Codes in the Torah

  Codes in the Torah

(The First 5 Books of the Hebrew Bible)
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This article originally appeared in the 1987 English edition of B'OR HA'TORAH and has been reprinted in a format, more appropriate for the internet. B'OR HA'TORAH retains full rights to the information printed in this article and if you wish to contact the publishers, SHAMIR, write to them at 6 David Yallin Street, Jerusalem or call them at 5385702. When making an international call to Shamir, dial 972-2-5385702.
Biography of the author that appeared in the 1987 English edition of B'OR HA'TORAH
Daniel Michelson was born in Riga, Latvia, in 1949 to parents who survived the Holocaust. In 1971 he graduated in Mathematics from Moscow University and emigrated to Israel. From 1972-1980 he taught Mathematics at Tel Aviv University, studied toward a doctorate and served in the army. Upon completing his PhD thesis in Applied Mathematics in 1980, he did postdoctoral research at the University of California at Los Angeles. In 1983 he was awarded fellowships from the Alfred P. Sloan Foundation and from the Igal Alon Foundation.
In the fall of 1983 Michelson returned to Israel and started to teach at the Hebrew University. There he learned about the codes in the Torah from his colleague and friend Dr. Eli Rips. This led him eventually, he says, "to do t'shuva".
Later he became involved in researching the Torah codes and lecturing about them (particularly in Los Angeles and Berkeley, California). Currently, Michelson is an Associate Professor of Mathematics at UCLA and at the Hebrew University. He lives now in Har Nof, Jerusalem, with his wife and three sons.
[Information Update! Michelson now (Jewish Year 5762=2002) teaches at the Weizmann Institute of Rehovot).]

What is Equal Interval Reading?

Let us eliminate the spaces between the words of the Torah (the Five Books of Moses) and consider the text as a sequence of letters. Now, starting from a certain letter, let us skip N-1 letters and read the N-th one, then again skip N-1 letters and read the N-th one, and so on. This will be called a reading with the interval N, The number N may also be negative, in which case the reading is backwards, Of course, besides the interval N one has to know where to start counting and how many intervals to go.

Let us make things more clear by showing a few examples. If we start with the first letter ת that appears in Genesis (i.e. the ת of בראשית, [= letter 6 in Figure 1 ]) and skip 49 letters, we arrive at the letter ו in תהום [letter 56, Figure 1]. Then we again skip 49 letters and arrive at the ר in וירא [letter 106, Figure 1] again skip 49 letters and arrive at the letter ה [letter 156, Figure1].

Thus we find that the word תורה (Torah) is spelled out by 50-letter intervals right in the beginning of Genesis.

The number 50 has several important meanings in Judaism. Every fiftieth year is a jubilee year; the Torah was given 50 days after the exodus from Egypt; and there are also 50 gates of wisdom.

The above example is a part of a larger pattern found by Rabbi Michael Dov Weismandel about 40 years ago ( Press here to see Figure 1 in a separate window). Namely, in Exodus (the second book of the Torah) the word תורה is again spelled out by a 50-letter interval beginning with the very first letter ת in the book (i.e. the ת of ואלה שמות letter 8 of Exodus, in Figure 1). In Leviticus (the third book) תורה is not spelled out in intervals: In Numbers (the fourth book of the Torah), the word תורה is spelled out backwards by the interval -50, starting with the letter ה (letter 14 in the chart for the Book of Numbers) in the first verse of the book. Finally, in Deuteronomy (the last of the Five Books of Moses) תורה is again spelled backwards, but this time starting from the fifth verse in the book (instead of the first) and at an interval of -49 (instead of -50).

What causes this deviation, and why is there no תורה spelled out at a 50-letter interval in the third book, Leviticus? The Vilna Gaon wrote in Aderet Eliyahu that Deuteronomy actually starts from the fifth verse, while each of its first four verses corresponds to the first four books. Indeed, the fifth verse reads: "On the other side of the Jordan, in the land of Moab, Moses undertook to expound the Torah. He said..." It is claimed that Moshe Rabbenu (Moses our Teacher) was given 49 out of the 50 gates of wisdom. Since the subsequent explanation of the Torah is given from the mouth of Moshe, the word תורה is spelled out at the beginning of Deuteronomy at an interval of -49.

We see that the system is symmetric. תורה is written out forwards in the first two books and backwards in the last two, but not at all in the middle book. Instead, in the third book (Leviticus) we find the ineffable four-letter name of G-d who gave the Torah spelled out by an eight-letter interval starting from the very first word in the book. (Eight equals seven plus one, and thus is closely related to 50=7*7+1, but this is another story upon which we shall not elaborate here.)

