2.1 - Introduction
2.2 - The 4 rows
2.3 - Multiples of 61 and the Cross
2.4 - The axes of the Cross
2.5 - The Lord on the Cross
2.6 - The Cross and the hexagram
2.7 - The Cross - some further observations
2.8 - The Urim and Thummim
2.9 - Addendum: 31 and 43: complements and related properties
- Appendix: The Hebrew spelling of the Lord's Name
Here, we continue our study of the breastplate grid in which the names of the sons of Israel appear in Mother Set (MS) order, viz Reuben, Simeon, Levi, Judah, Issachar and Zebulun (children of Leah), Gad and Asher (children of Leah's handmaid, Zilpah), Dan and Naphtali (children of Rachel's handmaid, Bilhah), Joseph and Benjamin (children of Rachel).
2. The 4 rows
Amazingly, the rows of the MS grid are based around three factors only, viz 31, 43 and 74:
At (a), we depict the MS grid, and at (b), the sums of the elements in each of the rows. Now observe the following:
Row 2 yields 961, or 31^2
The sum of rows 1 and 3 is 1333, or 31x43
The sum of rows 1, 2 and 3 is 2294, or 31x74
The sum of rows 2 and 4 is 1849, or 43^2
The sum of all 4 rows is 43x74
Clearly, we have a 'full house' for both 31 and 43; and to add to this elegant simplicity we find that 31+43=74!
One further observation concerns the sum 1333. Being fashioned from rows 1 and 3, the number 1333 consists of one 1 and three 3s. As with features touched on elsewhere, the nature of this 4 digit string bespeaks the Triune character of the Lord.
3. Multiples of 61 and the Cross
Columns 1 and 3 interact to display 3 paired CVs that are multiples of 61, thus:
As before, at (a) we depict the MS grid, and now the Cross with its flanking pairs at (b). The sums of these pairs - all multiples of 61 - occupy rows 1, 3 and 4 of the central column: in row 1, 305 = 5 x 61; in row 3, 61 = 1 x 61; and in row 4, 732 = 12 x 61. The sum of the 3 pairs is 1098, or 18 x 61. It will be observed that the remaining 6 CVs delineate the form of a Cross on the MS. Furthermore, this same sum of 1098 depicts the form of a halo on the Cross, thus:
At (b), we observe the sum of 466, 30, 101 and 501 to be 1098 - as before.
4. The axes of the Cross
On the MS the highest CV of 830 is found at the centre of the two axes of the Cross. It therefore partakes in the sums of both axes. The sums of the axes contain striking numero-geometrical features that are found to be related.
The following observations reveal:
The sum of the horizontal axis CVs is 961 = 31 x 31 = 31st rhombus/square.
The sum of the vertical axis CVs is 1953 = 31 x 63; and 1953 is the 62nd triangle.
Intriguingly, these figures are found combined in the following representation of the 62nd triangle within which the central triangle of order 30 (= 465) is embraced by a trio of triangles of order 31 (=496 = 3rd perfect number). Combining the inner triangle with any one of the outer triangles generates a rhombus of order 31 (= 31 x 31 = 961). Importantly, 496 is the sum of the leading CVs for each of the axes of the Cross, i.e. 466 + 30 = 496.
961 / 481 / 241 / 121 / 61 / 31
1953 / 977 / 489 / 245 / 123 / 62
5. The Lord on the Cross
Having found the form of the Cross to be a function of multiples of 61, we then observed the axes of the Cross to be a function of the number 31 and its multiples. Remarkably, the centre of the vertical axis is found to be a multiple of 121, the third member in the midpoint set starting at 31. This feature is observed as we make recourse to the figure of the Lord on the Cross.
Several numerical features are worth considering:
As stated, the sum of the vertical axis is 1331, the cube of 11 that also factorises as 11 x 121. Figuratively, this value represents the Lord's head and body on the Cross.
The remaining CVs sum to 131, and depict the Lord's hands nailed to the Cross.
Both 1331 and 131 are palindromes, their digits representing the Trinity stretched out in both directions. The allusion to the features of the number 31 - of which the axes of the Cross are a function - is unmistakeable.
