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 ofPHYSICAL 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. And using the chromatic scale, this tonic chord would be found with numbers 1,5,8,13. 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 numbers 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 Pisa, 1170 to 1230 A.D. travelled about the Mediterranean coastlands collecting information about mathematics. In 1202 he returned to Pisa and published a book entitled LIBER ABACI, in which he established the introduction of the Arabic notation in Europe; and provided a foundation for future development in arithmetic and algebra. Some years later, in 1220, he published another book calledPRACTICA GEOMETRIA, a treatise on Geometry. Out of these two books comes the study of the numbers we have quoted in this paper, 1,2,3,5,8,13, etc. which have ever since been attributed to Leonardo. It is known as theFIBONACCI SEQUENCE. The name derives from Filius Bonacci, meaning “son of Bonacci”, who was Leonardo’s father.
Another interesting area of study may be found by first drawing a rectangle of 89 units by 55 units, which makes aGolden Rectangle. Having done this, divide the rectangle into two portions, a square of side 55 units, and a smallerrectangle 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 aSPIRAL, 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 theInverse-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 at118, which, if it could be re-created, would have 191 Neutrons. If we calculate the ratio 191/118, we shall find it to be1.618! 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 toKNOW 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 Calvary has made it possible for us to be forgiven, redeemed, reconciled, and restored. Amen.