-Reish, Zayin-Shin, Het-Tav, Tet-Alef, Yod-Beit, Kaf-Gimmel, Lammed-Dalet, Mem-Heh, Nun-Vav, Samekh-Zayin, Ayin-Het, Peh-Tet, Tsadik-Yod, Koof-Kaf, Reish-Lammed, Shin-Mem, Tav-Nun, a set named by me Alef-Samekh.

All these pairs have appeared before at letter distance of eight, in the group that I've named Alef-Tet. This set that I'm calling Alef-Samekh contributes no new pairs of letters. This group Alef-Samekh is equivalent to the set Alef-Tet.

We could turn the dial again by one letter and create the next set of substitution pairs by adding on fifteen. However, we are seeing a pattern. We've been turning the dial counter-clockwise. Instead of counter-clockwise, let's turn the dial clockwise to see why we are finding no new pairs. Notice how the previous letter substitution at a distance of fourteen produces the equivalent substitution pairs at a distance of eight. This stand to reason. Eight plus fourteen equals twenty-two – the number of letters in the alphabet. The alphabet has been affixed in a circle, so whichever direction one turns the dial, a full turn returns the dial to the beginning setting.

So, let's reset the dial where its Alef is aligned with the Alef on the fixed ring. Now, let's turn the dial against the fixed ring a distance of seven letters clockwise. Since seven plus fifteen equals twenty-two, we expect that this turn of seven letters clockwise will place the dial at the same letter as turning it a distance of fifteen letters counter-clockwise. The outcome is that Alef on the dial is aligned with the letter Ayin on the ring as we expected. According to way that I have been naming the sets of pairings, I'm calling this set Alef-Ayin. The previous set, at a distance of fourteen, was called by me Alef-Samekh. This next set, being called Alef-Ayin, checks out.

Now let's visualize the equivalency by looking at letter pairs in the counter-clockwise direction. We are not moving the dial. We are visualizing what letter on the dial is aligned with the letter Alef on the fixed ring. We've set the Alef on the dial across from the letter Ayin. The adjacent letter to Alef on the dial in a counter-clockwise direction is Beit, which is opposite Peh in the ring. Further examination shows us Gimmel-Tsadik, Dalet-Koof, Heh-Reish, Vav-Shin, Zayin-Tav, Het-Alef. [These pairs are not contained in the new set. We are counting each of these as a distance in letters – a distance of seven as a complementary distance of fifteen in our regular direction of turning.] The letter Alef on the ring is paired with Het on the dial. Since we are not looking at the order of the pair of letters, which letter is on the dial and which is on the ring, this substitution set of pairs is the same as the group of pairs that I called Alef-Het. In addition, I've just listed a sample of pairs that I encountered in this set – Beit-Peh, Gimmel-Tsadik, . . . , Zayin-Tav. Each of these pairs is found within the set Alef-Het, although listed in opposite order. This set that I'm calling Alef-Ayin contributes no new pairs of letters. This group Alef-Ayin is equivalent to the set Alef-Het.

For the pairs of letters revealed by adding on sixteen, I'm going to reset the dial again where its Alef is aligned with the Alef on the fixed ring. Now, let's turn the dial against the fixed ring a distance of six letters clockwise since six plus sixteen equals twenty-two. The letter Alef on the dial lines up with Peh on the ring. I'm going to call this substitution set Alef-Peh, consistent with my naming convention all along. Where is Alef on the dial? Opposite the letter Peh – Alef-Peh. And where is Alef on the ring? Opposite the letter Zayin on the dial – Alef-Zayin. This set Alef-Peh is equivalent to the set Alef-Zayin, where Peh and Alef are already paired. So, this set that I'm calling Alef-Peh contributes no new pairs of letters.

For the pairs of letters revealed by adding on seventeen, I'm going to reset the dial again where its Alef is aligned with the Alef on the fixed ring. Now, let's turn the dial against the fixed ring a distance of five letters clockwise since five plus seventeen equals twenty-two. The letter Alef on the dial lines up with Tsadik on the ring. I'm going to call this substitution set Alef-Tsadik, consistent with my naming convention all along. Where is Alef on the dial? Opposite the letter Tsadik – Alef-Tsadik. And where is Alef on the ring? Opposite the letter Vav on the dial – Alef-Vav. This set Alef-Tsadik is equivalent to the set Alef-Vav, where Tsadik and Alef are already paired, for example. So, this set that I'm calling Alef-Tsadik contributes no new pairs of letters.

