Sacred Geometry - Part 2 In nature, we find patterns, designs and structures from the most minuscule particles, to expressions of life discernible by human eyes, to the greater cosmos. These inevitably follow geometrical archetypes, which reveal to us the nature of each form and its vibrational resonances. They are also symbolic of the underlying metaphysical principle of the inseparable relationship of the part to the whole. It is this principle of oneness underlying all geometry that permeates the architecture of all form in its myriad diversity. This principle of interconnectedness, inseparability and union provides us with a continuous reminder of our relationship to the whole, a blueprint for the mind to the sacred foundation of all things created. The Sphere Starting with what may be the simplest and most perfect of forms, the sphere is an ultimate expression of unity, completeness, and integrity. There is no point of view given greater or lesser importance, and all points on the surface are equally accessible and regarded by the center from which all originate. Atoms, cells, seeds, planets, and globular star systems all echo the spherical paradigm of total inclusion, acceptance, simultaneous potential and fruition, the macrocosm and microcosm. The Circle The circle is a two-dimensional shadow of the sphere which is regarded throughout cultural history as an icon of the ineffable oneness; the indivisible fulfillment of the Universe. All other symbols and geometries reflect various aspects of the profound and consummate perfection of the circle, sphere and other higher dimensional forms of these we might imagine. The ratio of the circumference of a circle to its diameter, Pi, is the original transcendental and irrational number. (Pi equals about 3.14159265358979323846264338327950288419716939937511...) It cannot be expressed in terms of the ratio of two whole numbers, or in the language of sacred symbolism, the essence of the circle exists in a dimension that transcends the linear rationality that it contains. Our holistic perspectives, feelings and intuitions encompass the finite elements of the ideas that are within them, yet have a greater wisdom than can be expressed by those ideas alone. The Point At the center of a circle or a sphere is always an infinitesimal point. The point needs no dimension, yet embraces all dimension. Transcendence of the illusions of time & space result in the point of here and now, our most primal light of consciousness. The proverbial "light at the end of the tunnel" is being validated by the ever-increasing literature on so-called "near-death experiences". If our essence is truly spiritual omnipresence, then perhaps the "point" of our being "here" is to recognize the oneness we share, validating all "individuals" as equally precious and sacred aspects of that one. Life itself as we know it is inextricably interwoven with geometric forms, from the angles of atomic bonds in the molecules of the amino acids, to the helical spirals of DNA, to the spherical prototype of the cell, to the first few cells of an organism which assume vesical, tetrahedral, and star (double) tetrahedral forms prior to the diversification of tissues for different physiological functions. Our human bodies on this planet all developed with a common geometric progression from one to two to four to eight primal cells and beyond. Almost everywhere we look, the mineral intelligence embodied within crystalline structures follows a geometry unfaltering in its exactitude. The lattice patterns of crystals all express the principles of mathematical perfection and repetition of a fundamental essence, each with a characteristic spectrum of resonances defined by the angles, lengths and relational orientations of its atomic components. The Square Root of Two The square root of 2 embodies a profound principle of the whole being more than the sum of its parts. (The square root of two equals about 1.414213562...) The orthogonal dimensions (axes at right angles) form the conjugal union of the horizontal and vertical which give birth to the greater offspring of the hypotenuse. The new generation possesses the capacity for synthesis, growth, integration and reconciliation of polarities by spanning both perspectives equally. The root of two originating from the square leads to a greater unity, a higher expression of its essential truth, faithful to its lineage. The fact that the root is irrational expresses the concept that our higher dimensional faculties can't always necessarily be expressed in lower order dimensional terms - e.g. "And the light shineth in darkness; and the darkness comprehended it not." (from the Gospel of St. John, Chapter 1, verse 5). By the same token, we have the capacity to surpass the genetically programmed limitations of our ancestors, if we can shift into a new frame of reference (i.e. neutral with respect to prior axes, yet formed from that matrix-seed conjugation. Our dictionary refers to the word matrix both