At this point, a skeptical reader would exclaim that the whole system is nothing but a coincidence and the above explanation about 50 and 49 gates of wisdom was cooked up to tie several unrelated appearances of the word תורה into a system. "I'm sure," this skeptic would continue. "You would be able to find such words and systems in any book." As the author of this review until recently was such a skeptic, the question of coincidence versus intentional design will be addressed most forcefully in this article. Meanwhile let us mention that on a statistical basis, the word תורה is expected to appear with any given interval N in Genesis about two or three times. This estimate is based on the total number of letters in Genesis (78,064) and on the amount of the letters

ת (4152), ו (8448), ר (4793), ה (6283)
in the book. Indeed, תורה appears three times in Genesis at the interval 50, which is what can reasonably be expected from any book of such length and of similar concentration of the letters ת,ו,ר,ה. There is however no reason why one of these three appearances should start with the very first ת 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 one in three million!

The above is one of hundreds of patterns found in the Torah by Rabbi Weismandel during World War Two. After his death in 1957 (~Jewish Year 5717), his students published a book entitled Torat Chemed that showed just a handful of his findings. The rest of his findings were lost. Of course at that time there were no computers. Instead, Rabbi Weismandel's deep knowledge of Torah guided him in deciding what and where to seek.

As for the length of the intervals-most of his examples use the numbers 50 or 26, the latter being the gematria of the ineffable four-letter name of G-d (26=5+6+5+10 =י-ק-ו-ק). (The letter ק replaces the letter ה here so as not to use the Name in vain.)

Later on, some of Rabbi Weismandel's followers continued his search, still working by hand. Rabbi Shmuel Yaniv and Avraham Oren and their students should be mentioned here. But the real breakthrough occurred in 1982 when the computer was put to work in this direction. Here most of the credit should be given to Dr Eli Rips of the Institute of Mathematics at the Hebrew University in Jerusalem. Rips was joined by Dr. Moshe Katz of the Haifa Technion and later by Doron Viztum of 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 under investigation is typed into the computer and stored there as an electronic file. A set of instructions then tells the computer to look for a certain word in the text at equal intervals in a given range.

For example, find all appearances of the word ישראל (Israel) in the first 10,000 letters of Genesis, at equal intervals ranging from -100 to 100. The computer then shows that the word is spelled out only twice, at intervals of 7 and -50, and is located in verses 1:31-2:3 (The relevant information is shown in Figure 2.
These are exactly the four verses that constitute the kiddush, which we recite over a cup of wine every Friday night to sanctify the Sabbath. This is astounding because seven and 50 are the only numbers related to Shabbat. Seven stands for both the Seventh Day of Creation and the seventh year of shmita when the land rests. After seven shmita cycles the land rests also on the fiftieth year, the jubilee year.

Is this merely coincidence? A simple calculation shows that the probability of the word ישראל appearing once at a given interval in the above verses is about 1 in 1200. The chance of two appearances at the intervals of 7 and 50 either backwards or forwards is about 1 in 400,000.

Another interesting example is shown in Figure 3. The text, Genesis 38, tells the story of Yehuda and Tamar, who gave birth to Peretz and Zerach. From the Book of Ruth we learn that Boaz was descended from Peretz. Boaz married Ruth and had a son Oved, who had a son Yishai, the father of King David. It is natural to ask, then, whether King David and his lineage are hidden in the verses about Yehuda and Tamar. And indeed we do find the names

בעז, רות, עבד, ישי, and דוד
spelled out backwards at the same interval of -49. Moreover, they all appear in chronological order! We have already mentioned the importance of 49 being the seventh shmita, which is followed by the jubilee year. The number 49 is also related to the counting of the Omer, which starts on the second day of Pesach and ends a day before Shavuot. Each day of the Omer is named according to the descending order of the lower seven sfirot. The forty-ninth day of the Omer is thus called Malchut sheb'Malchut (Kingdom of the Kingdom). Could there be a name that better fits David, the king of kings? Furthermore, Shavuot is the very day that David was born and died. It is also the holiday on which the Book of Ruth is traditionally studied.

Could this example also be a coincidence? It is easy to estimate the probability of such an event. After counting the total numbers of letters in Genesis 38 and calculating the relative proportion in which each letter of the alphabet appears in this total, we conclude that the probability of בעז appearing in our chapter at 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, 0.65, 0.76 and 0.25. The odds for all five names to show, up at a given interval are about 1 in 6500. If we also request that the names line up in chronological order, the chances are reduced to 1 in 800,000. Now, if the interval 49 is claimed to be as significant as -49, and as 50 and -50, then these three possibilities would increase the chances to 1 in 200,000. This is still quite an impressive number!

