Overview[ edit ] Contemporary discussion and controversy around one specific steganographic method became widespread in when Doron Witztum, Eliyahu Rips and Yoav Rosenberg published a paper, "Equidistant Letter Sequences in the Book of Genesis", in the scientific journal Statistical Science. The traditional WRR view of the codes is based strictly on their applicability to the Torah, and asserts that any attempt to study the codes outside of this context is invalid. This is based on a belief that the Torah is unique among biblical texts in that it was given directly to mankind via Moses in exact letter-by-letter sequence and in the original Hebrew language. To obtain an ELS from a text, choose a starting point in principle, any letter and a skip number, also freely and possibly negative. Then, beginning at the starting point, select letters from the text at equal spacing as given by the skip number. For example, the bold letters in this sentence form an ELS.
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It has been noted that when the Book of Genesis is written as two-dimensional arrays, equidistant letter sequences spelling words often appear in close proximity with portions of the text which have related meaning. Quantitative tools for measuring this phenomenon are developed. Randomization analysis is done for three samples. For one of them the effect is significant at the level of.
Key words and phrases. Genesis, Equidistant letter sequences, Strings of letters, Cylindrical representations, Statistical analysis. Since this tradition was passed orally, only few expressions that belong to the "hidden text" were preserved in writing Rabbenu Bahya, and Cordovero, ; actually almost all the words and the syntax are unknown.
Rabbi H. In a previous paper Witztum et al, , we developed a methodology for systematic and rigorous studies of the same nature; namely, for attempts to show objectively the existence of the"hidden text" in the Hebrew Pentaceuch.
This methodology was applied to study the "hidden text" of the Book of Genesis. The approach we have taken in our 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 possibilitities? 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. Suppose we are given a text, such as Genesis G. 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 -- i. 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. Our paper Witztum et al. There we developed 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 performed a randomization test and obtained a very small p-value, i. On Figure 1 we see a pair of words: the word private appearing as an ELS with skip with the word names appearing in the text Gn The rows in our table has letters only 21 of them are shown in Figure 1.
Note that G contains 78, letters. More examples are given in Appendix A. The measuring scheme for Phenomenon A see Witztum et al. In this paper we make certain choices of words to compare and perform similar randomization tests.
We obtain very small p-values; that is, we find that the results are highly statistically significant. Outline of the Procedure. In this section we describe the test in outline. In an appendix, sufficient details are provided to enable the reader to repeat the computations precisely, and so to verify their correctness.
We test the significance of the phenomenon on samples of pairs of related words. To do this we must do the following: define the notion of "distance" between any two words, so as to lend meaning to the idea of words in "close proximity"; 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 ; choose a sample of pairs of related words on which to run the test; and determine whether the statistics defined in ii are "unusually small" for the chosen sample.
Notice that the procedure here described is identical to that used in Witztum et al. Task i has several components. First, we must define the notion of "distance" between an ELS and an SL of the text in a given array; for this we use a convenient variant of the ordinary Euclidean distance. Third, a given word may occur many times as an ELS in a text; here again, a selection and amalgamation 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. 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, P1 and P2, which are defined and motivated in the Appendix Section A. Intuitively, each measures overall proximity in a different way. In each case, a small value of Pi indicates that the words in the sample pairs are, on the whole, close to each other. For details see Appendix, Section A.
Sample B1 is built on the basis of the Hebrew alphabet. Jewish tradition teaches, that these seventy descendents became the Seventy Nations which constitute Humanity. This concept is well known, and is usually found in biblical Encyclopaedias under the title "The Table of Nations" see for instance Encyclopedia Biblica, Finally, we come to Task iv , the significant test itself.
We apply the same procedure for all three samples. For Sample B3 we describe it here: for the other two samples the similar details are given in Appendix A. The list of Seventy Nations consists of 68 different names in two cases nations have the same name.
For each of the 68! The 68! If the phenomenon under study were due to chance, it would be just as likely that P1 occupies any one of the 68! Similarly for P2. This is our null hypothesis.
To calculate significance levels, we chose , random permutations of the 68 names; the precise way in which this was done is explained in the Appendix Section A. Each of these permutations determines a statistic together with P1, we have thus 1,, numbers. Define the rank order of P1, among these 1,, numbers as the number of not exceeding P1; if P1 is tied with other , half of these others are considered to "exceed" P1.
Let be the rank order of P1, divided by 1,,; under the null hypothesis, is the probability that P1 would rank as low as it does. Define similarly using the same , permutations in each case. For Sample B3 we performed an additional test with ,, random permutations. In this case only statistic was calculated. The time needed for the computation of for ,, random permutations is at present not within the reach of our possibilties.
After calculating the probabilities and , we must make an overall decision, for each sample, to accept or reject the null hypothesis. Thus the overall significance level or p-value for each sample, using the two statistics.
Results and Conclusions. In Tables 1 , 2 and 3 we present the results for the three samples. Table 1 shows the results for Sample B1. There we list the rank order of P1 and P2 among the 1,, corresponding and. Thus the entry for P2 means that for out of the , random permutations , the statistic was smaller than P2.
We conclude that for Sample B1 the null hypothesis is rejected with significance level p-value. The same calculations, using the same , random permutations, were performed for a control text V see Witztum et al. The text V was obtained from G by permuting the verses of G randomly. For details, see Appendix; Section A. Table 1 gives the results of these calculations too.
In the case of V, min is approximately. Table 2 shows the results for Sample B2 for G. The results are non-significant. We saw no reason to perform any further tests for this sample.
Table 3 shows the results for Sample B3. In part A of it, the results for G, as well for the control text V for , random permutations are summarized. In part B, the results for G for ,, random permutations are given.
The Remarkable Word Patterns in the Book of Genesis
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. Key words and phrases: Genesis, equidistant letter sequences, cylindrical representations, statistical analysis. 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.
For they found related names, events and dates from the nineteenth century! Only someone who knows the future could have placed them there! The three mathematicians submitted their findings to the prestigious scientific journal Statistical Science for publishing, Editor Robert Kass, said: "Our referees were baffled: their prior beliefs made them think the Book of Genesis could not POSSIBLY contain meaningful references to modern day individuals, yet when the authors carried out additional analyses and checks the effect persisted. The letters were placed into a two-dimensional array, which simply means, the letters were divided into rows, each row containing a certain number of letters.