### Reading the Torah with Equal Intervals

a review by Prof. Daniel Michelson
Department of Mathematics
University of California, Los Angeles

{Glossary of the character transcription is at the end of the file.}

At this point a sceptical reader would exclaim that the whole system is nothing but a coincidence and the above explanation with 50 and 49 gates of wisdom was "cooked up" to tie several unrelated appearances of the word T!RH into a system. "I'm sure", this sceptic would continue, "you would be able to find such words and systems in any book". Since the author of this review was, until recently, such a sceptic - the question of coincidence versus intentional design will be addressed most forcefully in this article. Meanwhile let us mention that on the statistical basis the word T!RH is expected to appear with any given interval N in Genesis about 2 or 3 times. This estimate is based on the total number of letters in Genesis (78064) and on the number of the letters T (4152), ! (8448), R (4793) and H (6283) in the book. Indeed, T!RH appears 3 times in Genesis with the interval 50 which is what one would expect from any book of such length and of similar concentration of letters T,!,R,H. There is however no reason why one of these three appearances should start with the very first T of the book and why this should happen both in Genesis and Exodus. As a matter of fact the probability of such a coincidence is about 1 in 3 million!

The above is one of hundreds of patterns found by Rabbi Weismandel in Torah in the time of WWII. After his death in 1948(?) his students published in the early fifties the book "Torat Chemed" where just a handful of his findings were exhibited. The rest of the findings were lost. Of course at that time there were no computers. Instead Rabbi Weismandel was guided by a deep knowledge of Torah as for what to seek and where to seek. As for the length of the intervals - most of his examples refer to the numbers 50 or 26, the last being the Gematria of the four letter name of God ('-H-!-H = 10+5+6+5 = 26). Later on, a few followers of Rabbi Weismandel continued the search, which was still done by hand. We should mention Rabbi Shmuel Yaniv, Abraham Oren and their students. But the real breakthrough occurred in 1982 when the computer was put to work. Here most of the credit should be given to Dr. Eli Rips from Institute of Mathematics, Hebrew University who was joined by Dr. Moshe Katz from the Technion, Haifa and later on by Doron Viztum from Jerusalem. Let us make it clear - the computer does not have an intelligence to find meaningful patterns. Instead it is used as a fast and accurate counting machine. The text being investigated is typed into the computer and is stored there as a file of integers. A set of instructions would then tell the computer to look for a certain word in the text with equal intervals in a given range. For example, find all appearances of the word I\$RAL- Israel (which in integer form spells 10 21 20 1 12) in the first 10000 letters of Genesis, with equal intervals ranging from -100 to 100. The computer then shows that the word is spelled out only twice, the intervals being 7 and -50 and is located in the four verses 1:31-2:3 (see fig.2).
We are stunned by the fact that these verses constitute our Kiddush recited every Sabbath evening over a cup of wine. Indeed 7 and 50 are the only numbers related to the Sabbath. The number 7 stands for the seventh day of creation and also for the seventh year - the year of Shmita when the land rests. Then, after 7 Shmita cycles the land should rest also on the Jubilee year - which is the 50-th year. Again, a coincidence? A simple calculation shows that the probability of the word I\$RAL to appear once with a given interval in the above verses is about 1 in 1200. The chance of two appearances with intervals of 7 and 50 either backwards or forwards is about 1 in 400,000. Another interesting example is shown in fig.3. The text which is Gen. 38 tells the story of Yehuda and Tamar. As the result of their affair, Tamar gave birth to Perez and Zerach. From the book of Ruth we learn that Perez started a lineage which led to Boaz. Boaz married Ruth and had a son Oved, which had a son Yishai, which was the father of King David. So it was a natural question to ask whether King David with his lineage is hidden in this chapter. Indeed, you find the names BYZ, R!T, YBD, '\$' and D!D spelled out with the same interval -49, moreover they all appear in the chronological order! We already mentioned the importance of 49 being the 7-th Shmita followed by the Jubilee. However 49 is also the last day of the counting of Omer which starts on the second day of the Passover and ends a day before Shavuot. Every day in this counting has a name and the 49-th is called MLK!T \$BMLK!T - kingdom of the kingdom. Is there a name which would fit David, the king of kings, better? Let us also mention that David was born and died on the very day of Shavuot and the book of Ruth is traditionally studied on this holiday. But maybe this system is another coincidence? It is easy to estimate the probability of such an event. As we count the total number of letters in Gen. 38 and the relative proportion of each of the letters of the alphabet, we come to the conclusion that the probability of the word BYZ to appear in our chapter with a given interval is 0.02. (That is assuming that on the level of equal intervals the text is random). Similarly, for the other four names the probabilities are 0.63, 0.065, 0.76 and 0.25. The odds for all 5 names to show up with a given interval are about 1 in 6,500. If we also request that the names line up in chronological order, the chances are reduced to 1 in 800,000. Now, if one would claim that the interval 49 is as important as -49 and the same for 50 and -50, these 3 possibilities would increase the chances to 1 in 200,000 - still quite an impressive number!