Let us turn to the third example in Figure 4 taken from the passage about Jacob's dream of the ladder reaching to heaven. When Jacob awoke from his sleep he said, "Surely the L-rd is present in this place, and I did not know it!" (Genesis 28:16). Where was this place? Rashi (the major commentator on the Torah) writes that it was Mount Moriah where the Temple later was built. Relying on the commentaries, Dr. Moshe Katz decided to check for the word מקדש (Temple). Indeed, מקדש appears through the important interval of -26, starting with the מ of the word מקום (Figure 4, letter 33) in the above verse. However, if we continue to count at -26 intervals after theש of מקדש, we find another five-letter word, התורה (the Torah) spelled forwards. Thus the two cornerstones of Judaism, התורה and מקדש, are spelled as one continuous sequence of nine letters at an interval of 26 (which is, to repeat, the numerical value of the Tetragrammaton). The probability of such an event (for a fixed position of the first מ ) is about 1 in 17 billion! In the same story we also find ציון (Zion) and מקום (place) spelled out at 26-letter intervals. The next example in Figure 5 (found by Moshe Katz) is related to Joseph's second dream in Genesis 37:9-10: "Here I had another dream and here the sun and the moon and eleven stars are bowing down to me". On hearing this, Jacob asked his son, "What is this 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 meant: "the mother [Rachel, represented by the moon in the dream] had already died, and Jacob did not know that it referred to Bilhah [Rachel's maid] who raised Joseph as if she were his mother." If we run on together three of the words within Jacob's rebuke,

הבוא חלמת אשר
we read within this string of letters
מתה רחל
(Rachel is dead.)

Now let's look for בלהה (Bilhah) in the same chapter. The computer found two appearances of this word, both starting with the ב in the word נבוא that comes directly after

הבוא חלמת אשר .
One occurs at an interval of -99 and the other at -156. We will not elaborate here on how the number 99 relates to the matriarchy; however 156 is the gematria of
Joseph's name יוסף namely ף=80 ס=60 ו=6 י=10
80+60+6+10=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 material presented, we again ask: are the above systems a mere coincidence or are they deliberately planned? The skeptic must concede that the odds for each individual system are very small. However, there are millions of different stories in which we can look for these patterns, so that occasionally some of them occur at small odds. Likewise in a lottery there are millions of players and a few winners. The truth of the matter is that only three to four people have been searching mainly the book of Genesis by computer for the last two years. They explored perhaps a few thousand words and systems with an astounding success ratio. Nevertheless, to counter the above argument on a statistical basis we have to find story-independent phenomena, which can be checked automatically by computer and compared with other texts. The following example will be used to demonstrate such a general phenomenon. This particular example is also important from a historical perspective since it marks the beginning of the "computer era" in the Study of Torah.

A "Hidden" Aaron in Leviticus

Our story starts in 1982. Avraham Oren of Kibbutz Sde Eliyahu was exploring the beginning of the first chapter of Leviticus manually to find out if אהרן (Aaron) is spelled out there at equal intervals. Why Aaron and why Leviticus? Leviticus discusses mainly the work of the cohanim, the priests; and Aaron, being the Cohen Gadol (High Priest) is the main hero of the book. Nonetheless, in the first open chapter ( Parasha P'tucha ) of Leviticus Aaron is not mentioned even once. Instead, the phrase "the sons of Aaron" is repeated four times. Being familiar with the work of Rabbi Weismandel, Oren naturally wondered if אהרן was hidden inside the chapter in equal intervals. And indeed he found quite a few. When he showed his findings to Dr. Eli Rips of the Hebrew University, the latter typed the whole chapter on the computer and asked it to find all appearances of אהרן therein.

The result of this search is shown in Figure 6.
There are all together 25 hidden "Aarons" not counting the explicit ones. The left side of Figure 6 shows the location of the start of all the hidden Aarons (namely the letter א of אהרן ) together with the equal length interval that separates the first letter from the second, the second from the third, and the third from the fourth. (As previously, negative numbers mean counting backwards.) In this example we are not selecting a specific meaningful interval like 26 or 50. Instead, the computer checks all intervals from 2 to 235 (the maximum possible in this chapter) forwards and backwards from every א, hunting the word אהרן.

When Rips received the results, he was overwhelmed by the large number of total appearances: 25. In the 716-letter long section, containing the first 13 sentences of Leviticus, there are

55 א's, 91 ה's, 55 ר's 47 נ's.
In a case of random distribution of these letters in the chapter, a statistician would expect only eight appearances of Aaron. Moreover, he would say that the probability of finding 25 or more hidden "Aarons" is about 1 over 400,000. That means that 400,000 pages of text like the one in Figure 6 would have to be searched until 25 or more hidden ) אהרן 's would be found on one page.

