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SACRED GEOMETRY PDF

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Sacred Geometry Pdf

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design are bound by the laws of sacred geometry, which is itself the Because sacred geometry reflected the universe, its pure forms and dynamic equilibriums . Science of by Seth T. Miller. Frank. Chester. SAcRED a New. Geometry .. Colorado, studied sacred geometry with Robert Lawlor (author of Sacred Geometry. ISBN Printed and bound in China Contents Introduction 4 The Practice of Geometry 7 Sacred Geometry: Metaphor of Universal Order 16 The.

AD in his Mathematics Useful for Understanding Plato Geometry deals with pure form, and philosophical geometry re-enacts the un- folding of each form out of a preceding one. It is a way by which the essential creative mystery is rendered visible.

T h e passage ffom creation to procreation, from the unmanifest, pure, formal idea to the 'here-below', the world that spins out from that original divine stroke, can be mapped out by geometry, and experi- enced through the practicc of geometry: Inseparable from this process is the concept of Number, and, as w e shall see, for the Pythagorean, Number and Form at the ideal level were one.

But number in this context must be understood in a special way. When Pythagoras said, 'All is arranged according to Number', he was not thinking of numbers in the ordinary, enumera- tive sense. In addition to simple quantity, numbers on the ideal level are possessed of quality, so that 'twoness', 'threeness' o r 'fourness', for example, are not merely composed of 2, 3, o r 4 units, but are wholes or unities in themselves, each having related powers.

Schwaller de Lubicz gives an analogy by which this universal and archetypal sense of Number can be understood. A revolving sphere presents us with the notion of an axis. W e think of this axis as an ideal or imaginary line through the sphere. It The twelfth-century archi- tecture of the Cistercian Order achieves its visual beauty through designs which conform to the pro- portional system of musical harmony.

Many of the abbey churches of this period were acoustic resonators trans- forming a human choir into celestial music. St Bernard of Clairvaux, who inspired this architecture, said of their design, 'There must be no' decoration, only proportion.

Hidden Nature

This icon can also be under- stood as an image of indi- vidual self-creation; for here, as in many medieval images of Christ, Tantric symbolism is evident. Christ holds the compass with his hand across the vital centre called the heart chakra, and from this centre he organizes the turmoil of the vital ener- gies contained in the lower chakras which are indicated on the body by centres a t the navel and genitals.

Geometry is symbolized here in both the individual and universal sense as an instrument through which the higher archetypal realm transmits order and harmony to the vital and energetic worlds.

Number in the enumerative sense corresponds to the measures and movements of the outer surface of the sphere, while the universal aspect of Number is analogous to the immobile, unmanifest, functional principle of its axis. Let us shift our analogy to the two-dimensional plane. If w e take a circle and a square and give the value 1 to the diameter of the circle and also to the side o f the square, then the diagonal of the square will always be and this is an invariable law an 'incommensurable', 'irrational' number.

It is said that such a number can be carried out to an infinite number of decimal places without ever arriving at a resolution. In the case o f the diagonal o f the square, this decimal is 1. With the circle, if w e give the diameter the value I, the circumference will also always be of the incommensurable type, 3.

The principle remains the same in the inversion: It is exactly at this point that quantified mathematics and geometry go their separate ways, because numerically we can never know exactly the diagonal of the square nor the circumference of the circle. Yes, we can round-off after a certain number of decimal places, and treat these cut off numbers like any other number, but we can never reduce them to a quantity. In geometry, however, the diagonal and the circumference, when considered in the context of formal relationship diagonal to side; circumference to diameter , are absolutely knowable, self-evident realities: J2 and 1: Number is considered as a formal relationship, and this type of numerical relationship is called a function.

The square root of 2 is the functional number of a square. Pi is the functional number of a circle. Philosophic geometry -and consequently sacred art and architecture - is very much concerned with these 'irrational' functions, for the simple reason that they demonstrate graphically a level of experience which is universal and invariable.

The irrational functions which we will consider rather as supra-rational are a key opening a door to a higher reality of Number. They demonstrate that Number is above all a relationship; and no matter what quantities are applied to the side and to the diameter the relationship will remain invariable, for in essence this functional aspect of Number is neither large nor small, neither infinite nor finite: Thus within the concept of Number there is a definite, finite, particularizing power and also a universal synthesizing power.

One may be called the exoteric or external aspect of number, the other the esoteric or inner, functional aspect. Let us look at the first four primary numbers in this spirit.

The number O NE can of course define a quantity; as, for example, one apple. But in its other sense, it perfectly represents the principle of absolute unity, and as such has often been used as the symbol to represent God. As a statement of form it can in one sense represent a point - it has been called the 'pointal' number, the bindu or seed in the Hindu mandala - or in another sense it can represent the perfect circle.

T w o is a quantity, but symbolically it represents, as we have already seen, the principle of Duality, the power of multiplicity. At the same time it has its formal sense in the representation of a line, in that two points define a line.

T HREE is a quantity, but as a principle it represents the Trinity, a vital concept which we will meet again later. Its formal sense is that of the triangle, which is formed from three points. With three a qualitative transition is made from the pure, abstract elements of point and line to the tangible, measurable state which is called a surface.