Let us turn to the third example in fig.4. We are in the beginning of the Parasha !'CA where it talks about the famous dream of Jacob with the stairway reaching the sky. As Jacob awoke from his sleep he said, "Surely the Lord is present in this place, and I did not know it!" (Gen. 28:16). Where was this place? Rashi (the main commentator of the Torah) writes that this was Mount Moriah where the Temple was built later on. Moshe Katz who was reading the commentaries decided to check for the word MQD\$ - the Temple. Indeed, the word does appear with a very important interval -26 starting with the M of the word MQ!M (place) in the above verse. However as you continue to count 26 letter intervals after \$ of MQD\$ you find another five letter word HT!RH (the Torah) spelled forwards. Thus the two cornerstones of Judaism HT!RH and MQD\$ are spelled as one continuous sequence of 9 letters with the interval 26 (which is, to repeat, the numerical value of Lord's name). The probability of such an event (for a fixed position of the first letter M) is about 1 in 17 billion! In the same story we also find the words C'!N (Zion) and MQ!M (place) spelled out with the interval 26.

The next example in fig.5 (found by Moshe Katz) is related to Joseph's second dream (Gen. 37:9-10) - Here I had another dream and here the sun and the moon and eleven stars are bowing down to me". On which Jacob answers, "What is the dream you have dreamed? Are we to come, I and your mother and your brothers, and bow low to you to the ground?". Rashi explains what Jacob had on his mind: "the mother (the moon) already died, while Jacob did not know that the moon refers to Bilhah (Rachel's maid) who raised Joseph as if she was his mother". As we stick together the words A\$R XLMT HB!A (which you have dreamed...), they spell RXL MTH (Rachel died). Now we are looking for the word BLHH (Bilhah). The computer found two appearances of this word on the same page, both starting with the same letter B next to the phrase A\$R XLMT - one is with the interval -99 and another with -156. We don't know exactly the meaning of 99 however 156 bears a direct reference to Joseph being the Gematria of his name ('!SP = 10+6+60+80 = 156).

There are hundreds of equally impressive examples which are not shown here due to the limited scope of this review. However, on the basis of the presented material we ask again the same question - are the above systems a mere coincidence or they are deliberately planned? Now the sceptic concedes that the odds for each individual system are very small, however there are millions of different stories which one could look for so that occasionally some of them occur with small odds. Likewise in a lottery there are millions of players and few winners. The truth of the matter is that there are 3-4 people who have been searching mainly the book of Genesis by computer for the last two years. They explored perhaps a few thousand words and systems while the success ratio was astounding. Nevertheless, to counter the above argument on a statistical basis one has to find "story-independent" phenomena, i.e. something which could be checked automatically by computer and compared with other texts. The following example will be used to demonstrate such a general phenomena. This example is also important from a historical perspective since it marked the beginning of the "computer era" in the study of Torah.

