The Prophetic Telegraph - No.50
The Golden Section
When I was a young man at
University, I studied some of the intricate laws of nature in the realms of
physics and astronomy. Some of the laws were simple to understand. Others were
far from simple. But all the time I found myself grappling with LAWS.
And without those laws, there would never have been a degree course in Physics.
Sadly I realised that most students were studying those laws out of a basic
interest, yes, but more so with a view to later employment prospects, rather
than to appreciate the evidence of the LAWGIVER.
Over the forty or so years
since then I have found my interest in the LAWGIVER taking
supremacy over the subject of PHYSICAL LAWS. However, I cannot
help but agree wholeheartedly with the Apostle Paul in Romans One, that God's
eternal power and godhead are apparent to all mankind through Creation, and
therefore man is without excuse if he fails to worship the CREATOR,
rather than the CREATION. There is one law of creation which,
to my way of thinking, takes precedence over all others in showing God's
handiwork. It is a simple law, and allows even those of primary school age to
discover it, and find it in virtually every aspect of life. And those whom God
has engraced with greater intelligence will find the law at work in many of the
more intricate areas of study. In this paper our intention is to display the
universality of the law of THE GOLDEN SECTION in such a way as to give praise to
our great God and Saviour. And we do so in recognition of the fact that we have
reached our FIFTIETH EDITION, which is commonly thought of as
a GOLDEN NUMBER.
One of the joys of a summer
walk in the fields and woods is to collect items of interest, in the following
way. If you live in heath land, with many varieties of conifers, then to
collect fir-cones would be a good way to start. Cones are of differing
patterns, but with a common design, having opposing spirals. A closer look will
show that the clockwise spirals flow at a different angle to the anticlockwise
spirals. And therefore, if you count the number of spirals in each of these two
directions, you will find a pair of numbers, such as 5 and 8. A good collection
of fir cones will produce other pairs of numbers. In my own collection, I have
found three sets so far, 3 & 5, 5 & 8, and 8 & 13. Notice the emergence of these
numbers - 3,5,8,13.
Other forms of botanical life contain spirals, and these may then be added to
the collection. For example, everywhere we tread we find in the grass small
white daisies with yellow centres. A magnifying glass is needed to count the
spirals in the yellow centre, but the effort is rewarded by finding 21 & 34. A dandelion clock is also rather
troublesome to count, especially if the breeze carries some away! But patience
will be rewarded by finding 13 & 21. And so, with just three examples of plant
life, we have discovered a series of numbers -,5,8,13,21,34.
A smart youngster would
soon spot the connection between these numbers. Each one is the sum of the two
previous numbers. The next two would therefore be 55 and 89. Returning from the walk, a look
round the garden will be rewarding. Take a large sunflower head, and you will
see that it is brilliantly designed. The spirals away from the centre have 55 in one direction, to 89 in the other! Try counting the
petals on some ordinary garden flowers. Marigolds have 13, Asters 21, Buttercups and Delphiniums 5, which is by far the commonest
group, though some Delphiniums have 8. Lilies and Irises have 3. Many different varieties of
daisies may be tested, and the numbers 34, 55, and 89 will be found. Once again, all
these special numbers are re-emerging. Using a pair of leather gloves, try
cutting a length of Gorse from a bush, and observe the way in which the spines are
arranged round the stern. Looking along the stem, mark those spines which
appear directly above one another. You will then observe that the are 21 spines in 8
revolutions
between those you've marked. A similar test using the needles on a branch of
fir will reveal 13 needles in 5 revolutions before the sequence is repeated. A
very obvious example of the opposing spirals is on the Pineapple. It always has
8 one way and 13 the other. I think enough examples
have been quoted to show that botanical design is NOT by chance, but is subject
to the simple law of a certain number sequence. I wonder how many of my readers
were aware of these numerical designs. Children love findings things out. Try
these tests on them, and see how they enjoy themselves!