In India the triangle was called the Mother, for it is the membrane or birth channel through which all the transcendent powers of unity and its initial division into polarity must pass in order to enter into the manifest realm of surface. The triangle acts as the mother of form. But three is yet only a principle of creation, forming the passage between the transcendent and the manifest realms, whereas FOUR represents at last the 'first born thing', the world of Nature, because it is the product of the procreative process, that is of multiplication: As a form, four is the square, and represents materialization.

The universality of Number can be seen in another, more physical context. We learn from modern physics that from gravity to electromagnetism, light, heat, and even in what we think of as solid matter itself, the entire perceptible universe is composed of vibrations, perceived by us as wave phenomena.

Waves are pure temporal patterns, that is dynamic configurations composed of amplitude, interval and frequency, and they can be defined and understood by us only through Number. Thus our whole universe is reducible to Number. Every living body physically: This Japanese Zen calli- graphic drawing beautifully shows 'creation' through the simple progression from the Unity of the circle, through the triangle, to the manifest form of the square.

The study of sound, as the ancients intuited, provides a key to the understanding of the universe. We've noted already that the ancients gave considerable attention to the study of musical harmony in relation with the study of mathematics and geometry. The origin of this tradition is generally associated with Pythagoras BC and his school, yet Pythagoras may be considered as a window through which we can glimpse the quality of the intellectual world of an older, eastern and mideastern tradition.

For this line of thinking, the sounding of the octave an octave is for ex- ample two successive 'Do's' on a musical scale was the most significant moment of all contemplation. It represented the beginning and goal of creation. What happens when we sound the perfect octave? There is an immediate, simultaneous coinciding of understanding which has occurred on several levels of being.

Without any inter- vention of thought or concept or image, we immediately recognize the recurrence of the initial tone in the form of the octave.

It is the same note, yet it is different; it is the completion of a cycle, a spiral from seed to new seed. This timeless, instan- taneous recognition more accurate than any visual recognition is universal among humans.

But something else has happened as well. A guitarist sounds a string. He next depresses this string with his finger exactly at its midpoint. He sounds the half-string. The frequency of vibrations produced is double that given by the whole string, and the tone is raised by one octave.

The string length has been divided by two, and the number of vibrations per second has been multiplied by two: Thus in this moment an abstract, mathematical event is precisely linked with a physical, sensory perception; our direct, intuitional response to this phenomenon of sound the octave coincides with its concrete, measured definition.

Hence we experience in this auditory perception a simultaneous interwovenness of interior with exterior, and we can generalize this response to invoke the possi- bility of a merger of intuitional and material realms, the realms of art and science, of time and space.

There may be another such moment in the created world, but the Pythagoreans did not know of it, nor do we. This is the essential spirit of the perception of Harmony, and for the Pythagoreans it was the only true supernatural moment: It was considered to be true Magic, an omnipresent and authentic mystery.

It was by means of geometry that the Pythagoreans poised themselves at this unique transition where heard vibration becomes seen form; and their geometry, as w e shall see, explores the relationships of musical harmony. Although interwoven in function, our two major intellectual senses, sight and hearing, use our intelligence in t w o completely different ways.

For example, with our optic intelligence, in order to form a thought we make an image in our mind. Hearing, o n the other hand, uses the mind in an immediate, unimaged response whose action is expansive, evok- ing a response from the emotive centres. Nowadays this emotive, sound-sensing faculty is usually associated with subjective, emotional, aesthetic o r spiritual experi- ences.

W e tend to forget that it is also involved when the reason perceives invariant relationships. Therefore when we place the auditory capacity at the centre of our sensory experience w e can become aware that it is possible to listen to a colour, or to a movement. This intellectual capacity is quite different from the 'visual', analytical and sequential one we normally employ. It is this capacity, which is associated with the right hemisphere o f the brain, that recognizes patterns in space, o r wholes o f any kind.

It can perceive opposites in simultaneity and grasp functions which to the analytic faculty appear irrational. It is in fact the perfect complement o f the 'left hemisphere', visual, analytic capacity, for it absorbs spatial and simul- taneous orders while the 'left' rational faculty is ,best suited t o grasp temporal, sequential organization.

T h e esoteric, functional aspect of Number, for instance, , would be apprehended through the 'right hemisphere' faculty, while the exoteric, enumerative aspect of Number is apprehended by the 'left'. This innate intellectual quality resembles very closely what the Greeks called Pure Reason, o r what in India was called the 'heart-mind'.

The ancient Egyptians had a beautiful name for it, the Intelligence of the Heart, and to achieve this quality of understanding was life's implicit goal. The practice of Geometry, while also utilizing the analytic faculty, uses and cultivates this audial, intuitive aspect of mind. For example, one experiences the fact of geometric growth through the image of the square with its diagonal which forms the side of a second square.

This is an unreasoned certainty absorbed by the mind from the actual experience of executing the drawing. The logic is contained within the lines o n paper, which cannot be drawn in any other way. As geometers, equipped only with compasses and straight-edge, we enter the two-dimensional world o f the representation of form.

