**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

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