In the world of natural phenomena, it is the underlying patterns of geometric form, proportion and associated wave frequencies that give rise to all perceptions and identifications. Therein lies our fundamental capacity to relate, to interpret and to know. In the biological realms, replication, continuity and diversity are established through the vehicle of genetic coding within the substance of the DNA. It is not just the molecules of carbon, oxygen, hydrogen and nitrogen that determine the replicating power of DNA, but its helix form and the geometric proportions and relationships within it.
Our ability to create music and interpret sounds depends on our body's capacity to mechanically interpret wave patterns, not through quantitative differences but proportional differences. All our bodily senses work because our nerve receptors are able to attune to variations in proportion of input frequencies. Our visual senses depend on the ability of the nerves of eye’s retina to tune in to the frequencies of light. Our sense of touch works in a similar way except the nerves embedded in our skin respond to a different range of frequencies, the vibrations associated with the shapes of the molecules of matter.
Even our sense of smell is reliant on our ability to interpret the geometric construction rather than the molecular substance of an odour. Our visual sense differs from our sense of touch only because the nerves of the retina are not tuned to the same range of frequencies as the nerves embedded in our skin. Hidden within the architecture of material existence are the geometric forms, the building blocks which make life possible as we know it.
When we immerse ourselves in a natural harmonious environment such as a beautiful forest glade, we tend to feel a sense of peace and well being. Of course there may be personal psychological influences in individuals that override that pleasant response, but most would agree that nature has a calming effect. It is not just the input to our visual and other senses, but something more like a feeling of connectedness. Our usual environment where we live and work, especially in a large metropolis, can be comparatively harsh to our senses. We occasionally need to restore self-balance by spending time in parks or gardens or the bush and experience that connectedness.
The architecture of the buildings we inhabit as our dwellings, our work places, our healing and education centres, and our churches and temples, also impact on our senses. Apart from any psychological influences, it is mainly the geometric spaces and the materials of the structures that influence our responses. Our sense of proportion tells us when a space or object is harmonious, and our mood responds accordingly. In much the same way we identify sounds as being pleasing or otherwise according to their wave frequencies and the proportional relationships between combinations of different notes. It is our spatial awareness that connects us or repels us with items that we normally perceive to be remote and external to us. It is a biological relationship that occurs at the cellular level between the myriad components of our physicality that triggers our sensual and emotional responses.
During many years of designing and building houses, I observed the responses of people to different geometric forms that I had incorporated into my structures. I observed that certain forms, as a consequence of their proportions or their geometry or both, seemed to evoke favourable responses. I also observed there was a 3-dimensional or non-square element to those spaces. On the other hand I observed little or no response to the conventional square spaces most commonly found in modern houses.
The ultimate non-rectangular environment is the environment created by nature, and to my mind that is the most peaceful and nurturing environment to be in. I wondered if there was some connection with the forms I was using and the hidden forms within nature. I have previously mentioned some of the geometric patterns that appear in plants, shells and minerals. Perhaps the geometric principles on which my house designs were based are similar to the principles that nature uses, possibly even the building blocks out of which the universe is constructed. Research into sacred geometry seemed to confirm this as a distinct possibility.
In trying to understand how the connection between nature and man made objects works, I considered the symbolic meaning of numbers and their associated geometric forms. For responses and experiences within the physical world to be truly meaningful and fulfilling, it is important to have understanding, in this case of the symbolism and power contained within numbers. It is in the understanding, whether brought about by intuition or logic (preferably both) that opportunities become available for transcendence. Without awareness, opportunities may be lost, for as has been said – "God gives us the nuts but He does not crack them for us".
I was particularly interested in the numbers 1 through to 6 because they were the numbers that were manifesting in the geometry of my designs. The power in these numbers appeared to me to be related to the six cycles of creation, each number representing a cycle and its purposeful influences on the evolution of consciousness as we know it. This is a notion that is presented in astrology and in the beliefs and dogmas of many spiritual practices. In Genesis, the first chapter of the Bible, it states that God created the earth in six days, and on the seventh day He rested. The word ‘day’ represents a cycle, thought by some to be around 24,000 years, the time it takes our galaxy to rotate through 360 degrees. In astrological terms, humanity is currently leaving the Piscean age and entering the Aquarian age, the sixth cycle, the cycle in this case being 2,000 years. There are cycles within cycles within cycles. Even in our measurement of time, six is the predominant number.
Going back to the time of Pythagoras, the number 6 was seen to be the first perfect number and the number of creation. The adjective "perfect" was used to describe numbers that are equal to the sum of all the smaller numbers that divide into them, in this case 6 = 1 + 2 + 3.