A linguist could counter that the letters of biblical Hebrew are correlated so that the language "likes" אהרן more than would be expected. Notice, however, that 12 אהרן's are going backwards, and it is not clear why the "forward" language should like them. And if so, then it should like other combinations of א,ה,ר,ן equally well. To check this, Rips took all 12 possible combinations (there are 2x3x4=24, but forward and backward count as one) and performed on them the same experiment as was done on the word אהרן. In the lower part of Figure 7 we see the results of the experiment.
The meaningless letter combination אהנר appears in the text 8 times, as does ארהן. The other results of 9,7,5 and so on center around 8 with a deviation of ±3 in a complete agreement with the statistics. Only אהרן stands out in contrast to this agreement.

The next experiment is shown in the upper part of Figure 7. As is well known, in Hebrew there is a short and a full spelling. Sometimes the same words are spelled fully and other times in short form within the Torah. A change in the spelling would make equal intervals non-equal. Thus there is no reason why the text should prefer

ן n
ר n
ה n
א
over
ן n+y
ר n+x
ה n
א
So we fix the numbers x and y and let the computer search for אהרן 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 אהרן's is shown in the table. We see that these totals vary from 2 to 15 with an average of 7.3 and a standard deviation of 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 at equal intervals.

But what about other words? Maybe they exhibit the same phenomenon? And what about other texts? For comparison Rips took all possible four-letter combinations of the Hebrew alphabet. As there are 22 letters, the total number of combinations, not distinguishing forward and backward spelling, is 22x22x22x22/2=117,128. He took each possible "word" out of the 117,128 possibilities-say אבגד- and performed the same experiment on it as he did on אהרן. Namely, he let the computer find the number of times that this word appears in our chapter and compared the result with the statistically expected number of appearances. Suppose that for אבגד these numbers are correspondingly 5 and 3. Then he computed the probability of having 5 or more appearances instead of the expected 3. The result was 0.185.

Now turn to the table in Figure 8.
The vertical axis shows the number of appearances of a word, while the horizontal axis shows its probability (on a logarithmic scale). The number 232 in the sixth row, third column shows that 232 words out of 117,128 appeared five times in the text and the probability of them appearing that many times was about 1/10. Thus the word אבגד was counted among the 232. And similarly for the other numbers. As the probability decreases and the number of appearances increases, there are fewer and fewer words in the table. The position of the word אהרן is shown by the circle. Obviously, Aaron is the winner of the competition!

There is just one other "word", יטעא (meaningless) with the same probability of 1/500,000 which appeared six times. Actually: all letter-combinations with a probability of less than 1/1000 turned out to be meaningless. There are also 12 words which appeared more often than אהרן but their probabilities are quite reasonable. Indeed, there are letters that appear more frequently and less frequently in our text. Words containing the more frequent letters should normally appear more often.

But what about other texts? For comparison, Rips took a fragment of the same length (716 letters) from the beginning of the novel Hachnasat Kala by Nobel Prize winning Israeli author Shai Agnon and ran the same experiment on it for 117,128 words. We see on the second chart in Figure 8b that the distribution of numbers is the same as above with only one exception - the circle which contained אהרן is now empty! Here again, no meaningful word passed a probability limit of 1/1000. This proves once again that the entire phenomenon of אהרן has nothing to do with the Hebrew language.

Perhaps the comparison with Agnon is unfair since his is a different, "modern" Hebrew. Ideally, a text should be chosen that is firstly canonic and secondly very close to the Torah. Professor Ben-Chaim of the Academy of the Hebrew Language came up with the excellent idea of using the Samaritan Torah. The Samaritans are thought to be the descendants of Kutim, the nations brought into Israel after the exile of the ten tribes in the seventh century BCE. Although influenced by the Jewish religion, they did not become part of the Jewish nation. There are still about 2000 Samaritans living in Nablus and Holon who possess a Torah different from ours. Actually, there are a number of differences among their own manuscripts, so it is hard to talk about an established Samaritan version. Nonetheless, a few years ago two Samaritans, the brothers, Tzdaka, published the most authentic version of the Samaritan text and compared it with our Torah. Figure 9 shows where in the beginning of the book of Leviticus, our text of the Torah differs in spelling and in wording from the Samaritan text.

