Divine Arrangements
The 17th Century French scientist and philosopher Blaise Pascal wrote in his Pensees that purely on the basis of probabilities it was advantageous to be religious, an assertion that has come to be known as Pascal’s Wager. If God existed the believer would be better off than the non-believer. If no God existed and death was final then the believer would then be no worse off than the non-believer. So by this rather pragmatic reasoning it is a safer bet to believe in the reality of God.
Pascal is also famously associated with Pascal’s triangle, a triangular array of numbers that has found many uses in mathematics, including, appropriately, probability theory. Although Pascal was not the first to discover the triangle named after him - it was known in India and China in Medieval times - he developed many of its applications, organising all that was known about it into his Traite Du Triangle Arithmetique.
Pascal himself was an intensely religious man, whose life in many ways exemplified Christian virtues. It is astonishingly apt, then, that the most sacred names of God in both ancient Hebrew and modern English, after numeration under a long-accepted system of gematria, correspond to symmetrical arrangements of numbers within Pascal’s triangle. This unusual feature of Pascal’s triangle suggests an influence on the spelling of these divine names by a Mathematician who knew of Pascal’s triangle long before Blaise Pascal - or anyone else - and wished to draw our attention to His presence in our lives.
Pascal’s Triangle
Pascal’s triangle is formed by the simplest of means, starting with the number 1, by summing adjacent numbers in every row of the array (empty spaces taking the value zero) then placing the sum below and between them to create the next row. The first step of this procedure is shown below.
There are two zeros either side of the one in row 0, and 1 + 0 = 1, so two ones are placed in the next row, which is termed row 1. The creation of the next two rows is shown below.
The procedure can be continued indefinitely to produce Pascal’s Triangle, the first eleven rows of which are shown below.
The aesthetic appeal of Pascal’s triangle is matched by its practical application in many fields of mathematics, such as algebra, number theory and probability theory. For example:
1. The sum of the numbers in each row is double that of the previous row. So the sum of row 0 is 1, that of row 1 is 2, that of row 2 is 4, etc, giving us the simplest geometric series: 1, 2, 4, 8, 16...
2. Each row gives the coefficients in the expansions of the binomial series (x + y)^n (n = 0, 1, 2, 3...), where the nth row gives the coefficients for that expansion. So (x + y)^3 expands to x^3 + 3x^2y + 3y^2x + y^3, the coefficients here being 1, 3, 3 and 1, which are the numbers in row 3 of Pascal’s triangle.
3. If we read the rows as numbers we obtain increasing powers of 11, as shown.
4. If we start with either number in row 1, then slice down diagonally through the triangle we obtain the counting numbers (1, 2, 3, 4...). Starting at either side of row 2, we get the series of triangular numbers (1, 3, 6, 10...). Next we have the tetrahedral numbers and after that we get the equivalents of the tetrahedron in higher-dimensional spaces.
This is just a glimpse of some of the mathematics Pascal’s triangle embodies, so the reader can begin to appreciate how important it is to mathematicians.
The Names of God
There are many names and titles by which the God of Israel is known.[1] However, we are concerned here with the two most important of these in the Hebrew and English languages: YHVH/LORD and Elohim/God. These are used in the Hebrew Bible far more often than any other name or title for our Creator.
YHVH/LORD
For Jews and Christians, by far the most sacred name for God is the Divine Name, also known as the Tetragrammaton (four-lettered name of God), for which the English transliteration is YHVH. [2] Its original Hebrew spelling is shown here:
YHVH is the personal name of God and its importance can perhaps be estimated by the fact that it is used no less than 6823 times in the Hebrew Bible [3] and that orthodox Jews are forbidden to speak it. It is often replaced in Scripture by the substitute word Adonai. In the NIV Bible and many other versions YHVH is usually translated into English as the upper-case
LORD
The lower-case “Lord” is used for Adonai.
Elohim/God
Undoubtedly the next most important Hebrew word for God is Elohim. Elohim is used 2570 times in the Hebrew Bible and in fact the first name or title given therein for God, being found in Genesis 1:1. This word symbolises God as Creator and Judge and is spelled in Hebrew as shown here:
It is actually the plural of Eloah,[4] but is used mainly with singular verb forms, adjectives, etc, so it is translated into English as
God
Two titles closely related to YHVH and Elohim are the digrammata, transliterated as Yah and El and comprising the first two letters of YHVH and Elohim.
Yah
Yah is thought to have derived from YHVH and can be regarded as a shorter form of this word. It is spelled as shown:
Yah is used over 50 times in the Hebrew Bible, for example in Psalm 68:4. It is also the ending in words such as ‘Hallelujah!’, ‘Elijah’, etc.
