The Golden Proportion: Extract taken from Robert Lawlor’s book Sacred Geometry.
p.42
‘It was the goal of many traditional teaching to lead the mind back to the sense of Oneness through a succession of proportional relationships. A proportion is formed from ratios, and a ratio is a comparison of two different sizes, quantities, qualities or ideas, and is expressed by the formula’ a:b.
A ratio then constitutes a measure of difference to which at least one of our sensory faculties can respond. The perceived world is then made up of intricate woven patterns of ‘differences which make a difference’.
Not only then is a ratio a:b the fundamental notion of all activities of perception, but it signals one of the most basic processes of intelligence in that it symbolises a comparison between two things, and is thus the elementary basis for for conceptual judgement.
A proportion, however, is more complex, for it is a relationship of equivalency between two ratios, that is to say: a is to b as c is to d, or a:b::c:d. It represents a level of intelligence more subtle and profound than the direct response to a simple difference which is the ratio. It was known by the Greeks as analogy.
When we think with four elements, that is with two different ratios, we locate our thoughts within the range of manifestation, of the natural world, since four is the number-symbol indicative of the finite, measurable, rational world of procreated form.
So, a:b::c:d is a general formula for four related elements: thus numerically 2:4 :: 3:6. When we further limit ourselves to three terms, that is when we lift ourselves one level to the realm of principles or activities (threeness), we find that the determination becomes more exacting through the reduction of the number of elements involved; thus a:b :: b:c. Here the extremes are bound together by a mean term b. The Greeks called this a continuous proportion of three terms, and this indicates a decided shift in the symbolisation of perceptual and conceptual processes. It is the perceiver himself (b) who forms the equivalency or identity between observed differences (a and b). The perceiver no longer stands outside the comparative activity as in the four term, discontinuous mode, which images the perceived difference as being separate ratios or distinctions. An example follows.
Our experience of the world is due to our organs of perception being sensitive to variations of the wave frequency patterns which surround and pervade our field of awareness. We distinguish a red cup from a green tablecloth only because our optic nerves set up a brainwave patterns which corresponds to the frequency patterns emanating from the cup and tablecloth. The perceiver then is the indispensable bond in the registration of these variations in external frequency patterns, interpreting and distinguishing them as objects such as the cup and tablecloth.
Many philosophers speak of reaching state of consciousness in which one is constantly aware of this integration and attunement between the apparent external vibratory field and the inner field of perception. This mode of perpetual awareness, which we find comparable to a three term, continuous proportion, was referred to by Sri Aurobindo as ‘knowledge by identity’, and regarded as an important stage in the process of spiritual development. While acknowledging an external source of experience we recognise that it is in a continual flow of relationship with our internal faculties of perception and cognition, and it is this relationship, not the external object itself, that we are experiencing. The objective world then is interdependent with the entire physical, mental and psychological condition of the perceiving individual, and constantly will be altered by his inward condition.
Is then, a three term proportion as close as we can approach the sense of unity with proportional thinking? The response to this question is No, for there is One, and only One proportional division which is possible with two terms. This occurs when ‘The smaller term is to the larger term in the same way as the larger term is to the smaller plus the larger’. And it is known as The Golden Proportion.’
Continued on pp.46-47
p.43
This page is deliberately left almost empty of writing and drawings, as I had planned a complex page on the reverse of this sheet, and did not want too much show through to interfere with the drawings there. So here I show only one particular aspect of the Golden Proportion, where the Whole is more than One
Geometry of the Planets
pp. 44-45
The Wooden Books series edited and designed by John Martineau provided an extraordinary treasure trove of information of many kinds. The edition ‘The Little Book of Coincidences’ was one of the first published of the series in 2001; and was researched and written, with exquisite illustrations by John Martineau himself.
The book draws attention to the amazing patterns in the sky made by the various planets as they journey round the sun. Some drawings concentrate on the geometric shapes made by the path of one planet; and others describe the relationships and conjunctions of two or more of the heavenly bodies. It is a remarkable discovery, and the text is concisely written and beautifully illustrated.
These two pages form a continuous double opening. So when considering how I was going to portray some of this information across the pages, I found an illustration of an Indian manuscript from Rajastan made around the18th century. The page is divided into squares with bars of quite heavy decoration. The squares are filled with signs and symbols, each representing to the Yogi ‘a different system or technique of thought for understanding the world and its structures’.
