Statistical Science 1994, Vol. 9, No. 3, 429-438 Equidistant Letter Sequences in the Book of Genesis Doron Witztum, Eliyahu Rips and Yoav Rosenberg Abstract. It has been noted that when the Book of Genesis is written as two-dimensional arrays, equidistant letter sequences spelling words with related meanings often appear in close proximity. Quantitative tools for measuring this phenomenon are developed. Randomization analysis shows that the effect is significant at the level of 0.00002. Key words and phrases: Genesis, equidistant letter sequences, cylindrical representations, statistical analysis. 1. INTRODUCTION The phenomenon discussed in this paper was first discovered several decades ago by Rabbi Weissmandel [7]. He found some interesting patterns in the Hebrew Pentateuch (the Five Books of Moses), consisting of words or phrases expressed in the form of equidistant letter sequences (ELS's)--that is, by selecting sequences of equally spaced letters in the text. As impressive as these seemed, there was no rigorous way of determining if these occurrences were not merely due to the enormous quantity of combinations of words and expressions that can be constructed by searching out arithmetic progressions in the text. The purpose of the research reported here is to study the phenomenon systematically. The goal is to clarify whether the phenomenon in question is a real one, that is, whether it can or cannot be explained purely on the basis of fortuitous combinations. The approach we have taken in this research can be illustrated by the following example. Suppose we have a text written in a foreign language that we do not understand. We are asked whether the text is meaningful (in that foreign language) or meaningless. Of course, it is very difficult to decide between these possibilities, since we do not understand the language. Suppose now that we are equipped with a very partial dictionary, which enables us to recognise a small portion of the words in the text: "hammer" here and "chair" there, and maybe even "umbrella" elsewhere. Can we now decide between the two possibilities? Not yet. But suppose now that, aided with the partial dictionary, we can recognise in the text a pair of conceptually related words, like "hammer" and "anvil." We check if there is a tendency of their appearances in the text to be in "close proximity." If the text is meaningless, we do not expect to see such a tendency, since there is no reason for it to occur. Next, we widen our check; we may identify some other pairs of conceptually related words: like "chair" and "table," or "rain" and "umbrella." Thus we have a sample of such pairs, and we check the tendency of each pair to appear in close proximity in the text. If the text is meaningless, there is no reason to expect such a tendency. However, a strong tendency of such pairs to appear in close proximity indicates that the text might be meaningful. Note that even in an absolutely meaningful text we do not expect that, deterministically, every such pair will show such tendency. Note also, that we did not decode the foreign language of the text yet: we do not recognise its syntax and we cannot read the text. This is our approach in the research described in the paper. To test whether the ELS's in a given text may contain "hidden information," we write the text in the form of two-dimensional arrays, and define the distance between ELS's according to the ordinary two-dimensional Euclidean metric. Then we check whether ELS's representing conceptually related words tend to appear in "close proximity." Suppose we are given a text, such as Genesis (G). Define an equidistant letter sequence (ELS) as a sequence of letters in the text whose positions, not counting spaces, form an arithmetic progression; that is, the letters are found at the positions n, n+d, n+2d, ... , n+(k-1)d. We call d the skip, n the start and k the length of the ELS. These three parameters uniquely identify the ELS, which is denoted (n,d,k). Let us write the text as a two-dimensional array--that is, on a single large page--with rows of equal length, except perhaps for the last row. Usually, then, an ELS appears as a set of points on a straight line. The exceptional cases are those where the ELS "crosses" one of the vertical edges of the array and reappears on the opposite edge. To include these cases in our framework, we may think of the two vertical edges of the array as pasted together, with the end of the first line pasted to the beginning of the second , the end of the second to the beginning of the third and so on. We thus get a cylinder on which the text spirals down in one long line. It has been noted that when Genesis is written in this way, ELS's spelling out words with related meanings often appear in close proximity. In Figure 1 we see the example of 'patish' (hammer) and 'sadan' (anvil); in Figure 2, 'Zidkiyahu' (Zedekia) and 'Matanya' (Matanya), which was the original name of King Zedekia (Kings II, 24:17). In Figure 3 we see yet another example of 'hachanuka' (the Chanuka) and 'chashmonaee' (Hasmonean), recalling that the Hasmoneans were the priestly family that led the revolt against the Syrians whose successful conclusion the Chanuka feast celebrates. Indeed, ELS's for short words, like those for 'patish' (hammer) and 'sadan' (anvil), may be expected on general probability grounds to appear close to each other quite