The "Aaron section" discussed above consists of the first 13 verses. (The differences between the two texts however, have been presented in Figure 9 up to verse 17.) In addition to an extra 20-letter phrase inserted into the tenth verse of the Samaritan text, there are 17 other points of difference (in the first thirteen verses). Otherwise, both texts tell the same story and in translation would read the same. So it was very interesting to see what effect these differences had on אהרן. And lo and behold, they destroyed 22 out of the 25 hidden "Aarons"! However, seven new "Aarons" surfaced. Thus the total became 10 instead of 25-in complete agreement with the statistics since the expected number is about 8 with a deviation of plus or minus 3.

At this point the skeptic 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 Akiva used to count the letters of the Torah. So apparently there was such a tradition."

Let us explore this line of thought. Suppose somebody, say some of the priests themselves, planted these "Aarons" into the text. But for what purpose? To impress later generations? Until discovered by Avraham Oren and Eli Rips, this secret was absolutely unknown. Moreover, had it been discovered 40 years ago, nobody would have been impressed by it. Indeed, we must make all the comparisons to realize how extraordinary this phenomenon is; and it was impossible to do this before the advent of the computer. So our skeptic backs up and suggests that maybe the whole system of the "Aarons" 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 even one success with a ratio of 1/400,000 is not at all remarkable!" We 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 of the "Aarons" was brought here not with the intention of showing another oddity but with the intention of demonstrating some general phenomena.

The Clustering Effect

After discovering the "Aarons", Rips obtained an electronic text of Genesis and started a systematic investigation. (The full electronic error-free text of the Torah became available to us only recently.) By the text of the Torah, unless stated otherwise, we always mean the traditional Ashkenazi Masoretic text published by Koren in Jerusalem. 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! Three of these nine appear in Genesis (which totals 78,064 letters). Further comparison may be sought in the various existent ancient manuscripts, for instance the 1,000 year-old Leningrad Codex written in Egypt (and named after the library that possesses it). 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 considered the "scientific text" of the Torah and is used by several universities for their databases. Clearly, even one missing or extra letter destroys the hidden words, which "leap" over it. However, the examples shown in this review appear in parts of Genesis not containing any of these contested letters and hence are not affected by them.

Let us now define the clustering effect. As we saw with "Aaron", the word was spelled explicitly (four times) in the same chapter where it appeared in a large concentration in equal interval form. Rips wanted to check whether the same phenomenon occurs with other words. Since it was not feasible to scan all the words, Rips started with the words at the beginning of Genesis. The text in Figure 10 consists of Genesis chapter 1 and 2 and Genesis 3 verse 1 as they appear in the Koren edition. The first 2956 letters (i.e. Genesis chapter1,2 and part of Genesis 3:1) have about 120 different words longer than two letters (not counting different grammatical forms). Each word was run by the computer to find where it appears at equal intervals. The intervals n were taken in a range from 2 to some N, both positive and negative.

The results of such a search for the word עדן (Eden) are shown in Figure 10. The word עדן is spelled out explicitly in three places as shown by the words highlighted in yellow. The red asterisks show the hidden "Edens" and the numbers at the left side of the chart indicate where the ע of עדן is located and the appropriate intervals between that letter and the next letter of the word.

The range of intervals N was taken to be 120. The number N is chosen in such a way that there is a reasonable amount of hidden words. So then, if one chooses N=240 there would be twice as many hidden "Edens" sprawling over the text and it would be difficult to see the clustering. Likewise for N=60 it would be too few words to make statistical estimates. We see that there are eight hidden "Edens" within the first 1815 letters of Genesis (Genesis 1:1- Genesis 2:3). The story of the Garden of Eden is told starting in the verse Genesis 2:4 starting at letter 1816. Here inside a segment of 379 letters 16 hidden "Edens" appear? What force has drawn them together? Maybe the three explicit "Edens" increase the local density of the letters ד ,ע , and ן so that there are more chances for the hidden ones? A computation like the one performed for Aaron shows that the expected number of "Edens" is about 5 and the probability of such a deviation is about 1 in 10,000. (We see another weaker cluster close to the end of Figure 10 where the Torah narrates the creation of woman.)

In Figure 11 there is a similar example with the word הנהר (the river). The word is mentioned four times explicitly as shown by the words highlighted in yellow. When run on the computer with intervals up to 80, it produces a cluster of 11 words that start within lines 76 to 101, while on a usual range of even 40 lines it appears about three times. [ Figure 11 has been divided up into 120 lines ].

Next in Figure 12 the word מקוה (gathering of water) is exhibited. There is a cluster of ten words around the explicit מקוה in the first 40 lines of Figure 12 while in the other two 40 line segments, in Figure 12, the word appears once or twice. Note that this time the hidden words do not cross over the explicit one so that the letters of the explicit מקוה could not cause the cluster.

