Table
1P. Results obtained for salvation
(ישע) with 3-ש
matrices.
|
Number of occurrences of
ישע at skips +2 to 25 in a matrix |
Number of matrices with
these occurrences |
Total number of occurrences
in matrices |
Number of symmetrical occurrences |
Number of the occurrences
in the central 3-column rectangle. |
|
0 |
6 |
- |
- |
0 |
|
1 |
17 |
17 |
4 |
*6 |
|
2 |
13 |
26 |
0 |
§4 |
|
3 |
9 |
27 |
*4 |
2 |
|
4 |
9 |
36 |
*7 |
3 |
|
5 |
1 |
5 |
0 |
1 |
|
6 |
2 |
12 |
**4 |
1 |
|
Total |
57 |
123 |
19 |
17 |
Note: An asterisk (*) indicates a matrix containing 2 items, while a section sign (§) indicates a matrix containing 3 items of the respective quantities.
Data obtained with matrices
containing 2 letters shin (ש)
in their central column
In order to confirm these results, matrices containing 2 letters shin and more than 3 letters shin have been prepared. The results obtained are summarized in the tables and figures below.
Table
2P. Results obtained for salvation
(ישע) with 2-ש
matrices.
|
Number of occurrences of
ישע at skips +2 to 25 in a matrix |
Number of matrices with
these occurrences |
Total number of occurrences
in matrices |
Number of symmetrical occurrences |
Number of the occurrences
in the central 3-column rectangle. |
|
0 |
3 |
- |
- |
*3 |
|
1 |
3 |
3 |
1 |
0 |
|
2 |
7 |
14 |
*3 |
0 |
|
3 |
6 |
18 |
0 |
0 |
|
4 |
4 |
16 |
1 |
0 |
|
5 |
0 |
- |
- |
0 |
|
6 |
1 |
6 |
0 |
0 |
|
Total |
24 |
57 |
5 |
3 |
Note: An asterisk (*) indicates a matrix containing 2 items of the respective quantities.
Data obtained with matrices
containing more than 3 letters shin (ש)
in their central column
Table
3P. Results obtained for salvation
(ישע) with more than 3-ש
matrices.
|
Number of occurrences of
ישע at skips +2 to 25 in a matrix |
Number of matrices with
these occurrences |
Total number of occurrences
in matrices |
Number of symmetrical occurrences |
Number of the occurrences
in the central 3-column rectangle. |
|
1 |
5 |
5 |
2 |
3 |
|
2 |
4 |
8 |
0 |
**¶8 |
|
3 |
3 |
9 |
2 |
1 |
|
4 |
2 |
8 |
**4 |
1 |
|
Total |
14 |
30 |
8 |
13 |
Note: An asterisk (*) indicates a matrix containing 2 items, while a paragraph sign (¶) indicates a matrix containing 4 items of the respective quantities.
The distribution of the number of letters shin in the central column of the matrices of this group was as follows:
|
4
letters ש |
5 letters ש |
6 letters ש |
7 letters ש |
|
9 |
3 |
1 |
1 |
Discussion
The average number of occurrences of the three-letter word salvation (ישע) in Hebrew appeared to be relatively constant, independent on the number of the letters shin in the central column:
2 letters shin in the central column (Table 2P): 57/24 = 2.38 occurrences per matrix;
3 letters shin in the central column (Table 1P): 123/57 = 2.16 occurrences per matrix;
>3 letters shin in the central column (Table 3P): 30/14 = 2.14 occurrences per matrix.
On the other hand, the occurrences in the central 3-column rectangle show a high dependence on the number of shins in the central column:
2 letters shin in the central column (Table 2P): 3/24 = 0.125 occurrences per matrix;
3 letters shin in the central column (Table 1P): 17/57 = 0.298 occurrences per matrix;
>3 letters shin in the central column (Table 3P): 13/14 = 0.929 occurrences per matrix.
These results could be expected – the higher number of shins leads to higher probabilities for occurrence of a word with a shin in its centre. The same is valid for the rate of symmetrical occurrences:
2 letters shin in the central column (Table 2P): 5/24 = 0.208 occurrences per matrix;
3 letters shin in the central column (Table 1P): 19/57 = 0.333 occurrences per matrix;
>3 letters shin in the central column (Table 3P): 8/14 = 0.571 occurrences per matrix.
It is evident from the above data that the rate of occurrence of both symmetrical and central ישע in all three cases is lower than 1. This is especially valid in the cases of 2- and 3-letter shin in the central column. Therefore, it seems that the most appropriate method for evaluation of the probabilities for symmetrical and central occurrences is the Poisson Distribution.