El
The second digrammaton is the root word from which ‘Elohim’ and many other sacred titles are derived and can be transliterated as ‘El’. The word is spelled as shown:
El is used 250 times in the Hebrew Bible and means ‘might’, ‘strength’ and ‘power’. It is used in constructs such as ‘El Shaddai’ (God Almighty) and others and ends many words with divine associations, such as ‘Immanuel’ and ‘Israel’.
One further Hebrew word and its associated character should also be mentioned: Aleph, the first character of the Hebrew alphabet, has long been associated with the mysteries of the Godhead. Aleph is formed as shown:
Aleph is the ‘father’ of the other twenty-one characters and reflects God’s fatherhood and primacy over all Creation. Aleph’s numerical value of 1 symbolises God’s Oneness. In Hebrew it is usually a silent letter, symbolising God’s quiet presence in our lives.
The classical Hebrew character for Aleph is composed of two yods and one vav, as shown.
The word ‘Aleph’ itself gives rise to the Hebrew word ‘Aluph’, which means ‘Lord’ or ‘Master’. Aleph is spelled in Hebrew as follows:
Numbering the Names
The bridge between the numbers within Pascal’s triangle and the words for God are Hebrew and English gematria. There are many systems of gematria. However, by far the most significant is the standard value system, which was the system of alphabetic numeration used to represent numbers in biblical times and therefore likely to have been the first system of gematria that was employed.
The table below shows the standard values of all the Hebrew and English names of God I showed above, along with those of the word Aleph, its Hebrew character and the Hebrew characters that constitute it.
Note: from now on I will substitute English transliterations for the Hebrew names, indicating in parenthesis if the word for which I have worked out the standard value is Hebrew or English.
The Background
All the imprints are the sums of groups of adjacent numbers within the first eight rows of Pascal’s triangle. Note that I have highlighted the central triangle of numbers; these are the focus for some of the most important encodings.
Since there are thirty-six small numbers within the array, having a range of values, finding a group of them that give a particular sum might not seem to be a remarkable feat. For instance the number 26, standard value of YHVH, is the sum of the numbers within the orange quadrilateral.
However, this figure is asymmetrical and sits off-centre within Pascal’s triangle. The numerical groupings I will show you are fundamentally different from the above figure and, in their symmetry and simplicity, bespeak something other than random selection. Specifically:
1. They are regular polygons, such as triangles, rhombii, etc, or symmetrical arrangements of polygons.
2. Their central axis is aligned with the central axis of Pascal’s triangle.
3. Almost all of the figures align with the rows and diagonals of Pascal’s triangle. A very few figures do cross over them but all of the figures have the first two properties.
Restricting groupings to regular polygons that align with the axis of the triangle drastically reduces the number of possibilities for selection and catches our attention because of the aesthetic appeal of the resulting combined object.
With that in mind let us now proceed.
The Arrangements
Yah
As we count from the top of the triangle the first number we come across is the standard and ordinal values of the first of the two digrammata, Yah.
Sum of numbers in green triangle = 15
Yah (Heb.) ..................................= 15
El
Counting down one further row we arrive at the standard value of the second of the digrammata, the Hebrew word transliterated as ‘El’
Sum of numbers in violet triangle = 31
El (Heb.) ......................................= 31
YHVH
The Tetragrammaton can be found by superimposing a triangle on the three numbers embedded at the centre of the first eight rows of Pascal’s triangle.
Sum of numbers in purple triangle = 26
YHVH (Heb.) ................................= 26
Given the primacy of this name in both Judaism and Christianity, I find it fascinating that its constituent numbers are found as a triangle of three numbers at the heart of this array, as if to emphasise its supreme importance. The individual letter values of the tetragrammaton are 10 (Yod), 5 (Hey), 6 (Vav) and 5 (Hey), which is almost identical to the constituents of this triangle of numbers.
From now on I will call this small central triangle “the YHVH triangle”.
Elohim
The standard value of the Hebrew word meaning ‘God’ is the sum of the numbers eclipsed by three rhombii centred on the YHVH triangle.
Sum of numbers in pale green rhombii = 86
Elohim (Heb.) ......................................= 86
This number is also the sum of the numbers either side of and below the YHVH triangle.
Sum of numbers in pale blue area = 86
Elohim (Heb.) ..............................= 86
Here we see that the two shapes are part of a larger block of numbers, representing the combined name YHVH Elohim (The Lord God). Given the close relationship between these two words - the two names for God that Jews hold most dear - the numerico-geometric correspondances seen here and in the previous illustration appear too fortuitous to be coincidence.