A "hidden" Aaron in Leviticus

Our story starts somewhere in 1982. Abraham Oren from kibbutz Sde Eliahu was exploring manually whether the word AHRN (Aaron) is spelled out with equal intervals in the beginning of Leviticus. Why Aaron and why in Leviticus? As everybody knows, Leviticus talks mainly about the work of the Cohanim - the priests, and Aaron being the Cohen Gadol (the high priest) is the main hero of the book. Nonetheless, in the first open chapter (Parasha Ptucha) of Leviticus Aaron is not men- tioned even once. Instead it repeats four times "the sons of Aaron". Abraham Oren was familiar with the work of Rabbi Weismandel, so it was natural for him to suggest that Aaron is hidden inside the chapter in the way of equal intervals. And indeed he found quite a few. When he showed it to Dr. Eli Rips from Hebrew University, the latter typed this chapter on the computer and asked it to find all appearances of the four letter word AHRN in the chapter. The result of this search is shown in fig.6. There are altogether 25 hidden Aaron's not counting the explicit ones. The numbers which point to the circled A's are the sizes of the intervals which should be counted from these A's in order to obtain the word AHRN. The negative numbers correspond to the backward counting. In this example we are not selecting any specific interval like 26 or 50. Instead the computer checks all intervals from 2 to 235 (the maximal possible in this chapter), forwards and backwards from every letter A andtries to find the word AHRN. As Rips looked at the results he was overwhelmed by the large number of total appearances: 25. Indeed, the chapter is 716 letters long out of which there are 55 A's, 91 H's, 55 R's and 47 N's. For a random distribution of these letters a statistical formula shows that the expected number of Aaron's in the text should be about 8 and that the probability of finding 25 or more Aaron's is about 1 in 400,000. That is, it would take 400,000 pages of text like the one in fig.6 until one would find 25 or more hidden Aaron's on a page. A linguist could charge that the letters in the language are correlated so that the Hebrew of the Bible may "like" AHRN more then expected. Notice, however, that 12 Aaron's out of 25 are going backwards and it is not clear why the "forward" language should like them. And if it does, then equally well it should like other combinations of the four letters A,H,R,N. So Rips took all 12 possible combinations (there are 2x3x4=24, but forward and backward count as one) and performed with them the same experiment as with Aaron.In the lower part of fig.7 we see the results of the experiment. The word AHNR (meaningless) appears in the text 8 times and so does ARHN. The other results 9,7,5 etc. center around 8 with a deviation of +-3 in a complete agreement with the statistics and only the AHRN stands out. The next experiment is shown in the upper part of fig.7. As well known, in Hebrew there is a short and full spelling. In Torah the same words sometimes are spelled full and other times short. If we change the spelling, the equal intervals become at once non-equal. Hence there is no reason why the text should prefer AHRN in the form A(n)H(n)R(n)N over A(n)H(n+x)R(n+y)N. Now we fix the numbers x and y and let the computer to search for Aaron with all possible n (i.e. from 2 to 235). The numbers x and y vary from -5 to 5 and for each pair x,y the total number of Aaron's is shown in the table. We see that these totals vary from 2 to 15 with the average 7.3 and the standard deviation 2.4. The number 25 corresponding to x=y=0 (i.e. equal intervals) is 7.4 standard deviations away from the average! So indeed, our text "likes" Aaron with equal intervals. But what about other words, maybe they exhibit the same phenomenon? And what about other texts? For comparison Rips took all 4-letter words, more precisely all 4-letter combinations in Hebrew alphabet. Since there are 22 letters the total number of combinations is 22x22x22x22/2 = 117,128.

At this point the sceptic is ready to admit that people could have done it deliberately. "You know", he says, "they had a lot of time to do this. The sages say that Rabbi Akiba used to count letters. So apparently there was such a tradition".