A link is forged between the most concrete form and measure and the most abstract realms of thought. By seeking the invariable relationships by which forms are governed and inter- connected w e bring ourselves into resonance with universal order. B y re-enacting the genesis o f these forms w e seek to know the principles of evolution. And by thus raising our o w n patterns o f thought to these archetypal levels, w e invite the force of these levels to penetrate our mind and thinking.

O u r intuition is enlivened, and perhaps, as Plato says, the soul's eye might be purified and kindled afresh 'for it is by it alone that w e contemplate the truth'. They are dynamic and active even among themselves.

Numbers, in the Pythagorean view, can be androgynous or sexual, procreators or progeny, active or passive, heterogeneous or promiscuous, generous or miserly, undefined or individualized. They have their attractions, repulsions, families, friends; they make marriage contracts. They are in fact the very elements of nature. The tools of u geometry and number represent the means to attain know- ledge of both external and internal space and time.

Lieben John. Sacred Geometry for Artists, Dreamers, and Philosophers

It seems to be the basic assumption of traditional philosophies that human intellectual powers are for the purpose o f accelerating o u r o w n evolution beyond the restraints o f the biological determinism which binds all other living organisms. Methods such as yoga, meditation, concentration, the arts, the crafts, are psycho- physical techniques to further this fundamental goal. T h e practice o f Sacred Geometry is o n e o f these essential techniques of self-development.

Each of the diagrams in the small squares represents a different system or technique of thought for understanding the world and its structures.

The first task of the spiritual aspirant confronting the varied contemplative paths is to harmonize the five universal constituents which compose his body earth, air, fire, water and prana. His clear cogni- tion of the outer and inner worlds is dependent upon the harmonious accord which he establishes between these elemental states in his own body and these same elements in nature. Each geometric cosmogram is meant to assist him in these attempts at liberation through harmonization.

I1 Sacred Geometry: Metaphor of Universal Order Whether the product of an eastern or a western culture, the circular mandala or sacred diagram is a familiar and pervasive image throughout the history of art. India, Tibet, Islam and medieval Europe have all produced them in abundance, and most tribal cultures employ them as well, either in the form ofpaintings or buildings o r dances. Such diagrams are often based on the division of the circle into four quarters, and all the parts and elements involved are interrelated into a unified design.

They are most often in some way cosmological; that is, they represent in symbol what is thought to be the essential structure of the universe: But what is most consistently striking about this form of diagram is that it expresses the notion of cosmos, that is of reality conceived as an organized, unified whole. Ancient geometry rests on no a priori axioms o r assumptions. Unlike Euclidian and the more recent geometries, the starting point o f ancient geometric thought is not a network of intellectual definitions or abstractions, but instead a meditation upon a metaphysical Unity, followed by an attempt to symbolize visually and to contemplate the pure, formal order which springs forth from this incomprehensible Oneness.

It is the approach to the starting point of the geometric activity which radically separates what we may call the sacred from the mundane or secular geometries. Ancient geometry begins with One, while modern mathematics and geometry begin with Zero. One of the most striking uses of the mandala is dome architecture, Islamic and Christian. The square represents the earth held in fourfold embrace by the circular vault of the sky and hence subject to the ever-flowing wheel of time.

When the incessant movement of the universe, depicted by the circle, yields to comprehensible order, one finds the square. The square then presupposes the circle and results from it. The relationship of form and movement, space and time, is evoked by the mandala. Here the mandala of Unity is inscribed on the hand, held in a ritual gesture, of a Japanese Buddhist deity. The mandala is the division of the Unity circle into the com- prehensible forms of square, hexagon, octagon, enneagon, etc.

But for thoughts to become activities and workings they require a will or force of intention, which is symbolized by the hand. The positions of the hand can be systematized to form a medium of communication mudra in which the gesture mirrors the various forces through which the dis- positions of creative mind come into manifest form.

I would like to consider in some detail these two symbolic beginnings, O n e and The primary geometric Zero, because they provide an exceptional example of h o w mathematical concepts forms are considered to be are the prototypes for the dynamics of thought, of structuring and of action. T h e origins of manipulating and construct- this symbol date back to sometime before the eighth century AD, when we have a ing these forms, will learn to record of its first written appearance in a mathematical text from India.

It is in- position itself in the essential teresting to note that during the century just prior to this time a particular line of poses of gesture-language. This school laid exclusive emphasis on the goal of obtaining personal transcendence and escape from karma through renunciation of the natural world, even to the extent o f mortification of the physical body. The goal of this highly ascetic pursuit was the attainment of an utterly impersonal, blank void, a total cessation o f movement within consciousness.

A description of it attributed to Buddha is 'a state o f incognizable, imperishable, selfless absence'. This single aspect o r possibility of meditative experience was held to be the ultimate goal of the created Universe as well as the goal of all individual spiritual development.

In retrospect this is now considered by many to be a dark period in the long, rich spiritual heritage of India, a decline from the previous tradition which upheld a spiritual significarrce in both the manifested and the un- manifested expressions of God, and whose tantric and yogic practices worked towards an intensification of'the relationship and harmonization between matter and spirit.