The universe is constantly evolving. From the human perspective we see change occurring in terms of time and space, but in other levels of reality time and space do not exist as we know them. They are obviously an essential feature of physical reality (as are polarities) in order that we can know that reality and function within it. We are also capable of entering other out-of-body realities in our psychic state. Creation takes place through multiple levels of reality, each level giving rise to the next at a lesser vibrational frequency, the physical universe occurring at the furthermost and densest level.
Because we cannot perceive other dimensional realities with our physical senses, but only through our psychic sensitivities, many people do not accept that they exist. Science is catching up, however, and scientists are recognising the probable existence of other dimensions and hidden universes. When we explore these dimensions we might find that life (consciousness) exists elsewhere but not necessarily in the material form that we are familiar with. Such beings, if they exist, are likely to create from thought rather than manipulating material objects as we do. One could speculate that humans also have this potential, and at some time in the future we will learn to create psychically as well as physically.
When we become more fully evolved at the completion of our sixth cycle, we will come to realise that we actually create our reality through our beliefs. We will eventually come to understand that we are the creators of the world as we see it, we create good and evil simply because we believe in them. It is our total absorption with the physical world and our rejection of the spiritual world that has brought this about. The creation process simply contains the potential for polarities, and we do the rest. I will not go into this in detail, but in the ensuing analysis of the sacred symbolism of certain numbers it is helpful to have a background understanding as to how creation works.
Be mindful, therefore, that it is not only the application of numbers and patterns that can stimulate the senses, but the intent that this be so. When we harness the powers of thought with these things in mind, we change our reality and the way we experience within it. The majority of us subscribe to specific belief systems based on such things as nationality, culture and religion. Thankfully, with some commitment and effort, we can override those belief systems and develop our own individual beliefs. So we need not be completely overwhelmed by the influence of the mass consciousness.
Our true existence is a blend of matter and spirit. When spirit is suppressed, it appears to us that it does not exist, but that is not so. However, by excluding spirit from our reality, our creative senses are limited, and the potential for resonance at refined frequencies of existence becomes impossibly elusive.
THE SYMBOLISM OF PI AND PHI
(Pi) and (Phi) are numbers that have been a source of fascination to mathematicians for thousands of years. Both are indeterminate or irrational numbers, ie they can never be expressed in a finite number of digits. They are numbers that go on forever without displaying any pattern or repetition. Any yet they both express specific geometric relationships – the relationship between the diameter of a circle and its circumference and its area, and the relationship between the side and the diagonal of a pentagon.
Phi is particularly fascinating, partly because it appears in mathematical expressions in many quite extraordinary ways. It also appears in nature as the mathematical basis of the spiral pattern seen in the leaf and petal arrangements of many plants, in mollusc shells such as the chambered nautilus, and in the patterns of billions of stars contained within galaxies. Its extraordinary qualities have been a constant source of inspiration to scientists, philosophers, artists, musicians and architects for many centuries. Because of its almost mystical properties it has come to be known as the Golden Ratio or the Golden Mean. It led the British scientist, Sir James Jeans (1847-1946), to famously pose the question – "Is God a mathematician?"
Euclid (300 B.C.) is considered to be the founder of geometry as a formal system for mathematical deduction. He defined a proportion derived from a simple division of a line in such a way that the ratio of the larger segment to the smaller segment is the same as the ratio of the whole line to the larger segment. Consider a line divided into two segments, the length of the larger segment being a, and the length of the smaller segment being b. Then the proportionality of the golden ratio is described in the equation phi = a / b = (a + b) / a. To quantify this equation numerically we take the length of the shorter segment (b) as 1 unit. Then a / 1 = (a + 1) / a. This produces the quadratic equation a squared – a – 1 = 0. The positive solution to this equation is a = (1 + square root 5) / 2 = 1.6180339887… phi = a / 1 = 1.6180339887… the Golden Ratio.
One of the intriguing properties of this number is that its square is produced by adding the number 1, and its reciprocal by subtracting the number 1. Using a calculator you can show that the square of 1.6180339887… is 2.6180339887… and the reciprocal is 0.6180339887…
Another way of defining phi is by the ratio of the side of a pentagon and its diagonal. The triangles contained within the pentagon have some interesting properties; the ancient Greeks were particularly interested in the association of phi with the pentagon and the pentagram. The Golden Ratio was considered to represent perfect harmony, and was used by them in many of their temples and works of art. It appeared similarly in ancient Egypt, including the Great Pyramid, although there is no evidence that the Egyptians were aware of its mathematical implications. It appears frequently in nature, as does the pentagon and the number 5, for example in the arrangement of leaves and petals in many flowers. Buildings which incorporate the Golden Ratio certainly tend to be pleasing to the eye compared with buildings constructed with less harmonious proportions. However, whether they impart a specific vibration that benefits the occupants I would not be prepared to offer an opinion. I have more definite views on the use of the pentagon, and of course, contained within the pentagon is.