Figure 13 demonstrates a similar effect with the word מקום surrounded by a cluster of eight hidden words, while in the second and third segments of 40 lines, there are all together four hidden words.

The results for long words are especially interesting. Obviously, the longer a word is, the smaller are its chances to be found in a text at a given interval. In Figure 14 three such words are shown: בהבראם (as they were created), החוילה (the Havila [river]) and המועדים (the appointed times). The six-letter word בהבראם was searched for by the computer over the whole book of Genesis (i.e. 78,064 letters) at equal intervals ranging from -300 to 300. It was found four times. At the interval 176, it clusters around the explicit word. Similarly, the word החוילה in the same range appears six times and clusters around the explicit word at the interval of 167. The seven-letter word המועדים was searched for in the book of Genesis at intervals from -10,000 to 10,000 ! It appeared only once, at the interval of 70, clustering exactly where the word is spelled explicitly. (By the way, there are 70 specially appointed times for holy days called מועדים in a year, as defined in Leviticus 23: fifty-two Sabbaths, seven days of Pesach, one day of Shavuot, one day of Rosh Hashana, one day of Yom Kippur, seven days of Sukkot and one day of Shmini Atzeret.)

But what about other words? Obviously, we cannot show here all of the results. However, about 40% of the words in the above three pages produced a strong clustering effect. Another 40% showed a moderate clustering. And the rest-no clustering. Part of the clustering is effected by the non-even distribution of letters. For example, when the word אדם (Adam) is mentioned in Genesis 2:5,7 the word אדמה (earth) appears nearby, adding the letters ם,ד,א to the text, thus increasing the likelihood of finding a hidden אדם. When we took a 3OOO-letter piece of text from Chaim Nachman Bialik's Hebrew novel Arie Ba 'al Guf, there was also a cluster effect although much weaker than in Genesis. Hence, in order to measure the "net" clustering, Rips suggested comparing the equal intervals with the non-equal ones in the same text, as was done with "Aaron" (see Figure 7.)

The next question is how to measure the clustering quantitatively. The simplest way is to specify in advance a neighborhood of the explicit word and then check how many hidden words appear in this neighborhood. It is clear, however, that for longer words the neighborhoods should be greater than for the shorter ones, and hence it is preferable not to compare words of different lengths. Finally, a controlled experiment was run for all three-letter nouns in Genesis 1 and 2, all together 50 words. The neighborhoods to be considered were 300 letters long (about eight lines) and centered on the explicit words. The total number of hidden words in these neighborhoods was 370 versus the expected 300, which was four standard deviations away from the expectation. The results for non-equal intervals were about the average.

Next the same experiment was performed on the Samaritan version. Here the results for the equal and non-equal intervals were about the same as the expectation.

Four standard deviations correspond to the probability of about 1/10,000. This is indeed a very small number. However, some statisticians may say that the text under investigation is too short. Besides, for three-letter words the non-equal interval test is very limited. That is, for the word עדן we consider the sequences

ן n+x
ד n
ע
with fixed x and all possible n. The number x should be small so that the non-equal intervals would be a small perturbation of the equal ones. For example, if x varies between -5 and 5 we have only ten different results to compare. If the word is longer, e.g. a five-letter word אבגדה the perturbed sequences are
ה n+z
ד n+y
ג n+x
ב n
א
so that with x,y,z in the same range of -5 to 5 there is a sample of 1330 different results.

Hence Rips suggested checking the clustering for five-letter words over the whole book of Genesis.This requires a prohibitive amount of computations, so Rips restricted himself to all four-letter nouns preceded by the definite article ה encountered in Genesis. The final list consisted of 86 words. Next Rips defined a probability function, which measured the clustering for each word. The definition is too technical to be presented here. Roughly speaking, the function attains the values between 0 and 1, is uniform for a random text and becomes small when a hidden word with a short interval appears close to the explicit one. Then for each word a "race" was performed in which the equal intervals competed with the non-equal perturbations. In the first "race" the numbers x,y,z were between -2 and 2, thus providing 5x5x5=125 "runners". The probability function was measured and the "runners" with the smallest value would win.

The results of the 86 "races" are as follows. In three instances the equal intervals defeated the non-equal ones. The words were המקנה (the livestock), החתמת (the seal) and הבהמה (the domestic animals). For 11 more words the equal intervals were among the top 10% of the "runners. These results are not impressive at all since the probability that 14 out of 86 instances would be in the upper 10% is about 1/20. Next, the three winners were "allowed" to compete with about 5000 "runners". Namely, the range of x,y, and z in the non-equal intervals was increased from [-2,2] to [-8,8] which produced 17x17x17=4913 "competitors". (It was too expensive to make such a "race" for all the words since it takes several hours of computer time to run a single word.) The words המקנה and החתמת were champions also in the big race. Now the combined phenomena of the 14 top 10% words and the two top .02% ones have a probability of 1 over 30,000.