The value of all 3 CVs forming the Lord's hands and head is 961 = 31 x 31. Comparing this to the vertical sum of 1331 = 11^3, we see the digits representing the Trinity persisting to a remarkable degree in the figure of the Lord crucified. Note that 11, index of the cubic number 1331, is also the prime index of 31.
The sum of the 4 CVs depicting the Lord crucified is 1462. Characteristically, the factors of this number are themselves palindromic, the pair 34 x 43 adding to the self-reflexive properties of the numbers featured above. As if to emphasise the connection we observe that the pair 34 and 43 concatenate as 3443, which itself factorises as 11 x 313.
As observed earlier, the rows of the MS are functions of the numbers 31 and 43.
The improbable persistence of the digits 3 and 1, and the self-reflexive properties of the numbers featured in the analysis of the Lord on the Cross, suggest a fundamental and symbolic integration of number and form as witnessed on the MS. The numbers 31, 43, 61 and 121 are so tightly interlocked that there can be little doubt that they are indexical of purposeful design.
6. The Cross and the hexagram
Already found to be a function of multiples of 31, the Cross displays further symmetrical symbolism around this same factor. Previously we observed the sum of rows 1 and 3 to be 1333 = 31 x 43. These rows sit astride the horizontal axis of the Cross, whose sum was observed to be 961 = 31 x 31. As a matter of interest, the two rows summing to 1333 further divide by a factor of 31, with symmetrical consequences, as below:
From the figure, above, the following may be observed:
This latter sum has some interesting permutations.
- Deleting the zero reduces the sum to 86, the CV of Elohim.
- Meanwhile, the factors of 31 and 26 are the CVs of El and YHWH respectively, thus perpetuating the references to the names of God.
- Curiously, the factors of 806 may be inverted without affecting the product, thus: 31 x 26 = 62 x 13 = 806
The features of rows 1 and 3, above, may be redrawn to depict a stylised hexagram - symbol of Israel - on the Cross.
7. The Cross - some further observations
The Cross may be further subdivided into the upper CV of 466 (symbolising the superscription posted above our Lord as He suffered upon the Cross), and the remaining 5 CVs - summing to 1618 - which depict the Cross itself in the form of a Tau (letter T). Here is the relevant diagram:
In a previous page, we saw how the combined CVs of the six sons of Leah - constituting the first two rows of the MS - gave a scaled approximation of . In like manner, using the same scaling factor of 10^-3, and applying it to the sum of the 5 CVs forming the T-shaped Cross, we are able to extract yet another close approximation to a mathematical constant, viz the golden ratio, known also as the Divine Proportion. As a number this ratio is termed phi in the United States, and tau, T, in the United Kingdom. The actual value of T is 1.6180339... As can readily be seen, the approximation depicted on the MS is as exact as allowable within the constraint of the parameters available. Perhaps more uncanny is the resemblance of the mathematical term T to the actual shape of the Cross depicting its scaled value of 1618. Ultimately, this mathematical symbol T comes to us as an abbreviation for the Greek word, temnw, meaning 'to cut'; which word we find incorporated in, peritomh, meaning 'circumcision'. Intriguingly, the Cross depicts this truth through the cutting off of Christ, as evidenced at Colossians 2:11,14.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610
Taking any term from the series, and dividing it by its predecessor, gives an approximation to T. For example, 377 / 233 = 1.6180257... As the series increments, the convergence to T becomes ever more exact. What is less known is that this feature applies to any number of consecutive terms in the series, providing the second set includes the term immediately preceding that with which the first set begins, whilst simultaneously omitting the concluding term of that first set. For example:
(F13 to F15)/(F12 to F14) = (233+377+610)/(144+233+377) = 1220/754 = 1.6180371...
From the Fibonacci series we get another absorbing aspect of the Cross as per the MS. Having obtained our approximation to T from the 5 CVs which form a T-shaped Cross, we observe that the remaining CV of 466 surmounting this Cross is the sum of three consecutive terms from the Fibonacci series, thus:
(F11 to F13) = (89+144+233) = 466
It follows that the matching term for an approximation to T would be:
(F12 to F14) = (144+233+377) = 754
The MS confirms this, as evidenced by:
1.618 x 466 754.