The result was that it achieved a specific name and symbol in both metaphysics and mathematics. In mathematics it came to be considered just like the other numbers, as a symbol which can be operated upon and calculated with. The name given to this concept in Sanskrit was sunya, meaning 'empty'. Some mathematical historians will argue that the exclusive claim of the Hindu notion of zero is not verifiable, claiming that before India, in Babylonia, Greece and in the Maya civilization a symbol was sometimes used to denote an empty column.

In a number such as , for instance, the empty column is where the zero is. But to mark an empty column is only a notational procedure, while on the other hand in Indian mathematics the zero is treated as a tangible entity, as a number. Aristotle and other Greek teachers had talked about the concept of zero philosophically, but Greek mathematics, fortified as it was by the Pythagorean teaching from Egypt, resisted the incorporation of zero into its system.

The Arabs, who functioned from the ninth to the fourteenth centuries as the transmitters of knowledge and culture from the ancient,, declining cultures of the far east and Egypt, carried this knowledge into the emerging ferment of Western Europe.

During these centuries they picked up the concept of zero along with nine other numerical symbols which had developed in India. The less mystical and more practical orientation of the Arab mentality saw in these symbols a practical device for facilitating calculation and recording large numbers, particularly numbers con- taining an empty column, such as Roman numerals, in use right through the Middle Ages, maintained a notation similar to that of Egyptian numeration in that both were based on groupings which did not require a zero to indicate the empty column: Egyptian 2 eee: The great eighth-century Arab mathematician, Al-Khwrizmi, carried the Indian numerals with zero to the Islamic world.

Then another years passed before the works of Al-Gorisma whose name became the basis of our word algorithm were brought into Europe through the Arabic settlements in Spain. His works were translated into Latin somewhere in the twelfth century.

Gradually this 'Arabic' number system was introduced into medieval Europe and began to support radical changes in Western science and thought. Some of the monastic Orders resisted the adoption of this system of decimal notation with zero, claiming particularly that zero was a device of the Devil.

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Among those who refused it was the Cistercian Order whose mystic and gnostic philosophy was the inspiration and foundation for the construction of the Gothic cathedrals, the cosmic temples of the Piscean Age.

But the merchants adopted the Arabic numerals and zero because they gave a mechanical ease to calculating operations and the recording of quantities. It was then through the mercantile impulse that zero took root. The consequences were enormous. First of all, within the structure of arithmetic itself, the additive basis of calculation had to be cast aside. Formerly the addition of one number to another number always produced a sum larger than either of the original numbers.

This was of course nullified by the utilization of zero. Other laws of arithmetic were also altered, so that we are now able to have operations such as the following: Here logic completely breaks down. The illogic of the symbol was accepted because of the convenience it afforded to quantitative operations.

Yet this breakdown of the simple, natural logic of the arithmetic structure allowed a complicated mental logic to take its place and invited into mathematics a whole range of numerical and symbolic entities, some of which have no verifiable concept or geometric form behind them.

Arising from the sixteenth century onwards, these entities include relative numbers i. The invention of zero permitted numbers to represent ideas which have no form. This signals a change in the definition of the word 'idea', which in antiquity was synonymous with 'form', and implies geometry. The theological impulse of the Indian mentality did not allow it to place zero at the beginning of the series.

Zero was placed after 9. It was not until the late sixteenth century in Europe, the dawn of the Age of Reason, that 0 was placed before 1, allowing for the concept of negative numbers. Not only has zero become indispensable in the mathematical system on which our science and technology depends, but it has become implicitly translated into our philosophy and theologies, our way of viewing nature, our attitudes towards our own natures and toward the environment.

We have seen how in India the adoption of zero was associated with a doctrine which negated the reality of the material world. The Sanskrit name for zero, sunya, meaning 'empty', became chifra in Latin, which carries the meaning of null or nothing. Needless to say, 'nothing' is a different concept from 'empty'.

Also at this period in India the Sanskrit word maya took on a new meaning. Originally it meant 'the power to divide' or the 'dividing mind', but at this time it came to mean 'illusion', or the material aspect of the universe as illusion. W e can see the reverse of this spiritual nihilism in the materialism of the West after the Industrial Revolution, when the spiritual aspect of reality came to be seen as illusory. The 'western' rationalistic mentality negated the ancient and revered spiritual concept of Unity, for with the adoption of zero, Unity looses its first position and becomes merely a quantity among other quantities.

The advent of zero allows one to consider anything below the quantitative number series as nil or of no account, while anything beyond the quantitatively comprehensible range becomes an extra- polation subsumed under the word God and deemed religious or superstitious.

Hence zero provided a framework in western thinking for the development of atheism or negation of the spiritual. From the point of view of the natural world, zero does not exist; it is a completely , mental entity. Yet the impact of this symbol was so great that it caused the sup- posedly empirical physics of the nineteenth century to adapt an atomic theory in which matter was modelled as composed of tiny building blocks, little spheres floating in a zero-empty void.

Twentieth-century nuclear phy- sics no longer conceives of the atom as a separate attracting and repelling particle, but instead it poses a field or matrix of interconnected, continually transforming energy fields of particles and patterns. Particles indistinguishable from process; matter indistinguishable from events. Likewise in the heavens, what was once thought to be a black, empty void with bodies floating in it is now known to be filled with substance-energy.