as a womb and an array (or grid lattice). Our language has some wonderful built-in metaphors if we look for them! The Golden Ratio The golden ratio (a.k.a. phi ratio a.k.a. sacred cut a.k.a. golden mean a.k.a. divine proportion) is another fundamental measure that seems to crop up almost everywhere, including crops. (The golden ratio is about 1.618033988749894848204586834365638117720309180...) The golden ratio is the unique ratio such that the ratio of the whole to the larger portion is the same as the ratio of the larger portion to the smaller portion. As such, it symbolically links each new generation to its ancestors, preserving the continuity of relationship as the means for retracing its lineage. The golden ratio has some unique properties and makes some interesting appearances: * phi = phi^2 - 1; therefore 1 + phi = phi^2; phi + phi^2 = phi^3; phi^2 + phi^3= phi^4; ad infinitum. * phi = (1 + square root(5)) / 2 from quadratic formula, 1 + phi = phi^2 * phi = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/...)))))... * phi = (sec 72)/2 =(csc 18 )/2 = 1/(2 cos 72) = 1/(2 sin 18 ) = 2 sin 54 = 2 cos 36 = 2/(csc 54) = 2/ (sec 36) for all you trigonometry enthusiasts ratio of segments in 5-pointed star (pentagram) considered sacred to Plato & Pythagoras in their mystery schools. Note that each larger (or smaller) section is related by the phi ratio, so that a power series of the golden ratio raised to successively higher (or lower) powers is automatically generated: phi, phi^2, phi^3, phi^4, phi^5, etc. * phi = apothem to bisected base ratio in the Great Pyramid of Giza * phi = ratio of adjacent terms of the famous Fibonacci Series evaluated at infinity; the Fibonacci Series is a rather ubiquitous set of numbers that begins with one and one and each term thereafter is the sum of the prior two terms, thus: 1,1,2,3,5,8,13,21,34,55,89,144... (interesting that the 12th term is 12 "raised to a higher power", which appears prominently in a vast collection of metaphysical literature) The mathematician credited with the discovery of this series is Leonardo Pisano Fibonacci and there is a publication devoted to disseminating information about its unique mathematical properties, The Fibonacci Quarterly Fibonacci ratios appear in the ratio of the number of spiral arms in daisies, in the chronology of rabbit populations, in the sequence of leaf patterns as they twist around a branch, and a myriad of places in nature where self-generating patterns are in effect. The sequence is the rational progression towards the irrational number embodied in the quintessential golden ratio. This most aesthetically pleasing proportion, phi, has been utilized by numerous artists since (and probably before!) the construction of the Great Pyramid. As scholars and artists of eras gone by discovered (such as Leonardo da Vinci, Plato , & Pythagoras), the intentional use of these natural proportions in art of various forms expands our sense of beauty, balance & harmony to optimal effect. Leonardo da Vinci used the Golden Ratio in his painting of The Last Supper in both the overall composition (three vertical Golden Rectangles, and a decagon (which contains the golden ratio) for alignment of the central figure of Jesus. The outline of the Parthenon at the Acropolis near Athens, Greece is enclosed by a Golden Rectangle by design.

The Square Root of 3 and the Vesica Piscis The Vesica Piscis is formed by the intersection of two circles or spheres whose centers exactly touch. This symbolic intersection represents the "common ground", "shared vision" or "mutual understanding" between equal individuals. The shape of the human eye itself is a Vesica Piscis. The spiritual significance of "seeing eye to eye" to the "mirror of the soul" was highly regarded by numerous Renaissance artists who used this form extensively in art and architecture. The ratio of the axes of the form is the square root of 3, which alludes to the deepest nature of the triune which cannot be adequately expressed by rational language alone. Spirals This spiral generated by a recursive nest of Golden Triangles (triangles with relative side lengths of 1, phi and phi) is the classic shape of the Chambered Nautilus shell. The creature building this shell uses the same proportions for each expanded chamber that is added; growth follows a law which is everywhere the same. The outer triangle is the same as one of the five "arms" of the pentagonal graphic above. Toroids Rotating a circle about a line tangent to it creates a torus, which is similar to a donut shape where the center exactly touches all the "rotated circles." The surface of the torus can be covered with 7 distinct areas, all of which touch each other; an example of the classic "map problem" where one tries to find a map where the least number of unique colors are needed. In this 3-dimensional case, 7 colors are needed, meaning that the torus has a high degree of "communication" across its surface. The image shown is a "birds-eye" view.