For the pairs of letters revealed by adding on eighteen, I'm going to use the previous procedure to turn the dial clockwise four letters. Eighteen plus four equals twenty-two. I'm going to call this substitution set Alef-Koof. This set Alef-Koof is equivalent to the set Alef-Heh, and it contributes no new pairs of letters.

Similarly, by adding on nineteen, I will find no new pairs of letters. The corresponding letter distance is three. Nineteen plus three equals twenty-two. I'm calling this next set of pairs Alef-Reish. Since both Dalet and Reish are at a distance of three from Alef, this set Alef-Reish is equivalent to the set Alef-Dalet.

Since both Gimmel and Shin are at a distance of two from Alef, they are also substitutions by adding on twenty. This next substitution set that I'm calling Alef-Shin contributes no new pairs of letters because it is equivalent to the set Alef-Gimmel.

Finally, both Beit and Tav are at a distance of one from Alef, so they are also substitutions by adding on twenty-one. I'm calling this final set Alef-Tav. When one is pairing Beit and Alef, one is also pairing Alef with Tav. I've written that out fully above with the first set of substitution pairs, Alef-Beit. Since the set Alef-Tav contributes no new pairs of letters, it is equivalent to the set Alef-Beit.

Moving the dial one more time aligns Alef on the dial with Alef on the ring. Remembering that this mishnah expects us to count how many pairs of letter substitutions are formed from the twenty-two letters of the Hebrew alphabet (see the sidebar), this move of the dial forms the case of no substitution pairs. No letter substitutes for itself. We have thoroughly visualized a lesson from this mishnah.

adding on - Cryptographers call the substitution system of this mishnah "adding on a fixed number." In the simplest case, letters have number values according to their order in the alphabet. In the Hebrew alphabet, Alef has the value one because it is the first letter of the alphabet. Beit has the value two because it is the second letter. The arithmetic of adding on one to Alef is 1 + 1 = 2. The letter with the value 2, Beit, is substituted for the letter with the value 1, Alef. Given that there are twenty-two letters in the Hebrew alphabet, if adding on gives a number greater than twenty-two, we subtract twenty-two. This is "modular addition" by a fixed number. For the Hebrew alphabet, the module is twenty-two – a total of twenty-two letters. For letter pair substitutions by adding on one, for example, Tav becomes paired with Alef. Tav's value is twenty-two. By adding on one, 22 + 1 = 23. Obviously, there is no twenty-third letter. We subtract the module twenty-two – 23 - 22 = 1. A cryptographer is also interested in the actual sum from adding on or the remainder after subtracting the value of the module.

They also call this a "mono-alphabetic cipher."  Letters are substituted only one time.

For a simple encryption like this, cryptographers would become used to calculating in their heads. Those who decrypt the code subtract the fixed number. Forward and back for cryptographers is the encrypting by the sender of a coded message and decoding back to the original by the recipient.

This mishnah, though, has us visualize a wheel with its rim so that we can calculate the total number of substitution pairs for twenty-two letters.

gates - When consonants are viewed as pairs of gate posts, an opening between posts represents a vowel.Two letters are like a gate for all the vowels to blow through.  These three elements – an opening consonant, its vowel, and a closing consonant – form a syllable. In my own words based on the lesson of this mishnah, I call the pair of letters which represent the gate posts "a syllable frame."  A syllable is the smallest unit of speech in any language, which is to say the smallest unit of meaning in a language.

Any of the twenty-two letters could serve as a post, and any letter can be paired with itself. Sometimes a language expresses itself with syllables that begin and end with the same consonant. English has the two words 'mom' and 'dad', for instance.

Using modern mathematics, our calculation for the number of syllable frames is 22 x 22 = 484. The opening letter of a syllable is any one of the twenty-two Hebrew letters. Pair each letter with itself or with one of the other twenty-one remaining letters and you have the mathematical expression 22 x 22.

The 231 gates – substitution pairs – of this mishnah form 462 different syllables, 231 x 2. However, we didn't count one set of pairs. This set was formed by pairing each letter with itself. We didn't count these pairs since no letter is a substitute for itself. Nevertheless, each letter with itself forms the frame for a syllable. As I just mentioned, English has the two words 'mom' and 'dad', for instance. This set of letters paired with themselves amounts to twenty-two unique syllables. Add the count of these twenty-two syllables to the 462 different syllables formed from this mishnah, getting 484. This is precisely 22 x 22, as expected.

Interestingly, an Aramaic word for gate is bava (BAH vah). The essence of this word for gate is the letter Beit paired with itself. The letter Beit is correlated with each gate post. (Beit is one of the Doublets, so it supports two related sounds – /b/ and /v/. Aramaic speakers add a short syllable /a/ to the root of the word to indicate that it is singular – a gate or one gate.)