Between a stellar body and the region surrounding it, there is a field continuum of which the star-body is simply a densification.

While weaning us from the nineteenth-century world view, both microcosmic and macrocosmic, today's science shows us a continual fluctuation and alternation between matter and energy, confirming that in the natural world there is no zero. The notion of zero also had its effect on our psychological conceptualizations. Ideas such as the finality of death and the fear of it, the separation of heaven and earth, the whole range of existential philosophies based on the despair and absurdity of a world followed by non-being, all owe much to the notion of zero.

We saw ourselves as separate individuals moving in a space which was other than ourselves, encountering in that space other beings separate from and other than ourselves. But these concepts are now also loosing their hold. We know now that we exist in groups, determined by various levels of energetic affinities, repelling, exchanging and absorbing through interconnected, subtle energetic communications. And our being extends outward through various energy fields to connect with larger fields.

We have had to learn that there is nowhere that we can dispose of the things we have finished using -that there is no zero drain in our sink; there is no factory pipe or hole in the ground that does not lead somewhere. Everything remains here with us; the cycles of growth, utilization and decay are unbroken. There is no throw- away bottle. With zero we have at the beginning of modern mathematics a number concept which is philosophically misleading and one which creates a separation between our system of numerical symbols and the structure of the natural world.

O n the other hand, with the notion of Unity which governs ancient mathematics, there is no such dichotomy. The notion of Unity remains, literally, unthinkable; simply because in order for anything to be, to exist, it must, in the very positive affirmation ofitself, negate that which it is not. Coldis only cold because it is the negation ofheat.

For a thing to be, its opposite must also be. There is then at the beginning of the created world a contingency of division of Unity into two.

With two, number begins. This same law governs our understanding, for in order to comprehend any objective state we must acknowledge and negate its opposite. Schwaller de Lubicz says, The Number One is only definable through the number two: The intelligence of things exists only through what we may call an original fractioning and the comparison of these fractions to one another, which is then only an enumeration of the aspects of Unity. Thus, unthinkable though Unity may be, both reason and spiritual experience compel the traditional thinker to place it at the beginning.

Everything that exists in his mathematical problem or in his universe is a fraction of the unknown One, and because these parts can be related proportionately to one another they are knowable. Sri Aurobindo says, At the origin of things we are faced with an infinite containing a mass of un- explained finites; an indivisible full of endless divisions, an immutable teeming with mutations and differentiations, a cosmic paradox is at the beginning of all things. This does not mean that the O n e is plural, o r can be limited o r described as the sum of the many.

O n the contrary, it can contain the infinite many because it exceeds all limitation or description by multiplicity, and exceeds at the same time all limitation by finite, conceptual oneness.

The Life Divine. Unity is a philosophic concept and a mystic experience expressible mathematic- r ally. The Western mentality, however, withdrew its discipline o f acknowledging a supra-rational, unknowable mystery as its first principle. But in rejecting this reverence to a single unknowable unity, our mathematics and science developed into a system requiring complex, interconnected hypotheses, imaginary entities such as those mentioned above, and unknown x quantities which must be manipu- lated, quantified o r equalized as in the algebraic form of thought.

So the unknown appears not just once but at every turn, and can be dealt with only by seeking quantitative solutions. O u r present thought is based on the following numerical and logical sequence: With zero in the centre, there is a quantitative expansion 1 , 2 , 3.

The system has a break-point, zero, disconnecting the continuum and dissociating the positive numbers from the negative balancing series. In the ancient Egyptian numerical progression, beginning with one rather than zero, all the elements are natural and real: All the elements flow out from the central unity in accordance with the law of inversion or reciprocity.

The Egyptians based their mathematics on this simple, natural series of numbers, performing sophisticated operations with it for which we now need complex algebra and trigonometry. W e have already seen the natural demonstration of this series in the physical laws of sound. The plucked string, when divided in half, produced double the frequency o f vibrations. Thus this series expresses the essential law of Harmony.

Much of post-Einsteinian physics seems to have this poise of mind as its basis, as inversion plays a major role in Relativity Theory, the Uncertainty Principle, and in such concepts as that ofBlack Holes. The idea of a continual interchange between matter and energy also requires this poise. Such metaphysical concepts as the immortality of the soul, rebirth and reincarna- tion are also more fully grasped by means of the notion of reciprocity.

T o the Egyptians, the nether world to which the soul proceeded after death was called the 'inverted world', the Dwat. The progression of inverse reciprocal elements supplies a mental basis for the notion of perpetual interchange through reversal. The natural progression of whole numbers with their The idea of the unknowable Unity at the beginning has been the basis of many inverse progression is a pat- philosophies and mythological systems.

While Shankhara, with the Buddhism of tern for the formation of the a certain. Hinduism has always rested on the notion o f the One, the Divine, w h o divided himself within himself to form his o w n self-created opposite, the manifested universe.

Within the divine self-regard, three qualities of himself became distin- guished: Sat immobile being , Chit consciousness-force and Ananda bliss. The original unity, represented by a circle, is then restated in the concept of the Real- Idea, the thought of God, which the Hindus called the bindu o r seed, what we call the geometrical point.