This layout seemed to me to be a very successful way of putting together a collection of individual but related pieces of information. The cross-bars are strong enough to form a grid which would separate yet hold together many different drawings - simple or elaborate - in a random order, so I did not need to make a careful rough draft. I made a rubber-stamp by cutting into an eraser to use as marking out the grid.
The Inner Planets
p.44
The top line of the squares depict The Signs for the Heavenly Bodies: firstly the Sun, Moon and Earth. It is interesting to note that, as seen from the Earth, the Sun and Moon appear to be the same size; this is confirmed by a total eclipse, where the Sun is completely blocked out by the Moon passing between the Sun and Earth. The length of the eclipse year was known by the Neolithic builders of the stone circles in Britain.
The second square shows the signs for the planets of the solar system. Only the visible planets were known in classical times; later Uranus was discovered in 1781, Neptune in 1846, and Pluto in 1905. The symbols used here are taken from and combine the symbols of the Sun, Moon and Earth.
Square 3.
This reveals the dance of Mercury and Venus. Their relative orbits round the sun are found by putting three circles together. The circle through their centres denotes the orbit of Mercury, and the outer circle that of Venus.
Square 4.
Arcs taken from a square drawn inside a circle frame the orbits of Mars and Jupiter.
Square 5.
The drawings and all the information here can be found in greater detail on pp. 24 &25. The original discovery of the geometric relationship of the Earth and Moon was discovered by John Michell, and published in his book ‘ The Dimensions of Paradise” Thames and Hudson 1988.
Square 6.
Earth and Jupiter: their relative orbits are unfolded by successive six- pointed stars within a circle.
Square 7.
The square is filled with the words ‘around and around’: which echoes the spinning of the image.
Square 8.
A verse written by John Donne: ‘Man has weaved out a net, and this net thrown upon the Heavens, and now they are his own’.
Square 9.
Mercury and Earth: relationships of their sizes and orbits are shown by 5 and 8 fold overlays.
Square 10.
Kepler’s inscribed Platonic Solids. Looking for a geometric or musical solution to the nesting of the orbits of the planets, he observed that 6 planets meant 5 intervals, he tried to fit the 5 Platonic solids between the orbits of their spheres.
Square 11.
The Star and the Pentagon. The Golden Proportional relationships are maintained as the inner stars scale down in size.
Square 12.
A list of the planets: their diameters in miles, their mean orbits around the sun, and period of days.
The Outer Planets
p.45
Square 1.
The chart shows the number of days between planetary ‘kisses’ (near approaches).
Thus 4 x Venus/ Earth = 3 x Mars/Earth. Many encounters are simple musical ratios i.e. 4:3 and 5:4.
Square 2.
This diagram is based on one of the stone circles of the Neolithic age, the curious flattened circles which were catalogued by Professor Alexander Thom - the discoverer of the Megalithic yard. This stone structure is to be found at Bar brook, Derbyshire, and belongs to the Type B in his grouping. The drawing starts with the familiar double circles within a larger circle, which in turn nest 3 smaller circles. A line is drawn up from the lower center point, and runs through the centre of each of the double circles.
The line to A denotes the length of the Eclipse Year = 346.62 days = 18.618 x 18.618.
The line to B denotes the length of the Solar Year = 365.24 day s= 18.618 x 19.618
The line to C denotes length days of 13 full Moons = 383. 89 days = 18.618 x 20.618.
The Golden Proportion is 1.618.
Info on Square 2 discovered by Robin Heath, pub.in Wooden Books
Square 3.
Earth and Saturn. A fifteen-pointed star gives their relative orbits and sizes.
Square 4.
Saturn and Uranus. Saturn’s orbit around the sun has a radius almost exactly 1/2 that of its neighbour Uranus. The equilateral shows the ratio 1:2 - the Octave.
Square 5.
The poem ‘ Here meet and marry many harmonies …’ is from The Land by Victoria Sackville West.
Square 6.
Mars and Saturn. The circumference of Mars’ orbit is the radius of Saturn’s. The circumference of Saturn’s orbit is the diameter of Neptune’d orbit.
Square 7.