Dimensionality The progression from point (0-dimensional) to line (1-dimensional) to plane (2-dimensional) to space (3-dimensional) and beyond leads us to the question - if mapping from higher order dimensions to lower ones loses vital information (as we can readily observe with optical illusions resulting from third to second dimensional mapping), does our "fixation" with a 3-dimensional space introduce crucial distortions in our view of reality that a higher-dimensional perspective would not lead us to? Fractals and Recursive geometries There is a wealth of good literature on this subject; it's always fascinating how nature propagates the same essence regardless of the magnitude of its expression...our spirit is spaceless yet can manifest aspects of its individuality at any scale. Perfect Right Triangles The 3/4/5, 5/12/13 and 7/24/25 triangles are examples of right triangles whose sides are whole numbers. The graphic above contains several of each of these triangles. The 3/4/5 triangle is contained within the so-called "King's Chamber" of the Great Pyramid, along with the 2/3/root5 and 5/root5/2root5 triangles, utilizing the various diagonals and sides. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The 5 Platonic solids (Tetrahedron, Cube or (Hexahedron), Octahedron, Dodecahedron & Icosahedron) are ideal, primal models of crystal patterns that occur throughout the world of minerals in countless variations. These are the only five regular polyhedra, that is, the only five solids made from the same equilateral, equiangular polygons. To the Greeks, these solids symbolized fire, earth, air, spirit (or ether) and water respectively. The cube and octahedron are duals, meaning that one can be created by connecting the midpoints of the faces of the other. The icosahedron and dodecahedron are also duals of each other, and three mutually perpendicular, mutually bisecting golden rectangles can be drawn connecting their vertices and midpoints, respectively. The tetrahedron is a dual to itself. The Archimedean Solids There are 13 Archimedean solids, each of which are composed of two or more different regular polygons. Interestingly, 5 (Platonic) and 13 (Archimedean) are both Fibonacci numbers, and 5, 12 and 13 form a perfect right triangle. Stellations of The Platonic & Archimedean Solids This is a stellation of a dodecahedron where each pentagonal face is capped with a pentagonal pyramid composed of 5 golden triangles, a sort of 3-dimensional 5-pointed star. Metatron's Cube Metatron's Cube contains 2-dimensional images of the Platonic Solids (as shown above) and many other primal forms. The Flower of Life Indelibly etched on the walls of temple of the Osirion at Abydos, Egypt, the Flower of Life contains a vast Akashic system of information, including templates for the five Platonic Solids. The background graphic for this page is a repetitive hexagonal grid based on the Flower of Life.

SACRED GEOMETRY IN BUILDING ---Buildings can have a profound influence on our health and our psychic and spiritual state of be-ing. Harmony and balance, light and colour, relationship to landscape, ecological sympathy, energy efficiency and geometric form are contributing elements of shelter which aspires to be nurturing rather than draining. We resonate at both cellular and consciousness levels with our environment. By creating an environment around us that is supportive to both our inner and our outer senses, we can enhance rather than alienate our human links with nature. Architecture, when employed as a means of embodying principles of universal harmony can sustain us rather than drain us, so that our homes become our havens, and our work places support our creativity.--- 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 mechanical interpretate 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. Even our sense of smell is reliant on our ability to interpret the geometric construction rather than the molecular substance of a perfume. 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. The architecture of the buildings we inhabit as our dwellings, our work places, our healing and education centres, and our churches and temples, has similar impact on our senses as the architecture of nature. 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. During many years of designing and building, 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 architecture. 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 Pythagorus, 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. When we are 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. It is not, however, essential knowledge for all potential pole frame house builders. 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. Thankfully, with some commitment and effort, we can do that on an individual basis, and 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. By excluding spirit from our reality, our senses are dulled, and the potential for resonance at refined frequencies of existence becomes impossibly elusive. The intention behind my architecture is not only to try and stimulate the senses, but to try to awaken the spirit within. I believe it is the simplicity behind the forms and the concepts that makes this a possibility.

THE SYMBOLISM OF PI AND PHI p (Pi) and f (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 – p the relationship between the diameter of a circle and its circumference and its area, and f the relationship between the side and the diagonal of a pentagon. f 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 mollusk 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 f = 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 a2 – a – 1 = 0. The positive solution to this equation is a = (1 + Ö 5) / 2 = 1.6180339887… f = 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 f 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 f 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 f , 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 f , but f can never be reached in absolute terms. In this respect, both numbers f and p 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 p , 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. p 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 f , 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 f . This is the purest form of proportion, and thus represents human consciousness in its grea-*test*-('") 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 f , 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 f 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 building incorporating a thematic hexagon brings with it the vibration of spiritual healing. 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. ---When architecture incorporates these symbols, they need only be implicit as a theme out of which the design evolves. The power within the symbols is vibrational, and the way they are used should optimally be compatible with the inhabitants. This is the role of the designer, to identify that compatibility. Numerology can be a useful tool here. When vibrational symbols are used as a rigid format springing from the mind, their power is limited. The mind is a tool. Only when thought is coupled with intent arising from the heart, does manifestation occur.--- This is the basic theory behind the influence that numbers have on our state of be-ing. These principles are able to be incorporated into the architecture of buildings with profound results, but as I have suggested, the effects they have are not the same for all people. In my essay, "The Nurturing House" I expand on this in more detail. I also deal with environmental factors that influence us vibrationally through the materials that are used in the structure of a building.