Rising or falling tones, as reciprocal arith- metic ratios, are applied to string-lengths. As Ernest McClain points out in The Myth of Invariance, Plato conceived the World-Soul as constitu- ted of reciprocal ratios iden- tical with those which, in Hindu mythology, create the musical 'drum of Shiva', the pulsating instrument of creation see p.

The bindu corresponds to the 'seed-sound idea' of the Tantras. T h e Divine transforms himself into sound vibration nada , and proliferates the universe, which is not different from himself, by giving form or verbal ex- pression to this self-idea. Ramakrishna summarized the scripture by saying, 'The Universe is nothing but the Divine uttering his own name to himself.

This transcendent Word is only a vibration a materialization of the Divine thought which gives rise to the frac- tioning o f unity which is creation. The Word saabda in Sanskrit, the logos of the Christians and Gnostics , whose nature is pure vibration, represents the essential nature o f all that exists.

Concentric vibrational waves span outward from in- numerable centres and their overlappings interference patterns form nodules of trapped energy which become the whirling, fiery bodies of the heavens. The Real-Idea, the Purusha, the inaudible and invisible point of the sound-idea remains fixed and immutable. Its names, however, can be investigated through geometry and number. This emitted sound, the naming ofGod's idea, is what the Pythagoreans would call the Music of the Spheres.

In ancient Egypt the primordial vibrational field called nada in India is called Nun, the primil ocean. It is the One imaged as undifferentiated cosmic substance, the source of all creation. Submerged within this primal ocean is Atum, the creator, who must first distinguish himself from Nun in order for creation to begin.

Atum is masculine, and analogous to Chit consciousness-force of the Indian myth. Atum is pictured in a state of total self-absorbed bliss. Some versions of the myth say that Atum is masturbating. His blissful self-contemplation provokes his ejaculation and this ejaculation catches in his throat, causing him to cough his own seed out of his mouth.

H e coughed and spit out Shu and Tefnut, who, together with himself, form the first triad o f the nine great Neteru or principles of creation. The hieroglyphic pith of a vibrating string sign for the mouth o is the same sign used to write the name of the supreme being, have a flattened, vesical RZ who, as creator, is known as Atum-RG.

Atum's projected seed enters into the form. This and other functional correlations with Egyptian myth have been developed by Lucie Lamy in Egyptian Mysteries. Today, in the field theory of modern astro-physics, the universe is conceived as an integral, incomprehensibly vast vibrating field of ionized, pre-gaseous plasma, an image not unlike that of the N u n or cosmic ocean of the Egyptian myth, or the Prakriti of the Hindu cosmology.

Within this field gravitational influences are triggered which cause a warp and densification into nodal patterns. The dis- equilibrium and turbulence caused by the newly formed galactic mass-centres under the forces of contraction releases compound ripples causing violent, abrupt changes in the pressure and density of the whole cosmic plasma.

These are referred to as galactic 'sonic booms', sonic because indeed the propagation of any sound is simply the rapid oscillatory pressure-density change in any medium. These whirling sonic shocks create a spin in the entire galactic cloud and within the inner regions set up by this spin the stars are born. This clearly restates the ancient image of universal creation through sound waves or the Word of God; science reaffirms that visible stars and galaxies are spiral blast patterns, residual imprints of standing shock waves from the thundering voice of the Universe.

Thus the most recent scientific model of creation is allied to the image given in ancient mythology, and both acknowledge an absolute singularity or Unity at the beginning. In terms of the orthodoxy of ancient mathematics, the symbols of mathematics should reflect the realities they describe.

With zero and the army of merely mental and statistical signs which followed from it, we are very far from having a system of mathematical symbols which corresponds to the pure geometric order of living space. The Division of Unity Those who use geometric figures to describe the beginning of Creation must attempt to show how an absolute Unity can become multiplicity and diversity.

Geometry attempts to recapture the orderly movement from an infinite formless- ness to an endless interconnected array of forms, and in recreating this mysterious passage from One to Two, it renders it symbolically visible. From both the metaphysical and natural points of view it is false to say that in order to arrive at two, you take two ones and put them together.

One only need look at the way in which a living cell becomes two. For One by definition is singular, it is Unity, therefore all inclusive. There cannot be two Ones. Unity, as the perfect symbol for God, divides itself from within itself, thus creating Two: Unity creates by dividing itself, and this can be symbolized geometrically in several different ways, depending upon how the original Unity is graphically represented. Unity can be appropriately represented as a circle, but the very in- commensurability of the circle indicates that this figure belongs to a level of symbols beyond reasoning and measure.

Unity can be restated as the Square, which, with its perfect symmetry, also represents wholeness, and yields to comprehensible measure. In geometrical philosophy the circle is the symbol of unmanifest Unity, while the square represents Unity poised, as it were, for manifestation.

The square represents the four primary orientations, north, south, east and west, which make space comprehensible, and it is formed by two pairs of perfectly equal yet opposi- tional linear elements, thus graphically fulfilling the description of universal Nature found in Taoist and other ancient philosophies. The square is the result of a crossing. The four orientations were related to the four constituents of creation: By definition the square is four equal straight lines joined at right angles.