Let us explore this line of thought. Suppose some people, say the priests themselves planted these Aaron's in the text. But for what purpose? To impress somebody later on? However, until discovered by Abraham Oren and Eli Rips this secret was absolutely unknown. Moreover, were it discovered 40 years ago nobody would be impressed by it. Indeed you should do all the comparisons to see how outstanding the phenomenon is - and this was impossible before the advent of computers. Did the author(s) of the book anticipate the computer era? And then a technical question - how did they do it? Suppose there was an existing text without Aaron's like the Samaritan Torah. Is it possible with a little editing to create the 25 Aarons? The author of this review actually tried to add another (26-th) Aaron to the existing 25 with no avail. But even if this is possible, there is a limit of how many words one can hide in a meaningful text. The 25 4-letter Aarons put 25x2=50 constraints on the 716 letter text (i.e. the distance between A and H is the same as between H and R and as between R and N - giving two constraints per word). It is hard to set a precise limit but we feel that one can't produce a meaningful story where 30% of its letters are tied up by constraints like those above. And this is not a question of personal ingenuity or whether the author had a computer at his disposal. The language has its set of words and grammatical rules, so mathematically speaking you are going to have more equations (constraints) then the unknowns (the words). Of course, if the author is creating the language simultaneously with the text - then the above limit does not apply.

These are indeed confusing questions. So our sceptic backs up and suggests that maybe the whole system with Aaron's is just another coincidence. "After all, why did you take the first chapter and why Aaron? There are so many chapters and so many important words you could have chosen so that one success even with a ratio of 1/400,000 is not outstanding at all!". We could reply that Aaron is the most important word in Leviticus and intuitively the first chapter has preference over the other ones. However the whole story with Aaron's was brought here not for the sake of showing another oddity but rather to demonstrate some general phenomena.

The clustering effect

After the discovery of Aaron's, Rips obtained an electronic text of Genesis and started a systematic investigation. (It was only recently that the full electronic error-free text of Torah became available to us). By the text of Torah, unless stated otherwise, we always mean the traditional Ashkenazi Masoretic text as published in socalled Koren edition. There is another text accepted among Yemenite Jews. These two versions were carried by two independent traditions for more than a thousand years. Yet, as we compare these texts, they differ only by 9 letters out of 304,805! Among the nine, there are 3 different letters in Genesis (of a total 78,064). Besides, there are several ancient manuscripts. One of them is called the Leningrad codex (because it is in the possession of a Leningrad library) and was copied 1,000 years ago in Egypt. As was shown recently by Dr. Mordechai Breuer in "Keter Aram-Tzova" this text differs from the Koren edition by 130 letters. Almost all of these 130 letters are contradicted by the majority of other manuscripts and, most important, by the Masoretic instructions. Nonetheless the Leningrad codex is called the "scientific text" of Torah and is used by several universities for their databases. Clearly, even one missing or extra letter destroys the hidden words which "leap" over this letter. However the examples shown in this review appear in parts of Genesis which are away from the doubtful letters and hence are not affected by them.

Our sceptic might be unimpressed by the probability of 1/30,000. Indeed, with Aaron's we already had 1/400,000. However this time the test was both word and segment independent. Namely, instead of a specific (though important) word Aaron we took a big "natural" sample and instead of the first chapter - the whole book of Genesis. One also should bear in mind that the lustering is only one aspect of the infinite information hidden in Torah in the way of equal intervals. There is no clustering for "Torah" in fig.1 or for "Israel" in fig.2. King David is not mentioned explicitly in fig.3 so we lose another story and likewise for the "Temple" in fig.4 and "Bilhah" in fig.5. One should really wonder that after all non-trivial patterns have been neglected there is still something to observe.

In the next section we will demonstrate another general idea which is common to many words and patterns.

The minimal intervals

When the computer searches for a certain word with equal intervals in a wide range of numbers it will find the word many times. Some of the intervals may be of special interest like the numbers 50, 26 etc. But what shall we do with the other ones? In the course of numerous experiments Rips observed that the short intervals tend to be more significant than the long ones, i.e. they appear more often in relevant places. We will present here one example of this phenomenon. The text in fig. 15 consists of Gen.2 (this is an enlargement of the third page of fig.10). Verse 9 reads: "And from the ground Hashem G-d caused to grow every tree that was pleasing to the sight and good for food with the tree of life in the middle of the garden and the tree of knowledge of good and bad". The names of the trees however are not mentioned in the chapter. So Rips suggested that perhaps these names are hidden in equal intervals. The book of Yehuda Feliks "The fauna and flora in the Torah" lists the names of all the trees which are mentioned in Torah. And all of these names - a total of 26, were found in the above chapter! Before the reader jumps out of his seat, let us explain that three- or four-letter words would normally appear with some intervals in a segment as long as ours (about 1000 letters). What is so exceptional here - is that most of the intervals (except for YRMN and LBNH) are very short. There is no other segment in Genesis of such length which contains so many trees with intervals less than 20. Based on the density of the letters in the chapter one could estimate the probability of the "orchard" phenomenon - the number is about 1 in 100,000!