The same experiment was performed also on the Samaritan text. Here only two words הקדשה (the harlot) and המגדל (the tower) -were in the top 10% and no word entered the upper 1%, Thus the Samaritan text behaves like a "normal" one.

Our skeptic might be unimpressed by the probability of 1/30,000. Indeed, with the "Aarons" 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 like 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 clustering is only one aspect of the infinite information hidden in the Torah through equal intervals. There is no clustering for "Torah" in Figure 1 or for "Israel" in Figure 2. King David is not mentioned explicitly in Figure 3 so we lose another story and likewise for the "Temple" in Figure 4 and "Bilhah" in Figure 5. It is quite amazing that after all nontrivial 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 Minimum Intervals

When the computer searches for a certain word at equal intervals of numbers, it will find the word many times. Some of the intervals may be of special interest like the numbers 50 and 26; 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; they appear more often in relevant places. We will present here one example of this phenomenon. The text in Figure 15 consists of Genesis chapter 2 and the first three verses of chapter 3.

Verse 9 reads: "And from the earth G-d caused to grow every tree that was pleasing to the sight and good for food, and the tree of life in the middle of the garden and the tree of knowledge of good and evil."

As the names of the trees are not mentioned explicitly in this chapter, Rips suggested that perhaps they are hidden there at equal intervals. He took all the 25 trees named in the Torah (as listed in The Fauna and Flora of the Torah by Yehuda Feliks) and found them in the above chapter! Before the reader jumps out of his seat, let us explain that three- or four-letter words would normally appear at 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 ערמן and לבנה) are very short. There is no other segment in Genesis of such length, which contains so many trees at intervals less than 20. Based on the density of the letters in the chapter, one could estimate the probability of this "orchard" phenomenon. It comes to 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 minimum 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 is expected to be published in Israel soon. We hope that our skeptic concedes that the equal interval phenomenon is not deliberate computer trickery but a reflection of a hidden design. We are far from understanding the rules of this design, in particular from understanding what stands behind the numerical values of all the different intervals.

In recent years some other coded systems were discovered (or rediscovered) in the Torah. Among others is the "multiples of seven" pattern of key words in each chapter appearing 7, 14, 21... times. Another rule discovered by the late Rabbi Suleiman Sasson states that every word which is repeated in Torah more than 80 times, appears for the eightieth time in a segment of text discussing a promise, covenant, marriage or purchase (i.e. different types of contracts). The equal intervals system is distinct from systems such as these in that it functions on the letter rather than on the word level and it contains apparently limitless information.

But who made this design? Midrash Tanchuma says Moses saw the Torah as a string of letters of black fire on white fire. This string of letters was not divided into words. As G-d dictated the Torah to him, Moses wrote it accordingly in the form of words and chapters. As Maimonides states in the introduction to his Mishneh Torah, Moses wrote out a copy of the Torah before his death for each tribe plus one to be kept in the Ark. Jews believe that the modern Torah text is the exact copy of the original (except perhaps a few letters, as suggested by comparing the Yemenite and Ashkenazi texts).

How does this affect biblical criticism? According to the theory of biblical criticism, the Torah is a patchwork of pieces written at different times by different authors. The pieces allegedly were put together during or after the Babylonian exile and then canonized. For example, Bible critics contend that Genesis 1 and Genesis 2 were written by different authors because Genesis 1 uses the name אלקים for G-d while Genesis 2 uses the Tetragrammaton. Hidden words like בהבראם which connect the first two chapters of Genesis then would have to be assumed by the critics as inserts made by the final "editor". When we counted the trees in Figure 15 and the most outstanding clusters like "Eden", 'the river" (Figures 10 and 11) and a few other systems with probabilities less than 1 out of 1000, we found that the number of letters employed by the hidden words is about 30% of the total. Should we then be forced to believe that this "editor" created all these codes with some small modification-and without any apparent reason?

"It is possible," says our skeptic, "that the ancients possessed some secret knowledge which we cannot comprehend. Take, for example, the great pyramids or the Inca temples." Whatever they knew, nobody would suggest that they could foresee the future (unless they had a time machine).