The words are taken from a hymn written by Joseph Addison ( 1672- 1719.
Square 8.
The Mean Orbits of Planets. The asteroid belt including Ceres divides the inner planets from the outer. The finding by Alex Geddes discovered the astonishing relationship between the four inner rocky planets and the four outer gas giants. Their orbital radii reflect each other either side of the Asteroid belt: Ve x Ma x Ju x Ur = Me x Ea x Sa x Ne - in millions of miles.
Square 9.
Jupiter and Saturn. Conjunctions and Oppositions against the Zodiac. Seen from the Earth, the two planets appear to meet and part every 20 years, forming a hexagram.
Square 10.
Mercury and Earth ( from the Inner Planets’ list). the relative sizes and orbits. Their physical sizes are in the same relationship to each other as are their orbits, accurate to 99.9 percent. This coincidence can be described in a variety of ways.
Square 11.
The Moon. There are roughly 28 days in the Lunar monthly cycle : 1 x 2 x 3 x 4 x 5 x 6 x7. Or more succinctly 4 x7. There are 13 x 28 cycles in a Solar year. 13 /4 makes a quarter.
Call the quarters a pack of cards. Add up the pack: ( 4x 1-10, + 4 x King13/ Queen12/ Jack11 = 364 + the Joker of 1 = 365 ) The number of days in a Solar year.
Square 12.
The Marriage of the Sun and Moon. The 2nd Pythagoras triangle gives side lengths of 12,13,5 units. The division of 2/3 ( a musical 5th) gives the exact number of Full Moons in a Solar year: -lying between 12 and 13 = 12 months + 10.875 days.
The Golden Proportion
p.46
On this page we have the counterpart of the drawing on p. 43, but here we have the Golden Proportion where the Whole is equal to One, dividing the line into two self-similar versions of Phi.
P.47
Continuing the extract from Robert Lawlor’s book Sacred Geometry.
‘In the second figure (seen on p. 46) we shift the value of Unity from the part to the whole (line), so its division must be less than One. In doing so, we will find the second and completely unique characteristic of Phi. (⏀). First term: a =1-b, second term: b = 1-a, third term: a+b=1.’
The following exposition is more clearly described in the 1st column of p. 47.
Lawlor continues with the following text.
‘If we once again use proportion as a model for the perceptual activity based on the recognition of differences, we have in this unique Golden Proportion ‘within’ Unity a case where the perceived difference (m that which we experience as an object) plus the perceiver of that object are symbolised as contained within a sustained awareness of an all-encompassing Unity.: a:b::b:1. This perceptual state corresponds to the goal of dynamic mediation.
The Golden Proportion is a constant ratio derived from a geometric relationship, which like Pi, and other constants of this type, is ‘irrational’ in numerical terms….and is first and foremost a proportion not a number - a proportion upon which the experience of knowledge (logos) is founded.
In a sense the Golden Proportion can be considered as supranational or transcendent. It is actually the first issue of Oneness, the only possible creative duality within Unity. For this reason, the ancients called it the Golden division, and the Christians have related this proportional symbol to the Son of God.
Why, it may be asked, cannot Unity simply divide into two equal parts? Why not have a proportion of one term a:a ? The answer is simply that with equality there is no difference, and without difference there is no perceptual universe, for as the Upanishad says ‘Whether we know it or not, all things take on their existence from that which perceives them.’
In a static equational statement, one part nullifies the other. An asymmetric division is needed in order to create the dynamics necessary for progression and extension from the Unity. Therefore the ⏀ proportion is the perfect division of unity: it is creative, yet the entire proportional universe that results from it relates back to it and is literally contained within it, since no term of the original steps, as it were, outside of a direct rapport with the original division of Unity.
This is the essential difference between the division of Unity by √2, and its division by ⏀. Through the creation of √2 we are immediately propelled outside the original square - leading further and further away from the original unity. There is no possible way for √2 to have an internal division of Unity. The ⏀ division retains in evolution, the image of the original perfection.’
After further discussion, Lawlor concludes:
‘The Golden Progression shows the possibility, not of a quantitive, statistical evolution, but instead of an evolution guided from within, an exaltation of the initial qualities of the Divine idea, passing directly from the abstract to the concrete or visible; where the manifest world is an image of the Divine.’