often, in any text. In Genesis, though, the phenomenon persists when one confines attention to the more "noteworthy" ELS's, that is, those in which the skip |d| is _minimal_ over the whole text or over large parts of it. Thus for 'patish' (hammer), there is no ELS with a smaller skip than that of Figure 1 in all of Genesis; for 'sadan' (anvil), there is none in a section of text comprising 71% of G; the other four words are minimal over the whole text of G. On the face of it, it is not clear whether or not this can be attributed to chance. Here we develop a method for testing the significance of the phenomenon according to accepted statistical principles. After making certain choices of words to compare and ways to measure proximity, we perform a randomization test and obtain a very small p-value, that is, we find the results highly statistically significant. 2. OUTLINE OF THE PROCEDURE In this section we describe the test in outline. In the Appendix, sufficient details are provided to enable the reader to repeat the computations precisely, and so to verify their correctness. The authors will provide, upon request, at cost, diskettes containing the program used and the texts G, I, R, T, U, V and W (see Section 3). We test the significance of the phenomenon on samples of pairs of related words (such as hammer-anvil and Zedekia-Matanya). To do this we must do the following: (i) define the notion of "distance" between any two words, so as to lend meaning to the idea of words in "close proximity"; (ii) define statistics that express how close, "on the whole," the words making up the sample pairs are to each other (some kind of average over the whole sample); (iii) choose a sample of pairs of related words on which to run the test; (iv) determine whether the statistics defined in (ii) are "unusually small" for the chosen sample. Task (i) has several components. First, we must define the notion of "distance" between two given ELS's in a given array; for this we use a convenient variant of the ordinary Euclidean distance. Second, there are many ways of writing a text as a two-dimensional array, depending on the row length; we must select one or more of these arrays and somehow amalgamate the results (of course, the selection and/or amalgamation must be carried out according to clearly stated, systematic rules). Third, a given word may occur many times as an ELS in a text; here again, a selection and amalgamtion process is called for. Fourth, we must correct for factors such as word length and composition. All this is done in detail in Sections A.1 and A.2 of the Appendix. We stress that our defintion of distance is not unique. Although there are certain general principles (like minimizing the skip d) some of the details can be carried out in other ways. We feel that varying these details is unlikely to affect the results substantially. Be that as it may, we chose one particular defintion, and have, throughout, used _only_ it, that is, the function c(w,w') described in Section A.2 of the Appendix had been defined before any sample was chosen, and it underwent no changes. [Similar remarks apply to choices made in carrying out task (ii).] Next, we have task (ii), measuring the overall proximity of pairs of words in the sample as a whole. For this, we used two different statistics p and p , which are defined and motivated in the Appendix (Section A.5). 1 2 Intuitively, each measures overall proximity in a different way. In each case, a small value of p i indicates that the words in the sample pairs are, on the whole, close to each other. No other statistics were _ever_ calculated for the first, second or indeed any sample. In task (iii), identifying an appropriate sample of word pairs, we strove for uniformity and objectivity with regard to the choice of pairs and to the relation between their elements. Accordingly, our sample was built from a list of personalities (p) and the dates (Hebrew day and month) (p') of their death or birth. The personalities were taken from the _Encyclopedia of Great Men in Israel_ [5]. At first, the criterion for inclusion of a personality in the sample was simply that his entry contain at least three columns of text and that a date of birth or death be specified. This yielded 34 personalities (the first list--Table 1). In order to avoid any conceivable appearance of having fitted the tests to the data, it was later decided to use a fresh sample, without changing anything else. This was done by considering all personalities whose entries contain between 1.5 and 3 columns of text in the Encyclopedia; it yielded 32 personalities (the second list--Table 2). The significance test was carried out on the second sample only. Note that personality-date pairs (p,p') are not word pairs. The personalities each have several appellations, there are variations in spelling and there are different ways of designating dates. Thus each personality- date pair (p,p') corresponds to several word pairs (w,w'). The precise method used to generate a sample of word pairs from a list of personalities is explained in the Appendix (Section A.3). The measures of proximity of word pairs (w,w') result in statistics p and p . As explained in the Appendix (Section A.5), we also used a variant 1 2 of this method, which generates a smaller sample of word pairs from the same list of personalities. We denote the statistics p and p , when applied to 1 2 this smaller sample, by p and p . 