Having commenced this article with an example of Rabbi Weismandel, let us conclude it with another example of his. The example concretizes the parallel between Moshe Rabbenu (Moses our Teacher) and Rabbenu Moshe ben Maimon (known also as Maimonides and as Rambam, his Hebrew acronym). The Rambam was born in Spain 852 years ago and later settled in Egypt, where he became a court doctor of Tzalach Ed-Din, There he wrote his most important work, the 14-volumed Mishneh Torah which classifies and clarifies all of the 613 Commandments. Figure 16 shows the beginning of the Mishneh Torah where Maimonides explains the origin of the Commandments and how they are classified in his 14 books.

In addition to having the same name, both Moshe Rabbenu and Moshe ben Maimon lived in Egypt and performed marvels before its rulers. (Maimonides's marvels were performed in his capacity as court doctor.) Furthermore, Maimonides's Mishneh Torah parallels Moshe Rabbenu's own Mishneh Torah, i.e. the book of Deuteronomy, which summarizes the entire Torah. (See the first example given in this article on the word "Torah" appearing at a 49-letter interval in Deuteronomy for an explanation of how this book differs from the others in the Five Books of Moses.)

The following finding is attributed to Nachmanides (the Ramban) who lived a few decades after Maimonides. The phrase

מצרים בארץ מופתי רבות
(Rabot Moftai B'eretz Mitzrayim-"My marvels may be multiplied in the land of Egypt") in Exodus 11:9 implies Maimonides. Indeed the first letters of the above four words form the word
רמבם (RaMBaM). [Reminder! the letter מ when it appears at the end of a word looks like this ם ]

"How beautiful," says our skeptic. "but you probably will find a 'Rambam' on every page." So we checked it out and found that this is the only "Rambam" acronym in the entire Torah.

But that's not the end of the story. Forty years ago when Rabbi Weismandel came across this passage, he asked himself whether some additional information about the Rambam was hidden in this passage at equal intervals. So he took the title of Rambam's MishnehTorah, spelled in Hebrew משנה תורה and searched for it. Since he had already discovered the "Torah" system at 50-letter intervals (corresponding to 50 gates of wisdom), he tried 50 again. And indeed, starting with the מ of Moshe in the verse just quoted, he found the word משנה at a 50-letter interval. The second word of the title תורה appeared much further on in the text at a 50-letter interval. The large gap the appearance of משנה and תורה apparently puzzled him, so he counted the letters between them. They came to 613. (This is the very number of Commandments in the Torah which were summarized by the Rambam in his Mishneh Torah! Read the introduction to his book in Figure 16. )

If you still wish to know the probabilities-the likelihood or such a משנה תורה starting with a given מ is 1 in 186,000,000. You could of course try some other מ, say 10 possibilities for מ in a neighborhood close to the "Rambam", And you could play with 613, counting between the ה of משנה and the ת of Torah or the מ of משנה and the ה of תורה and also include or exclude the first and the last letters in the counting, giving you six possibilities. With all this playing around, you can increase the likelihood to 1 in 3 million.

Now, what is the bottom line? Either the author of the Torah knew 2500 years in advance about Maimonides and the Mishneh Torah or the whole story is another coincidence with a probability of 1/3,000,000.

Unfortunately, when it comes to very small or large numbers, people often lose common sense. Let us suggest the following mental experiment. The opportunity is offered to play Russian roulette when one of six chambers of a pistol is loaded with a bullet. After the cylinder is rotated, one shoots the pistol at one's head. There is no other partner, and one must repeat the game 81 times. If the person dies-he dies. If he stays alive (and the chances are 1 in 3,000,000) he will have an exciting experience. Would our skeptic take the offer? Three thousand three hundred years ago there was another skeptic - Pharoah of Egypt. Our story from Exodus 11-12 is narrated after Pharaoh has already experienced nine plagues. He still was not convinced because, as the Torah says, "The L-rd hardened the heart of Pharoah." Should one wait for the tenth plague?


Further Suggested Reading

A Light Unto The Nations
A Partial List of Topics
# Rabbi Akiva’s Proof For the Existence of Hashem[G-d]
# The Experience of People Who Were Clinically Dead And Came Back to Life
# The Biblical Story of Creation in Light of Scientific Discoveries
# Most of The Prohibitions of The Bible Do Not Apply To Gentiles
Judaism 101
A place where every Jew can feel at home
Designed to provide comprehensive information base of basic Jewish terms, concepts and practices in the hope that everyone, no matter their nationality, faith or personal creed, will find these pages helpful, informative, and spiritually uplifting.
    If you are Jewish, it is worthwhile to attend a special seminar sponsored by Yeshiva Aish Hatora where the latest statistical evidence is offered (that has convinced even military officials involved in deciphering codes) that some meaningful codes do exist in the Bible.

    To find out information about the seminar (Discovery program) go to http://www.aish.com/
 
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