Close-packed reflective spheres clustered in this concentric shell structure produce an optical distribution network that links the Golden Mean and Pi. The Golden Mean is expressed in the 52 degree angle pyramid structure and Pi is expressed in the ratio of the diameter to the circumference of each sphere of course; but it is also expressed in the ratio of the sizes of spheres in the 10 spheres within 10 spheres concentric shell structure that I discovered. This concentric shell structure could continue to expand with many shells and still retain the same ratio between shells. It was not until 1991 that I was able to build and photograph the images inside a ten-within-ten (2 shell) structure. I used 10 - 6" diameter spheres and 10 - 19" diameter hemispheres. The structure is shown here. These are photographs of the interior of the 10 spheres within 10 spheres cluster shown above. The photographs that I made from this 10-within-10 sphere structure revealed the inherent limitations of photography for this work. The final images were not what I was seeing with my eyes while directly viewing the interior of the structure. But this approach did reveal a more complete face form and I also realized that I would need at least 10 more spheres (about 60" in diameter at $1500 each) to complete the enclosure and remove the remaining gaps in the images. Also, the lighting system was limited to the exterior and it was very difficult to control the positioning, color and brightness. So, in 1992, I purchased a computer to model these structures with ray-tracing software which enabled me to investigate more thoroughly the relationship between this cluster geometry and the archetypal images generated therein. During this tour into cyberspace, I could take a camera into the sacred chamber central to the concentric shells of reflective spheres which are simulated by a computer program (Real 3D by Realsoft Oy, Finland) that most accurately renders the effects of real-world light sources and records the patterns generated by multiple reflections on metal surfaces. With this method of investigation, I am able to more conveniently control the many variables which led to these discoveries and conclusions: 1.) At least 40 spheres (4-10 ball shells) are necessary to enclose the central area and fill in the gaps in the images. 2.) Most of the lights should be point sources placed at the points of contact between the innermost spheres of the structure; although additional symmetrically-paired, point-lights are necessary in the central area. 3.) Some of the innermost spheres are reduced in size and they can float within certain areas in the central space. Figure 12 shows typical positions and sizes. 4.) The camera position and field of view as shown in Figure 12 produces the most convincing image of the human form. Figure 12 5.) The image of this artificial anthropoid that is produced in these structures can be animated when the positions and sizes of certain spheres are modulated as shown in the three animations listed below.

Human Form From Sacred Geometry The computer image shown here was made from the camera position and field of view shown in Figure 12. Figure 14 is from the same camera position (shown in Figure 12) but the lens set at a very wide angle. Figure 14 Figure 15 is from the same camera position (shown in Figure 12) with the camera lens set to zoom in. Figure 15 This is a stereo image and it shows the interior of the cluster in 3D. Stare through the images with your eyes focused at a distant point and the two images will turn into three images and the center image will appear in 3D. In 1996, I produced an animation Sacred Spaces (6 minutes) which has been screened in many national and international film festivals and it has won some awards (see resume). I have also produced Flesh Tones (5.5 Minutes) another animation completed in 1997 and I have produced many high resolution images for prints/slides which I have presented with the video animations at lecture presentations. These images are stills from the animations and they demonstrate the variety of image generating that is inherent in this system. I have concentrated on exhibiting the videotape documentation of my work thus far because it is the most portable presentation format that describes this research most completely. Conclusion This interdisciplinary research has taken me into many related areas of study. Geometry, Optics, Ancient History of Art and Religion, Computer Imaging, Photography, Animation, Graphic Art and of course Sculpture are the major connecting disciplines that have contributed to this work. I feel that I have rediscovered some of what was a highly developed understanding of Mankind's relationship to the Universe and this knowledge was utilized and documented in the geometry of ancient structures. Sacred Geometry, the study of the unity of the cosmos, demonstrates relationships between Number and Space and the Human Form. It was used in the construction of ancient glyphs and monuments thereby preserving the knowledge of these principles of Natural Law for future generations. This construction of reflective spheres may embody the technology that produced the animated images of the deities in the temples of antiquity. The Tree of Life which is a graphic representation of the interaction between cosmic forces is shown in Figure 22. It is found in many ancient texts of the Kabbalah. Figure 22 I realized that The Tree of Life graphic can also represent the 10 sphere cluster made with 2 5-ball pyramids. The construction of this structure of clustered metal reflective spheres (offering bowls) is well within the capability of many ancient cultures and with the addition of a few glass lenses, these images could be projected onto walls or into smoke. Perhaps there is some Truth behind the smoke and mirrors of Ancient Religion ... perhaps it is geometry ... Sacred Geometry. This research which is documented in four computer animations Sacred Spaces, Flesh Tones, Our Mothers and Sacred Spaces 2, in color computer prints, and in color slides has given me new insight into the motives that may have inspired the construction of The Great Pyramid.