A word for gate in Arabic is bebun (BEH boon). (Today's languages have come to us from early speech communities. In the case of Arabic, Aramaic, and Hebrew, the early speech communities were relatives or neighbors, so similar words appear in today's languages.)

We established that twenty-two beginning letters associated with any one of these same letters as a closing letter form 484 unique syllable frames. Any vowel can fit between the two consonants. The Book of Formation counts the vowels of Divine speech as ten, just as the twenty-two letters are the consonants of Divine speech. Altogether, we arrive at 4,840 unique Divine syllables. This larger figure also begins to address the issue that I raised at the beginning of the Companion on this page. How can we account for the complexity of Creation compared to the countable Statements, words, and letters of Genesis? Even though all 4,840 syllables do not appear in the Creation narrative, the nature of language carries this potential variety.

turns forward and back - Two posts create symmetry and polarity.  We have one gate but two uses – two syllables with the same vowel.  We can leave confinement within the periphery through a gate, and we can enter the confinement. The two gateposts of letters spell one syllable going out and another syllable coming back in. In terms of 4,840 syllables, we can speak about syllables that correspond to leaving the periphery – moving "forward" – and  syllables that correspond to turning back into the periphery. As an idea, we may not be aware of any practical application. Nevertheless, the Book of Formation presents material for meditation. The 4,840 syllables when polarized forward and back amount to 9,680 entities.

Companion – World

wheel - corresponds well with the opening of the entire book, "The Almighty created his world in three realms – s'farim." A wheel creates a realm. For this mishnah, we have been visualizing a round periphery and attaching letters to its s'far – its edge.

this wheel turns forward and back - In this mishnah we are visualizing the building of a wheel and its rim. This reflects visualizing building any balanced periphery. Without good balance, this wheel will not turn forward and back well. Let us build gates around a perimeter, but not 231. Let's start by building twenty-two gates just as we formed a dial with twenty-two notches for the twenty-two letters of the Hebrew alphabet, and we formed a fixed ring around the edge of this dial.

We set out one post on the perimeter. Let's associate it with the letter Alef. Now, looking at the first gate post from within the perimeter, set up a second post to the left and associate it with the letter Beit. We don't have to place it to the left. But, considering that Hebrew is written from right to left and considering that we are inside the perimeter, placing new posts on our left will result in a perimeter of twenty-two posts in the standard order of the Hebrew tradition.

Actually, someone who is outside the perimeter will see the Hebrew letters in left-to-right order. The Book of Formation prepares us for a full visualization, both from inside and outside the perimeter. Note how these two viewpoints are also consistent with the mishnah's words "forward and back."

We now have set up two posts which delineate one gate. A third post forms a second gate. At this rate, the last post, the twenty-second one, will have only marked the twenty-first gate. However, when we visualize a circular perimeter, the post for the last letter is next to the first post that we erected.

Since we started by setting up two gate posts for the first two letters of the Hebrew alphabet, let's return there to find the center of an ideal periphery for the other posts, a balanced wheel.

This mishnah told us to visualize a wheel, and a wheel is balanced around an axle at the wheel's center. People who build wheels – wheelwrights – can tell us all that we need to know about placing our gate posts or about a balanced wheel. However, we can experiment ourselves, although I don't see how we can visualize. We can build a small model of twenty-two gates, though.

For our model, we are visualizing gates that are as close to the same width as possible. How do we prepare to place the last post where it forms a gate with the first post that is just as wide as the first gate? This mishnah helps us to find where to set each gate post in its ideal place. We will discover that the twenty-two gate posts form a circle with equal gate widths if we place each of the gate posts about three and one half gate widths from our center in the gated enclosure. We just visualized a periphery of twenty-two gates for framing the twenty-two letters and each gate is as wide as any other.

In terms of mathematics, we'll call the width of the first gate the unit measurement of the circumference of the circle. The distance from any gate on the periphery to the center will be a radius of three and a half units, with a small margin of imperfection. The distance from a gate on one side of the circle / wheel to the gate at exactly the opposite side is the diameter, which is almost exactly seven units long. Actually, the circumference of this circle could hardly be closer to twenty-two units.

This ratio between the length of the circle's circumference and its diameter is a value called pi – π. Ancients discovered that its value is slightly more than 3 – more by one part in ten.

Following the prescription of this mishnah, we have created twenty-two gates for "affixing" the twenty-two fundamental letters to a wheel – a periphery – where each letter is virtually equally distant from its neighboring letters as any other letter is from its neighboring letters.