For those with even the
smallest knowledge of music, the following observation may be made. Look at the
piano keyboard, and see how the OCTAVE is composed. There are 13 notes, which are divided into 8 white and 5 black. Furthermore, the black notes are
arranged as a pair and a triplet. Here then are the numbers 2,3,5,8, and 13. The tonic chord of Doh, Me, Soh,
and Doh' -in the key of C major, is composed of white notes numbered 1 3,5,and 8. The keyboard design is based upon
certain frequencies of sound which are pleasing to the ear, and the result
'just happens' to require the use of this number sequence! Without the
botanical studies, the musical sequence might have been missed, but now it
stands out with clarity. Using the lower humbers in the sequence as generated
by the musical scale, we have now found the existence of all the following - 1,2,3,5,8,13,21,34,55,89.
For those who study
microbiology, we suggest investigating the double helix of the DNA
molecule. Even
here, this number sequence will show up. Where else can we find it? The answer
is, in many places.
Try dividing one of the
numbers by the one before it.. Leaving aside the first few numbers, we get the
following.
13/8 = 1.625
21/13 = 1.615
34/21 = 1.619
55/34 = 1.618
89/55 = 1.618
and every larger pair of
numbers gives the same ratio, 1.618. This number is known as THE GOLDEN SECTION,
and it crops
up in the fields of ART and ARCHITECTURE. An artist will know, almost by instinct, that a rectangle of certain
proportions is most pleasing to the eye. When a number of people are asked to
draw such a rectangle, and the ratio of the side-lengths is calculated, the AVERAGE will turn out to be somewhere about
1.6. Furthermore, the positioning of
major features WITHIN a painting is often found to manifest this
Golden Section by measuring distances to the sides, or top and bottom of the
picture. Many ancient paintings, as well as major architectural designs, can be
shown to incorporate the Golden Section.. For example, the Parthenon in Athens was built according to the Section,
though part of the front gable end is now missing, and has to be reconstructed
on a photograph to check this fact.
The reason for using the
Golden Section is clearly related to the PLEASURE it gives to us as human
beings.. In the same way, the keyboard harmonies are related to a PLEASING
combination of frequencies. Why is this? What is it about the design of the
human ear that creates the sensation of PLEASURE when using certain
combinations of frequency? And what is it about the human eye that demands
observance of the Golden Section to create PLEASURE? In the field of human
physiology relating to the ear, I have only limited knowledge, and so I cannot
answer the first question. But I have read widely on the structure of the human
eye, and one feature stands out as being most remarkable. When we read a page
of print, our eyes look along the page to follow the words. This is because,
although the eye sees quite a wide panorama, ACUTE CLARITY OF VISION is limited to a very small area
directly in front of our field of view. The reason for this is due to the
structure of the RETINA, which is the light-sensitive surface at the
rear of the inside of the eyeball. It is composed of two types of nerve
endings, known as RODS and CONES,
because of their shape. The Cones are our daytime light-receivers, and the Rods
are best for dim light. But the Cones are concentrated more and more towards
the small spot directly behind the lens, where we see most clearly.
There is a slight
depression at this point, and the Cones are most tightly packed here, and this
is where the image falls of the word or words on the page towards which we are
focussing our vision. Outside this small depression the clarity falls away
quite sharply. Now here is the remarkable fact. This slight depression in the
Retina, known as the FOVEA CENTRALIS, is OVAL in shape, and the ratio of the
width to the height is approximately 1.6! In other words, we have been
created to receive beauty and right-proportion in accordance with a simple law,
the Law of the GOLDEN SECTION! No doubt in due course a similar fact will be
observed for the COCHLEA in the Inner Ear, which receives sound
frequencies. But whether this discovery is to be made or not, the fact remains.
We have been created to enjoy harmonies according tu the Law of the Golden
Section.
Now, lest we feel superior
to our distant ancestors, as often modern man does, the following facts emerge.
The
Great Pyramid in Egypt was constructed very definitely using the Golden Section. Hence the
ancient Egyptians were aware of the Golden Section as far back as the time of
the great Flood of Noah's day. In fact, a study of ancient architecture and
literature reveals quite a wealth of examples of the use of this Golden Number.