The Fibonacci sequence is an arithmetic series beginning with any two numbers. Each succeeding number is taken as the sum of the two preceding numbers. As the sequence progresses, it is seen that the ratio of any two successive numbers approximates to , and the further down the sequence the closer the ratio becomes to 1.6180339887… As the series is carried to infinity, the closer one comes to the absolute value of phi , but phi can never be reached in absolute terms. In this respect, both numbers phi and pi represent the search for absolute truth, or love, or God. We can strive to get closer and closer to these abstracts, but we can never finally reach them or define them absolutely. In absolute terms they are unknowable.
The following assessment of the symbolism behind certain numbers is the basis on which I believe vibrational influences are contained within building structures. It is not an exact science of course, as there are so many variables, but the principles themselves hold true, only more as potential than fact. As in all things it is the individual’s perception that determines the experience. The influences that certain geometric symbols impart are not measurable other than by use of the human senses, both physical and psychic. Unless the senses are alert to experiencing at emotional and psychic levels, there is unlikely to be a conscious experience. Also, because different individuals are at different stages of spiritual development, certain symbols will resonate more comfortably with others. As six is the number of the new cycle we are entering, it is the most appropriate for the coming age, but it will not be everybody’s choice.
ONE BEGETS TWO - THE ESSENCE OF DUALITY
The circle is the geometry that defines pi , and the circle is traditionally taken as representing God. Before the universe was created there was nothing else but God. Beyond the boundary of the circle nothing existed, not even time and space. The circumference of the circle is that which encompasses all that is, and the area is all that is. All geometric forms that evolve out of prime numbers are contained within the circle, for it is the circle that defines their symmetry. Implicit within a circle, and its most fundamental product or creation is its diameter. There are an infinite number of possible diameters, but they are all identical to each other. So really there is only one diameter. The diameter is a straight line – it defines two points, its end points, both of which lie on the circumference located at extreme opposite positions. The line represents a single object, unity, and its end points symbolise duality or polar opposites. However, one in the sense of absolute exclusivity as distinct from one as a unit amongst many, is incomprehensible. Only God is absolute unity. Try to imagine that you alone existed within an infinite void – you would be incapable of identifying even yourself as there would be no datum point for recognition to occur. One therefore is meaningless, as only God is absolute unity, and one represents creation unmanifested.
The diameter of a circle could be perceived as one, but in reality is two, which demonstrates the hidden nature of duality. Two enables the first step in the creation of consciousness to occur; it allows recognition to occur. A straight line, in this case the diameter, has two end points, identical but opposite. Two gives rise to creation because on can recognise the other. Because a straight line has two end points, flow or recognition of something else can occur. It can be seen now that duality, represented by the number 2, is the most fundamental principle of creation. Nothing can exist without its opposite. Thus the diameter of a circle symbolises the first and most fundamental manifestation within creation. Pi represents, therefore, the relationship that exists between God and His creations. All geometric forms arise out of this basic principle, the straight line containing two end points.
Creation is the process of manifesting something out of nothing. To understand this in another way, let us consider the process in reverse. If we take two equal and opposites, +1 and –1, and put them together we have nothing. +1 –1 = 0. Creation is expressed as 0 = +1 –1, which demonstrates how something can be made out of nothing. Imagine an infinite flat, featureless plain, stretching forever, with nothing to distinguish your position, as every point is the same as every other point. Time and space would be meaningless. Now let us take a scoop of the material of the plain from its surface, and place that material beside the depression left by the removal of the material. We now have two equal and opposite features, a hole and a mound. We have created something out of nothing.
This shows that duality is an essential feature of creation, and in our dense reality that is no exception. Polar opposites are an essential part of our reality in order that we can identify who we are by enabling us to discern and make judgements. We can only know ourselves by knowing others. We can only know good by perceiving evil. We can understand the infinite by perceiving the finite. Duality enables us to learn and grow by perceiving polarities and acting as we think is right for us. But in truth, duality is not absolute but only a state of potential, for we create what is right or wrong through what we believe. Justice becomes a judgement based on what we perceive to be injustice. One cannot exist without the other, for simply by believing in justice we infer the existence of injustice. We all exist at the mental level within a specific belief system that is based on what we feel, what we see, and what we are told. Personal judgements come into play, but social and religious judgements arising out of the group consciousness within which we identify tend to wield the strongest influence. Belief systems vary from one person to another, from one culture to another, from one religion to another, and from one nation to another. However, ultimately there is only What Is.