But a more important definition is that the square is the fact that any number, when multiplied by itself, is a square. Multiplication is symbolized by a cross, and this graphic symbol itself is an accurate definition of multiplication. When we cross a vertical with a horizontal giving these line-movements equal units of length, say 4 for example, we see that this crossing generates a square surface: The principle can be transferred symbolically to the crossing of any contraries such as the crossing of male and female which gives birth to an individual being, o r the crossing of warp and weft which gives birth to a cloth surface, or the crossing of darkness and light which gives birth to tangible, visible form, or the crossing of matter and spirit which gives birth to life itself.

So the crossing is an action-principle which the square perfectly represents. The word Nature means 'that which is born', and all birth into nature requires this crossing of opposites. So the square came to represent the earth, and as such symbolized the conscious experienc-eof finite existence, ofwhat is born into Nature. This brings us to the problem of whether the sides of the square are curved or straight: This objective consciousness might be seen as a reduced velocity of the universal con- sciousness, and has as its instrument the cerebral cortex in man.

The Indians called this power of isolation and arrestation of the ever-moving universal Becoming tapas. The Greek philosopher Heraclitus likened it to a paralysis o f vision such as one experiences when stung by a scorpion. H e called objectivization the 'scorpion sting'.

The Buddhist and Hindu philosophers were concerned lest human consciousness become fascinated o r preoccupied by this segmented perception of reality. T o use a familiar Buddhist analogy, Time is like a necklace of square beads of tangible objects, o r moments o r events, and to be absorbed by this succession o f limited frames is maya o r illusion, whereas only the inner thread o f the necklace, the un- imaginable continuum, is reality.

Pythagoras, however, taught that the experience of life in a finite, limited body was specifically for the purpose of discovering and manifesting supernatural existence within the finite.

One's concentration, then, should also be on the finite itself, to discover h o w this finite could contain intrinsically a power to express the infinite. This does not mean concentration on finite, material effects, but on the abstract principles revealed in the finite world, and the Causes which create and support this embodiment.

Hence Pythagorean mathematics were limited to whole numbers, that is, definable, arrested states, and sought after universal expressions within the measurable, geometric frame o f the square, a profound symbol of finite perfection. The following Workbook is the first ofnine such sections in this volume, intended to take readers step by step through the principal drawings and concepts of Sacred Geometry.

It is suggested that readers take compasses and straight edge and draw for themselves, following the instructions given adjacent to the drawings, each of the figures in the Workbook sections. It is also advisable to use graph paper for these drawings, so that verification for certain relationships can be obtained by simply counting the grid squares.

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Workbook 1 The square cut by its diagonal; J Z Drawing 1. Draw square A B G F. Join C Eo and D continuing the line until it cuts the arc at E. Draw EB perpendicular to AB. Repeat the process of Drawing 1.

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With centre J swing an arc equal to the side of square 2. Draw square 3, MKHA. In a similar manner construct squares 4, 5, etc.

The relationship of the side to the diagonal of each square, and of each square to the next larger square, is identical to that of square 1 to square 2.

This can be written as: JI J2Z: T or, in general terms, ab. This type of progression is called a 'geometric progression', where the numerator, when multiplied by the denominator of the second Drawing 1.

Using the same method as in Drawing 1. With B denominator of the first relationship. This law of as centre and BA as radius, swing an arc of at least cross multiplication between sets of numerators and half of a circle to determine points H and J.

Using denominators holds true for any ratios in the the same method as in Drawing 1. The area of square 2 is exactly twice that of the primary square. This is intuitively evident by the larger square containing four identical triangles, whereas the primary square contains only two. The side of the primary square square 1 is 1, and that of square 2 is J2.

The diagonal of square 2 is equal to 2, exactly twice the side of the primary square. This relationship can be written as: J2 -- diag: Even as the squares increase in size, their root-diagonal relationships remain proportional identities. Drawing 1. With centre] swing an arc equal to the side of square 2. Extend the sides A] and H] until they intersect the arc at K and M.

I The side of a square is called its root. The side of the primary square square 1 is' 1, and that of square 2 is. This represents a variation of the relationship between proportion and progression previous geometric progression, but extended in we are reminded of the alchemical axiom that the opposite direction of diminution.

With B immutable component proportion as well as a and C as centres and radius EB equal to half the volatile, mutable component progression. Draw The relationship between the fixed and the line EF, intersecting the sides of square 1 at G.

Draw square BHFG square manifest, be it in the physical world or in the 2. Repeat this process, constructing squares which world of mental images and conceptions, belongs progressively diminish according to the geometric to the ever-flowing progressions of constant progression, 2, 4, 8, 16, 32 etc. Our science errs in express the creation of Two from Unity the attempting to attach fixed, absolute laws and initial square , and a consequent proliferation of definitions to the changing world of appearances.