Amazingly, 754 is the value of the Lord's name in Hebrew: = Jesus Christ. Adding to the mystery, if we multiply this sum of 754 by another mathematical absolute, p, the product is 2368.76... Removing the decimal appendage leaves the integer 2368 = Jesus Christ in Greek, IhsouV CristoV.
There remains yet one more feature of the Cross on the MS; this time the revelation relates to an artefact within the breastplate itself.
8. The Urim and Thummim
At Exodus 28 we read that the breastplate was to be a repository for the mysterious Urim and Thummim. Little is known of the identity of this artefact other than its being a recognised medium for judicial enquiry of God, worn and used by the High Priest alone. The Hebrew words mean 'lights and perfections', and are written [aurim v'thummim]. Their CVs are 257 and 496, respectively, and, significantly, these are found at the extremities of the Cross.
Here it may be seen that the CVs of Zebulun and Joseph (contents of the purple cells) total 257 - the CV of 'Urim'; and the CVs of Simeon and Judah (contents of the green cells), 496 - the CV of 'and Thummim'.
Concerning the latter sum, 496 has a mathematical presence as third perfect number. In light of this fact, we observe that the Hebrew word [v'Thummim] has the meaning 'perfections'.
9. Addendum: 31 and 43: complements and related properties
Inseparably linked on the rows of the MS grid, 31 and 43 share further numerical and geometrical properties suggestive of a deeper complementarity, thus:
- Deletion of the zero
- 31 x 13 = 403
- 43 x 7 = 301
- By deleting the zero, the two products complement one another. We note also that the respective factors of 13 and 7 double as first hexagram/hexagon pair.
- Triple of 8s
- Adding the products above we get 403 + 301 = 704 = 8 x 88.
- As per the MS, 43^2 - 31^2 = 1849 - 961 = 888.
- Trinity and triangles
- a) 31 x 43 = 1333, whose string of 4 digits is composed of one 1 and three 3s
- b) The 31st triangle sums to 496, and the 43rd to 946 - a digit permutation of 496.
- Midpoint structure
- The midpoint structure of each number extended to the 3rd level reveals the following:
- 31 / 61 / 121
- 43 / 85 / 169
- It will be observed that 121 is the square of 11, and 169 the square of 13.
- From this observation a more fundamental relationship is intimated.
- Following this lead, we note the interaction of 11 and 13 on the MS.
- Cube and centre
- a) cubes: the 11th cube, at centre, is supplemented by the first column to produce the 13th cube; and,
- b) centre: 1331, the cube of 11, represents the concatenation of 13 and 31.
- In this capacity it offers a tantalising reference to John 1:1.
- John 1:1
- The CV of John 1:1 is 3627. It has the palindromic factors 13 x 3 x 3 x 31, which telescope into the form 1331.
- 3627 further factorises as 403 x 3 x 3
- Finally, 3627 factorises as 39 x 93. Remarkably, concatenating these factors produces the number 3993 = 1331 x 3.
- This last feature provides us with an interesting observation.
- Genesis 1:1 and John 1:1
- Concatenation of the palindromic factors of the premiere creation verses reveals a cubic connection.
- a) The factors of Genesis 1:1 concatenate as 3773.
- 3773 = 343 x 11, with 343 = 7th cube.
- b) The factors of John 1:1 concatenate as 3993.
- 3993 = 1331 x 3, with 1331 = 11th cube.
- From the above we observe that 343 depicts 43 reflecting in both directions from the centre.
- Similarly, 1331 depicts 31 reflecting in both directions from the centre.
- Birthplace of our Lord
- A final observation on the pair 31 and 43 reveals the birthplace of our Lord. Bethlehem lies at latitude 31 degrees 43 minutes North. While longitude measures - based on the location of Greenwich, England - can be considered arbitrary, latitude measures are absolute, being founded upon the geometry of the Earth's poles and equator.
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