Poisson distribution is described by the general formula
λn
P(n) = ——— e-λ
n!
P(n) is the probability for n items to be found in a matrix if the probability follows the Poisson distribution. λ is called distribution parameter and it corresponds to the overall number of occurrences of a quantity spread among the total number of matrices. e is the base of the natural logarithms.
In order to use this formula for the estimation of probabilities for n = 4 (symmetrical occurrences) and n = 3 (central occurrences) in first place we must check whether the data obtained agree with the formula.
According to the Poisson distribution, the expected number of matrices that do not contain an item (that is, n = 0) will be e-λ; the expected number of those containing one item will be λe-λ, etc.
Let us explore the data obtained for symmetrical appearances in the
matrices containing 3 letters shin
in their central column. λ, as calculated above, is 19/57
= 0.3333. It is evident from the fourth column (coloured in plum) in Table
1P that the number of matrices with two symmetrical occurrences of salvation (ישע) is 4. Hence the number of matrices with one occurrence is 19 – 4×2
= 11. The number of those, which do not contain this item, is 42. The results
are summarized in Table 4P. Deviation in terms of (Expected – Observed)2/Expected
is given for the purpose of calculation the chi-squared (χ2)
criterion.
Table 4P. Expected values according to Poisson distribution and actual data obtained with matrices with 3 letters shin (ש) in their central column containing symmetrical occurrences of salvation (ישע) at skips +2 to 25.
|
Number of occurrences in
a matrix |
Expected |
Observed (O) |
Deviation |
||
|
Proportion |
Number (E) |
O - E |
(O – E)2/E |
||
|
0 |
0.7165 |
40.8 |
42 |
+1.2 |
0.03529 |
|
1 |
0.2388 |
13.6 |
11 |
-2.6 |
0.49706 |
|
2 |
0.0399 |
2.3 |
4 |
+1.7 |
1.25652 |
The calculations with the central occurrences in the matrices containing 3 letters shin in their central column are summarized in Table 5P.
Table 5P. Expected values according to Poisson distribution and actual data obtained with matrices with 3 letters shin (ש) in their central column containing central occurrences of salvation (ישע) at any skips. λ = 17/57 = 0.2982.
|
Number of occurrences in
a matrix |
Expected |
Observed (O) |
Deviation |
||
|
Proportion |
Number (E) |
O - E |
(O – E)2/E |
||
|
0 |
0.7422 |
42.3 |
43 |
+0.7 |
0.01158 |
|
1 |
0.2213 |
12.6 |
12 |
-0.6 |
0.02857 |
|
2 |
0.0330 |
1.9 |
1 |
-0.1 |
0.00476 |
|
3 |
0.0033 |
0.2 |
1 |
||
The same type of calculations for the matrices containing 2 letters shin in their central column are presented in Tables 6P and 7P.
Table 6P. Expected values according to Poisson distribution and actual data obtained with matrices with 2 letters shin (ש) in their central column containing symmetrical occurrences of salvation (ישע) at skips +2 to 25. λ = 5/24 = 0.2083.
|
Number of occurrences in
a matrix |
Expected |
Observed (O) |
Deviation |
||
|
Proportion |
Number (E) |
O - E |
(O – E)2/E |
||
|
0 |
0.8119 |
19.5 |
20 |
+0.5 |
0.01282 |
|
1 |
0.1691 |
4.1 |
3 |
-1.1 |
0.29512 |
|
2 |
0.0176 |
0.4 |
1 |
+0.6 |
0.90000 |
Table 7P. Expected values according to Poisson distribution and actual data obtained with matrices with 2 letters shin (ש) in their central column containing central occurrences of salvation (ישע) at any skips. λ = 3/24 = 0.1250.
|
Number of occurrences in
a matrix |
Expected |
Observed (O) |
Deviation |
||
|
Proportion |
Number (E) |
O - E |
(O – E)2/E |
||
|
0 |
0.8825 |
21.2 |
22 |
+0.8 |
0.03019 |
|
1 |
0.1103 |
2.6 |
1 |
-0.8 |
0.22857 |
|
2 |
0.0069 |
0.2 |
1 |
||
These calculations for the matrices containing more than 3 letters shin in their central column produced the results presented in Tables 8P and 9P.