Conclusion

We started with the "Torah" of Rabbi Weismandel, went through the examples of "Israel", "King David", "Temple-Torah", "Rachel with Bilhah" ,to "Aaron", then to the clustering effect in general and to the "orchard" and the minimal intervals phenomenon. There are many more fascinating examples and stories which could not be included in this limited review. A book with much of this material should soon be published in Israel. We hope that our sceptic also concedes that the equal interval phenomenon is not an imagination of a few "phony" people or a deliberate trickery with a computer but a reflection of a hidden design. We are far from understanding the rules of this design, in particular - what stands behind the numerical values of all the different intervals? In recent years there were some other coded sys- tems discovered (or rediscovered) in the Torah. Let us mention the multiples of seven, where the key words in each chapter appear either 7 or 14 or 21 etc. times. Another rule discovered by the late Rabbi Suleiman Sasson states that for each word which is repeated in Torah more than 80 times, its 80-th appearance is in a segment which talks about a promise, covenant, marriage or purchase (i.e. different types of contract). The distinction of the equal intervals is that they appear on the letter rather then the word level and that they contain apparently limitless information.

But who made this design? Nachmanides writes in the introduction to his commentaries on Torah that Moses saw the Torah as a letter string of a black fire on the background of a white fire. This string of letters was not divided into words. As G-d dictated the Torah to Moses, he (Moses) wrote it accordingly in the form of words and chapters. As Maimonides states in the introduction to Mishne Torah, Moses wrote the Torah before his death - one copy for each tribe and one to be kept in the Ark. It is believed that the modern Torah text is the exact copy of the original (modules maybe few letters, as suggested by the comparison of the Yemenite and Ashkenazi texts). This is what Judaism claims.

Three thousand three hundred years ago there was another sceptic - Pharaoh was his name. Our story in Ex. 11-12 is told after Pharaoh had experienced nine plagues. He was still not convinced because, as Torah says, "The Lord had stiffened the heart of Pharaoh". Should one wait for the tenth plague?

Here are some explanations for the attached material. Since we don't have Hebrew letter printer the Hebrew letters in the article have been represented by similarly sounding English letters or similarly looking characters. The correspondence is as follows

```  English letters/characters	       Hebrew letters

A				    Alef
B				    Beit
G				    Gimmel
D				    Dalet
H				    Hey
!				    Waw
Z				    Zain
X				    Chet
+				    Tet
'				    Yud
K				    Kaf
L				    Lamed
M				    Mem
N				    Nun
S				    Samech
Y				    Ain
P				    Pey
Q				    Kuf
R				    Reish
\$				    Shin
T				    Tav
```

Please treat with respect the sheets with the text of Torah. If you wish to dispose them, they should be buried by Chevre Kaddisha. If you don't know how to do it, bring it to some synagogue. The article is expected to be published in the Journal "Be-or Ha-Torah", in English.

The research of the codes in Torah and the publication of the material requires substantial financial support. We believe that the discovery of the codes will have a strong impact on society and will illuminate the eyes of those who are not indifferent to the truth. If you wish to support the research, please send your contribution to the foundation "Forum for Cultural and Educational Exchange", 1100 S. Carmelina Ave. Los Angeles, CA 90049. The number of the foundation for tax deduction purposes is 23 7134525.

Recently I was informed: The article "Codes in the Torah" (which I first found at several places on the net) is originally from a publication called "B'Or Torah" which is published by Shamir, PO Box 5749, Jerusalem, Israel, phone 02-223-702. The article was published in No. 6, 1987

I inquired and was told by a university official from the UCLA:
"Prof. Michelson was here from 1981 to 1988, according to our records."