This mishnah speaks to 231 gates. However, it only suggests visualizing gates, so I associated 231 gates with twenty-two gates which compound themselves into 231.

This mishnah did not instruct us that the gates are equally spaced. However, it did describe a wheel, and this teaches us to visualize a circle that is well balanced.

Not incidentally, though, this mishnah has demonstrated an association between twenty-two letters and the value seven. As I explained, a periphery of twenty-two units corresponds to a diameter of seven units. It's not surprising that a subset of seven of the twenty-two Hebrew letters belongs together as one category – the Seven Doublets. This is an organic value, an essential quality, of the unified alphabet.

[231 gates - If we had made a model wheel with 231 gates of equal width, its circumference is 10 ½ times larger than the circumference of twenty-two gates. It follows that the diameter of this wheel is about 73 ½ units, giving a radius of about 36 ¾ units. I have yet to find a lesson from this literal reading of this mishnah.]

diameter and radius– The Book of Formation need not to be precise. Precision is not a quality found in the Universe. A circle is an idealized ellipse – a very special case where both foci are located at the same point. The form of an ellipse, though, pervades the Universe (actually three-dimensional ellipsoids).

wheel - Also a sphere. Spherical geometry is a geometry of the Universe on both the large scale and the small scale. Plane geometry (Euclidean geometry) – a subject in the curriculum of American high schools – comes to us from the ancient science of land measurement. In fact, the word 'geometry' comes to us from two Greek words – geo and metron. The Greek word metron means 'measure'. The word geo refers to the Earth.

On the human scale, the Earth's surface can be treated as though it were flat. In this plane geometry – surveying land – parallel lines do not meet. On the surface of the Earth, though, lines which begin parallel at the equator approach each other as they move away from the equator. Eventually they meet at the pole. In spherical geometry, parallel lines do meet.

forward and back - To a viewer outside the Earth's frame of reference, the closer side of the Earth spins in one direction while the direction of spin on the other side is opposite.

Also, a point on the circumference of the Earth appears to move faster at the center of the circumference than it does as it approaches the viewer's edge. At the moment of transition to retrograde movement, a point on the circumference appears to stand still and then speed up in the opposite direction.

turns forward and back - Measuring waves is the basis of the science of trigonometry. This word 'trigonometry' comes to us from two Greek words – trigonon and metron. The word trigonon refers to three sides. Triangles are the basis of trigonometry. Again, these triangles are pictured on a plane. The sum of the angles of a triangle is 180 degrees. However, in the properties of a sphere, the sum of angles varies. Also, sides of triangles are not necessarily straight. Living in the Universe, we experience functions which move up and down, back and forth, and side to side – possibly all at the same time.

The author of the Book of Formation – the Patriarch Abraham – recognized that he lived in Universe with spherical properties (actually ellipsoidal).

forward and back - The foci of an ellipsoid have a dynamic relationship of moving "forwards" and "backwards' relative to each other.

Companion – Time

When 231 pegs or posts are set up outdoors circularly around an observer with the observer precisely in the center, he or she can devise units of time.  At night, a person keeps track of the length of time for a planet or star to pass from one gate to another.  This becomes a unit of time.

For example, the transit of the planet Venus close to the horizon between gates establishes a local unit of time.  The star Sirius also passes close to the horizon for part of the night during winter (when and where Sirius can be viewed).  This establishes a unit of time that is roughly equal to the duration of Venus's transit through the same number of gates.  Sirius is more useful for viewing through the gates when it is first visible as the summer ends and during the late spring nights than during winter months.  During the winter, Sirius's transit across the sky is so high above the horizon that ancients used tall megaliths to measure its transit rather than using ordinary gate posts.

Natural units of time have been used to calibrate water clocks and sand timers.  Society depends on standard measurements for numerous purposes, among them for commerce.  The Patriarch Abraham (who composed the Book of Formation) lived in a society which had been using standard units of time, weight, and measurement long before he was born.

Of course, the accuracy of natural standards of measurement is not sufficient for use with the technology of the last couple centuries.

The Book of Formation teaches us in this mishnah that measurement of time can be derived from units of length.

Companion – Consciousness

turns forward and back - This also refers to "running (out) and returning." See Mishnah I.4, part 2 – "if your heart hastens, return to the One."

forward and back - "Running (out) and returning."  The Prophets use variations of this theme to describe humanity's relationship with G-d.  Sometimes we draw away from Him.  Sometimes we approach.  Sometimes G-d draws away, and sometimes He approaches us.

* Definitions and Pronunciation

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