Appendix A - A closer examination of the Genesis/A4 enigma.

[1] - Here, reading from right to left, are the eight words of Hebrew that open the Book of Genesis (and the Bible!):

The first seven represent Genesis 1:1, and the eighth, the first word of the second verse. Bracketed above the text are the values of the numerals represented by the Hebrew letters; and below the text, the numbers expressed by the words - in each case, the sum of its letter-values. Let this ordered set of eight word-values be called G.

[Earlier pages have drawn attention to certain geometrical implications of G: the first seven total 2701 (73rd triangular number), all eight total 3003 (77th triangular number), the sum of 6th and 7th is 703 (37th triangular number), and the sum of the last five is 1801 (25th numerical hexagon.]

The following table of differences involving the members of G sheds much light on the phenomenon currently under investigation:

Each of the 28 entries in this table is derived from the values heading the row and column in which it occurs by subtracting the smaller from the larger. Thus, 710 = 913 - 203, and so on.

99 and 105 (the metric dimensions of the panels displayed in Figure 10) are observed to figure prominently in this table, thus:

401 - 302 = 302 - 203 = 395 - 296 = 99

407 - 302 = 401 - 296 = (296 - 86)/2 = 105

The remaining differences - except those involving 913 - may each be simply expressed in terms of these parameters, thus:

6 = 105 - 99; 12 = 2.(105 - 99); 93 = 2.99 - 105; 111 = 2.105 - 99; 117 = 3.105 - 2.99; 192 = 3.99 - 105; 198 = 2.99; 204 = 99 + 105; 210 = 2.105; 216 = 3.105 - 99; 309 = 2.105 + 99; 315 = 3.105; 321 = 4.105 - 99.

[Note the use of the period (.) here to signify 'multiplied by']

But we also observe that if 500 be subtracted from each of the differences involving 913 the residues fall in with the same scheme! Thus,

710 - 500 = 210 = 2.105; 827 - 500 = 327 = 5.105 - 2.99; 512 - 500 = 12 = 2.(105 - 99); 518 - 500 = 18 = 3.(105 - 99); 506 - 500 = 6 = 105 - 99; 617 - 500 = 117 = 3.105 - 2.99; 611 - 500 = 111 = 2.105 - 99.

[Of passing interest, 216 and 512 (the cubes of 6 and 8, respectively) are seen to be included in the table of differences.]

Clearly, these eight values - representing a natural alternative reading of the first eight Hebrew words of Genesis - which we might have expected to be numerically independent, are here shown to belong together! These observations lead us to consider a simple visualisation of G

[2] - Figure A3(a) depicts a small region of an infinite lattice comprising two coplanar and mutually-perpendicular sets of parallel lines spaced at 99 and 105 units, respectively. Associating an integer value with any of the points of intersection, and adopting the usual convention with respect to sign, a fixed pattern of integers is established for the complete lattice. Choosing this reference value to be 500 units we find that all but the first of the eight numbers of interest occur in a tightly-knit group in its vicinity with a cube of cubes, 512, in close attendance. It should not pass unnoticed that these - along with all other values represented in this lattice - are of the form, (in other words, on division by 3, each leaves a remainder of 2.

A second plane lattice [Figure A3(b)], parallel with the first, and spaced 500 units above it, includes all integers of the form, - the reference value, 1000, lying vertically above the 500 of the first. 913 (the first word of the Bible) is seen to be close at hand, and also sums of pairs and doubles of G, thus: 808 = 401 + 407, 703 = 401 + 302 = 407 + 296, 790 = 395 + 395, and so on. Clearly, this neighbourhood is rich with evidence of the closely integrated nature of these opening words of Holy Scripture! In addition, the second component of the 'friendly' pair, 1184/1210, is observed to be in the vicinity - its partner being a simple multiple of 37 and of 296.

[It is worth observing, (a) that a region of either of the lattices described may be represented by a number of segmented A4 sheets (see Figure 10) laid side by side on a flat surface; and (b) that 913 may be expressed as the sum of 401 and 512 (the cube of 8) - both of which are found nearby on the lower lattice.]

To complete the picture, we may envisage a third lattice - situated 500 units below the first - one embracing all integers of the form, , ie all multiples of 3. Between them, these three lattices represent the complete domain of the integers (infinitely recurring - as Section 7 of Appendix 2 makes clear!). The matter is summarised in Figure A4.

[3] - A consideration of the foregoing relationships leads to the following parametric representation of the members of G:

The parameter values - all small integers - are defined in the following table:

For further developments, see Appendix B .

[4] - One further aspect of the Genesis/A4 Enigma concerns (a) the nominal area of an A4 sheet [ = one sixteenth of a square metre (see Figure 3)], and (b) the specified ratio of its sides, viz. A square of side 250mm has, therefore, the same area as an A4 sheet.

Supposing L to be the actual length of A4 in mm, and W to be its width, then

Since the ISO standard specifies that dimensions be expressed to the nearest mm, we confirm the nominal size of A4 to be 297 x 210 mm - these dimensions related by a simple division of the metre (the unit upon which the A-series is based) and the 4th root of 2!

Now an important constant as far as man is concerned is ie the twelfth root of 2. This is the ratio of frequencies represented by the interval of the semitone (western music's smallest indivisible step) in the scale of equal temperament, and it is upon this that the world's masterpieces of musical composition are based (eg Bach's B minor Mass, Beethoven's 'Ninth', Mozart's operas, etc). This now becomes part of the A4 link with the opening words of Genesis, as the following analysis reveals:

Note here that the orientation of the lambdas indicates the 'direction' of the associated minor thirds. This is made clear in the following example:

G (the set of the opening eight words of Genesis) may now be expressed as 'rounded up' values of expressions involving minor thirds, as follows:

It should be remembered that these relationships arise from the scale of 'equal temperament' - itself an approximation of the so-called 'natural scale' which had severely restricted musical composition prior to Bach (1685 - 1750).

Vernon Jenkins

27.1.00

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