Note that the triangle of numbers above the coloured block sums to 15, which we have already noted is the numerical value of Yah.
God
The standard value of the English word ‘God’ is the sum of a rhombic-shaped cluster of numbers proceeding from the apex of the triangle.
Sum of numbers in orange rhombus = 71
God (Eng.).......................................= 71
Lord
Astoundingly, the name ‘Lord’ is the standard value of the remaining numbers in the first eight rows of the triangle!
Sum of numbers in yellow pentagon = 184
Lord (Eng.).........................................= 184
As with the shapes representing the combined Hebrew title, these form a larger block, in this case a triangle superimposed upon the entire first eight rows! The representation of the combined title “LORD God” by a triangle is particularly appropriate for Christians and once more suggests intelligent design.
Aleph
The first letter of the Hebrew alphabet has a numerical value of one and is obviously represented by the digit at the apex of Pascal’s triangle.
Aleph (Yod, Vav, Yod)
As I showed above, the classical Hebrew character for aleph is composed of two yods and a vav. These sum to 26, which is the value of the Hebrew word YHVH, shown earlier, and is therefore represented by the same trio of numbers within Pascal’s triangle.
Sum of numbers in purple triangle = 26
Yod, Vav, Yod (Heb.) ....................= 26
There is an even more auspicious correlation of this name with Pascal’s triangle, however. The individual letter values are 10 (Yod), 6 (Vav) and 10 (Yod), which are the precise values of the numerical elements of the triangle! In both character and triangle the Vav/six sits between the Yods/tens.
Yod = 10, Vav= 6, Yod = 10
It actually looks like the construction of the letter Aleph has been modelled on the YHVH triangle. This is particularly appropriate, of course, because of Aleph’s status as first among the Hebrew characters.
Aleph (Name)
The standard value of the name ‘Aleph’ is 111, which is also known as the full value for the letter Aleph. The number 111 is the sum of the numbers occluded by the pentagon shown here.
Sum of numbers in cerise pentagon = 111
Aleph (Heb. (full))..........................= 111
The individual letters of ‘Aleph’ are Aleph, Lamed and Pey, which individually have the numerical values 1, 30 and 80. The pentagon easily splits into precisely these numbers. Looking down the pentagon we also see that the three letters are represented in the same order in which they are found in the word: Aleph, Lamed, Pey.
Sum of numbers in cerise triangle .........= 1
Aleph (Heb.).........................................= 1
Sum of numbers in purple trapezium = 30
Lamed (Heb.)......................................= 30
Sum of numbers in blue trapezium ....= 80
Pey (Heb.).........................................= 80
Here divine mystery and mathematics fuse into these extraordinary representations of Aleph, the ‘father’ of the Hebrew Aleph-Bet.
Summary
The first eight rows of Pascal’s triangle can be symmetrically subdivided into triangles and other simple figures, the numbers within which sum to the standard values of the two most sacred divine names in Hebrew: YHVH and Elohim. Two shorter words derived from them are also represented, along with the gematria of the letter Aleph and its constituent parts. Finally, the English words ‘LORD’ and ‘God’ are also represented. These numbers could not have been encoded within Pascal’s triangle (which could not be altered even by God) but within the words themselves, the spellings of which must have been guided to conform via gematria with the numerical properties of the triangle.
The figures are symmetrical, align with the central axis of Pascal’s triangle, in two cases fit together to form larger shapes, reflect the first eight rows of the array and its internal structure - particularly the central “YHVH triangle” - all of which suggests foresight, planning and a desire to communicate with us. Whether or not Pascal’s wager is a good argument for maintaining religious faith, it is beautifully apt that the remarkable mathematical array that also bears his name is the setting for these symmentrical arrangements, proclaiming the reality of our Creator, aligning with the fundamental truths of mathematics and displaying the awesome power of a God who can reveal himself in this profound and meaningful way.
Bill Downie 28/7/10
Notes
1. Some of the imformation on the names of God in this page comes from the excellent website Hebrew For Christians.
2. Sometimes the letter ‘W’ is used. However, this is probably incorrect, as there is no ‘w’ sound in Hebrew.
3. Interestingly, the number 6823 is a numerical anagram of the standard value of the Greek name ‘Ihsous Xristos’ transliterated as Jesus Christ, which is 2368!
4. It seems to me that this allows for the Christian concept of the Trinity, which is also suggested by the triangular form of most of these “imprints”.