Table 8P. Expected values according to Poisson distribution and actual data obtained with matrices with >3 letters shin (ש) in their central column containing symmetrical occurrences of salvation (ישע) at skips +2 to 25. λ = 8/14 = 0.5714.
|
Number of occurrences in
a matrix |
Expected |
Observed (O) |
Deviation |
||
|
Proportion |
Number (E) |
O - E |
(O – E)2/E |
||
|
0 |
0.5647 |
7.9 |
8 |
+0.1 |
0.00127 |
|
1 |
0.3227 |
4.5 |
4 |
-0.5 |
0.05556 |
|
2 |
0.0922 |
1.3 |
0 |
-1.3 |
1.30000 |
Table 9P. Expected values according to Poisson distribution and actual data obtained with matrices with >3 letters shin (ש) in their central column containing central occurrences of salvation (ישע) at any skips. λ = 13/14 = 0.9286.
|
Number of occurrences in
a matrix |
Expected |
Observed (O) |
Deviation |
||
|
Proportion |
Number (E) |
O - E |
(O – E)2/E |
||
|
0 |
0.3950 |
5.5 |
6 |
+0.5 |
0.04545 |
|
1 |
0.3668 |
5.1 |
5 |
-0.1 |
0.00196 |
|
2 |
0.1703 |
2.4 |
2 |
-0.4 |
0.06667 |
|
3 |
0.0527 |
0.7 |
0 |
+0.1 |
0.01111 |
|
4 |
0.0122 |
0.2 |
1 |
||
Conclusion about the Suitability of the Method
The criterion most widely used for the verification of a hypothesis for a distribution pattern is the chi-squared value. Let us check the results using this criterion. The deviations shown in the last two columns in the tables have been corrected for the classes with expected items below 0.4. Such corrections are recommended for the bordering classes in case the expected number E is < 1. The recommendation in such cases is to enlarge the class combining it with the neighbouring one. In our cases, these values were combined with the previous class and hence the observed values have been summed up in the same way. Then, aggregate values for the deviation have been calculated. Chi-squared value is calculated according to the formula
(Observed – Expected)2
χ2 = ∑——————————
Expected
Summing is performed for all classes in a group.
The χ2 values calculated are summarized in the following table.
|
|
χ2 for
symmetric occurrences |
χ2 for
central occurrences |
|
2-ש
matrices |
1.21 |
0.26 |
|
3- ש matrices |
1.79 |
0.04 |
|
>3-ש
matrices |
1.37 |
0.13 |
It is obvious from the figures in the table that the Poisson distribution hypothesis is confirmed. On the other hand, no dependence between the number of symmetric and central occurrences in a matrix can be inferred from Tables 1P – 3P and the graphic presentations below them. Also, definitely no dependency could be derived between the number of central occurrences and the overall number of occurrences at skips +2 to 25 in a matrix. On the other hand, it is quite naturally that the higher is the number of the latter, the higher should be the probability for one of them to be a symmetric one. But we have constructed our proposition on the basis of preliminarily unknown overall number of occurrences at skips +2 to 25 in the matrix observed.
Therefore, the Poisson distribution can be applied for the calculation of the probabilities for symmetrical and central occurrences of the 3-letter word salvation (ישע) in the 51×21 matrix shown on Figure 4 and then the probability for the concurrent event.
Calculation of probability
Note: Because the probabilities are approximate, in all calculations of their values, the differences between matrices containing 20 and 21 rows have been ignored. The chances for 21 rows are of the order of 1.03, or just 3%, higher than the mean value.
The probability for 4 random symmetric occurrences of salvation (ישע) at positive skips 2 to 25 in a 51×20 or 21 matrix is
Ps(4) = (0.3333^4/4!)×e^-0.3333
= 3.685×10^-4
or Ps(4) = 1/2,714.
The probability for 3 random central occurrences of salvation (ישע) at any skips in a 51×20 or 21 matrix is
Pc(3) = (0.2982^3/3!)
×e^-0.2982 = 0.00328
or Pc(3) = 1/305.
Hence the overall probability for the random occurrences of the items salvation (ישע) complying with the requirements according to the test is
Ps(4) × Pc(3)
= 3.685×10^-4 × 0.00328 = 1.209×10^-6
or P = 1/827,000.
But a biased and skeptic one may claim that we haven’t specified positive skips for the symmetric occurrences. Let us check the probability in this case.
The overall number of occurrences of ישע at - 2 to - 25 in the Torah is 1470. This is only 2% higher than 1441 - the number of the occurrences at the respective positive skip. Therefore, we can reasonably expect a doubled rate of occurrences, or λ = 0.3333×2 = 0.6667. In this case, however, the number of occurrences is already five. Thus the probability for 5 random symmetric occurrences of salvation (ישע) at skips ±2 to 25 in a 51×20 or 21 matrix should be
Ps(5) = (0.6667^5/5!)e^-0.6667
= 5.635×10^-4
or Ps(5) = 1/1,775.