3 4 Finally, we come to task (iv), the significance test itself. It is so simple and straightfoward that we describe it in full immediately. The second list contains of 32 personalities. For each of the 32! (pi) permutations (pi) of these personalities, we define the statistic p 1 obtained by permuting the personalities in accordance with (pi), so that Personality i is matched with the dates of Personality (pi)(i). The 32! (pi) numbers p are ordered, with possible ties, according to the usual order 1 of the real numbers. If the phenomenon under study were due to chance, it would be just as likely that p occupies any one of the 32! places in this 1 order as any other. Similarly for p, p and p. This is our null hypothesis. 2 3 4 To calculate significance levels, we chose 999,999 random permutations (pi) of the 32 personalities; the precise way in which this was done is explained in the Appendix (Section A.6). Each of these permutations (pi) (pi) determines a statistic p; together with p, we have thus 1,000,000 1 1 numbers. Define the rank order of p among these 1,000,000 numbers as the 1 (pi) (pi) number of p not exceeding p; if p is tied with other p, half of 1 1 1 1 these others are considered to "exceed" p. Let rho be the rank order of p, 1 1 1 divided by 1,000,000; under the null hypothesis, rho is the probability 1 that p would rank as low as it does. Define rho, rho and rho similarly 1 2 3 4 (using the same 999,999 permutations in each case). After calculating the probabilities rho through rho, we must make 1 4 an overall decision to accept or reject the research hypothesis. In doing this, we should avoid selecting favorable evidence only. For example, suppose that rho = 0.01, the other rho being higher. There is then the 3 i temptation to consider rho only, and so to reject the null hypothesis at 3 the level of 0.01. But this would be a mistake; with enough sufficiently diverse statistics, it is quite likely that just by chance, some one of them will be low. The correct question is, "Under the null hypothesis, what is the probability that at least one of the four rho would be less than or i equal to 0.01?" Thus denoting the event "rho <= 0.01" by E, we must find i i the probability not of E, but of "E or E or E or E." If the E were 3 1 2 3 4 i mutually exclusive, this probability would be 0.04; overlaps only decrease the total probability, so that it is in any case less than or equal to 0.04. Thus we can reject the null hypothesis at the level of 0.04, but not 0.01. More generally, for any given delta, the probability that at least one of the four numbers rho is less than or equal to delta is at most 4delta. i This is known as the Bonferroni inequality. Thus the overall significance level (or p-value), using all four statistics, is rho := 4 min rho. 0 i 3. RESULTS AND CONCLUSIONS In Table 3, we list the rank order of each of the four p among the (pi) i 1,000,000 corresponding p. Thus the entry 4 for p means that for i 4 precisely 3 out of the 999,999 random permutations (pi), the statistic (pi) p was smaller than p (none was equal). It follows that min rho = 4 4 i 0.000004 so rho = 4 min rho = 0.000016. The same calculations, using the 0 i same 999,999 random permutations, were performed for control texts. Our first control text, R, was obtained by permuting the letters of G randomly (for details, see Section A.6 of the Appendix). After an earlier version of this paper was distributed, one of the readers, a prominent scientist, suggested to use as a control text Tolstoy's _War and Peace_. So we used text T consisting of the initial segment of the Hebrew translation of Tolstoy's _War and Peace_ [6]--of the same length of G. Then we were asked by a referee to perform a control experiment on some early Hebrew text. He also suggested to use randomization on words in two forms: on the whole text and within each verse. In accordance, we checked texts I, U and W: text I is the Book of Isaiah [2]; W was obtained by permuting the words of G randomly; U was obtained from G by permuting randomly words within each verse. In addition, we produced also text V by permuting the verses of G randomly. (For details, see Section A.6 of the Appendix.) Table 3 gives the results of these calculations, too. In the case of I, min rho is approximately 0.900; i in the case of R it is 0.365; in the case of T it is 0.277; in the case of U it is 0.276; in the case of V it is 0.212; and in the case of W it is 0.516. So in five cases rho = 4 min rho exceeds 1, and in the remaining case 0 i rho = 0.847; that is, the result is totally nonsignificant, as one would 0 expect for control texts. We conclude that the proximity of ELS's with related meanings in the Book of Genesis is not due to chance. ------------------------------------------------------------------------------ Table 3 (pi) Rank order of p among one million p i i --------------------------------------------------------------- p p p p 1 2 3 4 ---------------------------------------------------------------- G 453 5 570 4 R 619,140 681,451 364,859 573,861 T 748,183 363,481 580,307 277,103 I 899,830 932,868 929,840 946,261 W 883,770 516,098 900,642 630,269 U 321,071 275,741 488,949 491,116 V 211,777 519,115 410,746 591,503 ----------------------------------------------------------------

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