THREE IS THE POWER BEHIND CREATION
The next step in creation, moving on from 2 points on the circumference of a circle, is 3 points. When these are in symmetry with each other, they define the equilateral triangle. Three linear structural elements connected together at their end points form a rigid frame – a fundamental engineering principle. The triad or trinity in esoteric terms expresses a rigid structure, and demonstrates the stability within creation by virtue of three fundamental principles. Christianity recognises the principles of the Father, the Son and the Holy Spirit as the essential components of spiritual faith. Within Hinduism, the Upanishads describe the three faces of God as Brahma, Shiva and Vishnu, representing active intelligence, will and love. Esoteric occultism refers to the three aspects of the Solar Logos as will, love and form, or monad, soul and personality.
All subsequent steps in creation contain triangles. The 3-dimensional version of the triangle is the tetrahedron, a solid object contained within a sphere, but one which is pure of form because it contains four faces all of which express the trinity. It has no diagonals, no shortcuts between its defining points. Although it appears to be a 3-dimensional object, it is really multi-dimensional, and represents consciousness on higher planes of reality wherein all manifestations are aware of their existence within the trinity. It also demonstrates how every single addition (point) in creation gives rise to another triangle as part of a larger rigid framework, the basis of construction of a stable universe. The ancient Egyptians recognised the power of triangles as symbols of spirit by making them the faces of the pyramids, but they gave them an earthly base, the square. The triangles pointed the way from earth to spirit in heaven, which is why they were so popular as tombs.
I have heard of people who tried to live in a pyramid shaped house, but found it impossibly uncomfortable – not surprising!
FOUR IS MANIFESTATION INTO MATTER
Next we have four, the square, symbolising creation in matter. The square defines the four directions of north, south, east and west, and in its 3-dimensional form, the cube (contained within a sphere) defines the 3 directions of 3-dimensional reality. The square and the cube represent physical manifestation unevolved. The potential that lies within them is implicit. For example, the square's diagonal implies another square, which in turn contains a diagonal with an even larger square. Continued on, this creates a spiral which demonstrates the dynamic energy within all creation. It symbolises involution and evolution, progress towards the infinite, both inwards and outwards. Four also represents the first product of the procreative process, that is of multiplication 2 x 2 = 4.
The almost exclusive use of the square and rectangle in western architecture reveals an obsessive focus on the material world.
INVOLUTION - FIVE IS THE POTENTIAL CONTAINED WITHIN MATTER
The next step is five, bringing expanded consciousness into matter by virtue of phi, the golden proportion. Five points equally spaced on the circumference of a circle give rise to the pentagon, within which lies the pentagram or 5-pointed star. This is generally accepted as representing man – the head, two legs astride, and two arms outstretched. The diagonal of a pentagon, in proportional relationship with its side, defines . This is the purest form of proportion, and thus represents human consciousness in its greatest and most harmonious state. All creative expressions that emanate from man can be evaluated to the degree in which they incorporate the golden proportion.
Consciousness, like phi, cannot be defined in absolute terms. Perfect consciousness in matter is something to be sought. An evolved consciousness recognises perfect harmony, but in the imperfect state there is very little status attributed to harmony. Make what you will of the state the world is in at the beginning of the twenty-first century. It comes back to perception, and perception is linked to consciousness.
Five and are dominant in the structure of many living forms. The exotic flowers of love, the rose, the orchid, the azalea and the passion flower, are all governed by pentagonal symmetry, as are the flowers of all edible fruit-bearing plants. They demonstrate the gifts of the Creator within the realm of the purely physical. The United States central administration is housed in a building known as the Pentagon because of its shape. It signals American aspiration to bring harmony to the world.
EVOLUTION - SIX HARNESSES THE POWER OF SPIRIT AND BRINGS IT INTO MATTER
Whereas 5 expresses the potential for harmony hidden within the world of matter, 6 is explicit in its representation of symmetry and balance. The hexagon has all its diagonals passing through the centre of the circle and equal to the diameter. It has six axes of symmetry. Within it lies the hexagram (6-pointed star) and its 3-dimensional form the star tetrahedron. The hexagram is made up of two superimposed equilateral triangles, one inverted. The trinity has replicated itself in the world of matter as represented by the top inverted triangle. The lower triangle is a reflection of the upper one. The hexagram thus presents the challenge for mankind to recognise its own divinity.
A similar relationship between God and His creations in matter can be seen in the Vesica Piscis, where two circles overlap. The area of overlap defines the perfect blend of matter and spirit. A connection exists in these two symbols because the vesica is the geometric basis for establishing the hexagram - the intersection points of the circles define the star points.. The Vesica can also be seen as the two aspects of God within the material level of creation - masculine and feminine, the polar opposites in humanity arising out of duality. Two circles apart define separation between male and female, but in the overlapping they are integrated. The Vesica thus expresses the powerful potential in mankind to achieve harmony through the blending of the masculine and feminine qualities. The overlap area has traditionally been seen as a fish or the Piscean Christ.
Copyright John Koch. All rights reserved.