In this case, the overall probability for the random occurrences of the items salvation (ישע) complying with the requirements according to the test is
Ps(5) × Pc(3)
= 5.635×10^-4 × 0.00328 = 1.848×10^-6
or P = 1/541,000.
As it appears, even a skeptical approach to the evaluation of the probability will lead to a value of the order of one in a half million.
Interpretation of the Rate of Appearances of Heaven
(השמים) in the Plain Text and the
Encoded Occurrence
Another good example in my view is the word heaven (השמים). The estimation of the rate of appearance in the plain text is valid because the only passage in the whole Torah where God explicitly says “I have spoken from heaven” is in the matrix.
I tried to expose this fact in the matrix shown on Figure 3. However, there are shown only four occurrences because I retained the central place for the column containing the year 2006 (תשרקתשרקו) in order to emphasize its importance. In fact, a matrix with a lower numbers of columns could be constructed (see Figure 5), containing five appearances of heaven (red ovals) in the plain text as well as the encoded word (blue ovals) at skip, which, remarkably, is 5. The year 2006 is in black ovals. The matrix is 67 columns long. Let us estimate the probability for finding at random 5 words השמים in such a matrix.
Heaven (השמים) appears 81 times in the plain text of the Torah. The whole text of the Torah is about 223 times 67-columns long matrices at skip 14965 (containing either 20 or 21 rows). If the 81 words השמים are considered randomly spread, the probability for finding one of them in a matrix of the specified parameters will be 81/223 = 0.3626. In fact, they are not evenly spread: 13 of them (16 %) are within the first 2650 letters of the text. Also, there are several pairs of these words (less than 10 pairs) that are at distances below 100 along the text. Two of these pairs are found in Figure 5. If we accept the approach of finding pairs, however, the calculated probability will prove to be even lower compared to the random-spread approach.
Because we know the likelihood for a random appearance of heaven in the matrix, the simplest way of estimation the probability for finding five words in the matrix is to raise this known value to the power of 5:
P(5) = 0.3626^5 = 0.006268
or P(5) = 1/160
There is also the encoded word heaven at skip 5 in the matrix. There are 7 occurrences of heaven (השמים) at skips ±2 to 5 in the Torah. Therefore, the probability for finding one of them in the matrix will be merely
P(H) = 7/223 = 0.03139
or P(H) = 1/31.9
Hence the overall probability for a random occurrence of five words in the plain text and an encoded word at skip ±2 to 5 appears to be
P(5) × P(H) = 0,006268×0.03139=
0.0001967
or PHeaven = 1/5,080
Another interesting matrix I managed to construct while preparing this document is the absolutely symmetrical one shown in Figure 6. The designations are the same as in Figure 5. The matrix is of skip 14964 and contains 63 columns. There are four words heaven in the plain text plus the encoded word. The amazing fact is that the number of letters in ovals to the left and to the right of the geometrical centre is the same: 16. There are two letters in the central column: tav (ת) and mem (מ). They form the two-letter word תם at skip -104748. This is what is written for two words of the same spelling in the Wilson’s Old Testament Word Studies [3]:
םתּ (tam) perfect, plain, undefiled, upright;
םתּ (tom) full, integrity, perfect, perfection, simplicity, Thummim, upright, uprightly, uprightness.
Although a short one, this word fits rightly in its place. The meaning is in full harmony with the idea of conclusion and perfection that emanates from this strange code.
The perfection is underlined by the symmetric vertical positioning of the השמים that gives the mem to form the 2-letter word and the encoded one in the bottom row.
But there is more in the central column. Already a 3-letter word is formed there including these two letters, but at skip +104748:
תתּמ (matat) gift, to give, reward [3].
General Conclusion on Probabilities
The probability for the concomitant finding of the items discussed in this paper can be calculated as the product of the probabilities at each stage. Thus, even if we confine to the matrix in Figure 4, where are two of the words heaven at skip +1 wholly contained in the matrix and the encoded one at skip 5 and the findings about salvation, and carry out the adjustments required for the passing from a 67-column matrix to a 51-column one, yet a skeptical evaluation will produce a probability as low as
P
= 1.848×10^-6×0.2760^2×0.02385 = 3.358×10^-9
or P = 1/300,000,000.
The combined probability for salvation and heaven appears to be one in three hundred millions! And yet many of the crucial words found in the matrix haven’t entered so far!
Lyuben
Piperov
July 15th 2004
Sofia, Bulgaria
Contact Lyuben Piperov at
l_piperov@yahoo.co.uk