In more recent times, an
Italian mathematician by the name of Leonardo of
Another interesting area of
study may be found by first drawing a rectangle of 89 units by 55
units, which
makes a Golden Rectangle. Having done this, divide the rectangle into two portions, a square of side 55
units, and a
smaller rectangle of 34 by 55 units. It will be seen that this small rectangle is
another Golden Section. Now divide this smaller rectangle in the same way,
making a square of side 34, and a rectangle of 21 by 34 units, which is again a Golden Section.
This process may be continued until a very small square of side 1 by 1 is
formed. Now take a compass, and placing the point at the lower corner of the 55
by 55 square, draw a quadrant of a circle, radius 55. Then repeat this process
for each smaller square, making sure that each quadrant-end joins up with the
next. The result of these operations will be a SPIRAL, more strictly known by
mathematicians as a LOGARITHMIC SPIRAL.
It will be seen at once
that this form of spiral is found in many places in life. Sea shells
immediately spring to mind, whether they be of the Chambered Nautilus type, or the Cockle Shell type. Each grows according to the Logarithmic Spiral. But then again we find it in the
shape of an elephant's tusks, the horns of sheep and goats, the claws of birds, our finger nails (if allowed to grow long enough to
appreciate the law), and the crown of hair on our heads, most easily seen on a young child
as the hair begins to grow.
But all these examples were
taken from life-forms. The law also operates in the formation of Spiral Galaxies, due to the Inverse-Cube Tidal
Force Law. If
the Law of the Golden Section is found operating at Galactic size, it may come
as a shock to know that it also operates at the subatomic, or nuclear size! All
elements contain in their nuclei PROTONS, (which are positively charged),
and NEUTRONS,
(which have no electric charge.) Some of the elements of smaller size, near the
beginning of the periodic table, have equal numbers of Protons and Neutrons.
For example, Carbon has SIX of each. But as the complexity of the Atom
grows, the Neutrons tend to outnumber the Protons. Thus,
Aluminium has 13 Protons to 14
neutrons;
Iron has 26 Protons to 30
Neutrons;
Silver has 47 Protons to 60
Neutrons;
Gold has 79 Protons to 118
Neutrons;
and Uranium has 92 Protons to
146 Neutrons.
In the last fifty years
Nuclear Physicists have been able to reproduce Atoms of even higher numbers,
Atoms which are normally Radioactive, and which disintegrate over periods of
time. As a result it has been possible to investigate the properties of
elements up to 105 Protons. What has emerged is the evidence for a completed
period of elements, ending at 118, which, if it could be re-created, would have 191 Neutrons. If we calculate the ratio 191/118, we shall find it to be 1.618! This is the EXACT Golden Section.
Space prohibits any further analysis of this amazing Law. I believe sufficient
has been said to establish the truth, that God has used this Law throughout His
creative schemes, from the size and shape of Spiral Nebulae, to the ratio of
Protons and Neutrons in the Nuclei of Atoms, and in the world of living things,
from the numbers and arrangements of leaves, petals, buds and shoots, to the
shape of sea shells, tusks, horns, and claws of animals, the crown of hair on
our heads, (and other features of the human body not included in this paper),
to the Musical Scale, and Right Proportion in Art and Architecture, based on
physiological apparatus. created within us to be able to enjoy these things.
And what more can we say?
Is this the evidence of Evolution? Surely Darwinism must recede as a
poverty-stricken theory with no substantial foundation, when viewed alongside
the grand work of the LAWGIVER and CREATOR of the UNIVERSE. Thank God for the EYES and EARS He gave us, whereby we can
appreciate and enjoy these things. Thank God that the main evidence for this
Universal Law comes from the observation of simple things around us, like
daisies, fir cones, pineapples, sea shells, things that little children can
collect and enjoy, and that their parents can use to point them to God at a
very early age. Truly we have a wonderfully wise Creator God, and being created
in His image, we are able to KNOW and to RESPOND to that fact. May we enjoy His
works, and give Him the praise He deserves. May we be equally thankful to God's
beloved Son, whose death at