The history of science shows us perpetually The square divided by its diagonal provides an discarding or revising one world model after archetypal model for geometric proportions and another. Because of the disturbingly unstable progressions of this type, that is 1: But the the progression. A fixed, proportional increase or unchanging, generative principles remain, and our rate can be the generative pattern for other contemporary rejection of them is taking place infinitely expanding geometric progressions, for only because we have sought for the permanent in example, I: In this geometric demonstration of the metaphysical.

Commentary on In Drawing 1. The side of the original square, called its 'root', is given the value Workbook 1 of 1 since it is the first or primary unit. The simple act of drawing the diagonal has given rise to 2, not because the square has been divided in half, but because square 2 is implied, since the diagonal of square 1 is the root of square 2, and square 2 is exactly double in area to square 1. The reader may justifiably wonder why, having arrived at the symbol of the square, we must further consider the square built on its diagonal; for that matter, why consider this diagonal at all?

This requires that we define the cause-effect relationship as seen in contemplative geometry. Once the four-cornered square has been drawn, one has implicitly all that is necessary to draw the square's diagonal lines.

Moreover this diagonal line like any straight line is implicitly the side or root of a square. In other words, we are obliged to think through or make explicit that which is implicit in any geometric figure. A form is a geometric system and like any system, biological, chemical or other, it must be seen in the unfolding continuum of its components in their cause-effect relationships. The movement from implicit to explicit is similar to that from cause to effect.

It is only in the arbitrary mental world that cause can be separated from effect, while in the natural world they are inseparable: If we carry this logic further we see that the square surface also only exists in a continuous relationship to a cubic volume, of which it forms one of the six faces. In contempla- tive geometry the attempt is always to follow the complete movement from the purely abstract, two-dimensional world of line, then plane, as it becomes explicit in the actual world of three-dimensional volume.

T o return to our square, two paradoxes have been revealed in the act of its division by the diagonal. The first lies in the uncanny coinciding of the two functions of root and diagonal in the geometric moment of the square root of 2. The same line unit is both root and diagonal, the paradox ofsameness and difference. This simultaneity of function yields three seemingly contradictory yet geometrically true relationships: J2 diagonal ' , root root ' diagonal diagonal: The square root of 2 is an irrational function and a universally applicable relation- ship.

Since everything in the natural world undergoes change, this root, being invariant, is by definition supernatural or supra-rational, that is to say it is a symbol of the archetypal realm. The Pythagoreans are said to have referred to the incom- mensurable numbkrs as 'unutterable'.

We can be assured that it was neither secrecy nor puerile piety which led them to so designate them. It was, on the contrary, the keen discretion of an intellect aware of and maintaining a relationship between Number and cosmic realities. N o matter how vast the numerical relationships become, this proportion a: This progression can extend toward diminution as well as toward vastness through a bisection of the square, concurrent with a numerical expansion through the power of the diagonal of the square.

The square root of 2 thus represents the power of multiplicity which can extend itself both towards unlimited expansion and towards utterly minute finiteness. This figure perfectly represents the growth pattern of cellular fission in living organisms. Not only number but form proliferates from the division of Unity. In this geometric analysis of the Parthenon by Tons BrunCs from his book The Secrets of Ancient Geometry, it can be seen that the architecture is governed by the relationship between side and diagonal in a series of squares.

Each of the squares is in relationship to the larger square enclosing it in the ratio of 1 to ; therefore the whole proportional system is based on the functional relationship of J2 to 1 to 1. What appears is a reptilian entity. This creature is very frightening and fierce looking, but i can assure you it can do you no harm.

I know this sounds mad, but it is true and very real. After you get used to interacting with the reptilian, you can move on to the next entity. To interact with the second being you must rotate the flower by 3odegrees and repeat the steps. This creature is even more frightening than the first, so be prepared. Again, this creature cannot harm you in anyway.

Somehow these entities are in suspended animation. They are real but they do not and cannot move. I know within freemasonry, they call the first entity "the khaibit man". You need to have been initiated into the 10th degree to be aware of the khaibit man. As part of the initiation into the 10th degree, the khaibit man is conjured up in front of you.

I am not a freemason, but my father has been a freemason for over 40 years. He has been through the 10th degree. When I made this discovery and drew the complete flower, I was living in Edinburgh. I phoned my father who was living at home in Kirkwall http: When I told my father what I had seen, he could not believe what I was saying. He told me that the only way I could know what I was telling him was if I had been through the 10th degree.

Obviously I'm not a mason and hadn't been through the 10th, so from that day on my father has told me everything he knows about freemasonry. He also told me that his granny my great granny Fox told him when he was 16 years old, that someday one of his children would discover something that would affect the whole world. He is now convinced that the complete flower is this discovery.

You may know that the complete flower contains the kabbalah v s tree of life, the fruit, the egg and the seed of life pic. The complete flower also contains the three dimensional metatron cube pic6 , which holds all the Platonic solids pic7. Not just the building blocks of life, but the building blocks of creation itself. Metatron Cube 7. Platonic Solids I found this symbol on a knights templar tombstone pic. Knights Templar tombstonefrom St Magnus cathedral in Kirkwall I believe this symbol is a representation of the egg of life pic.

The egg of life left The second rotation of the first layer of the complete flower of life right I was born and bred and now live back in Kirkwall. At no time did I ever get taught at school, that there was any knights templars who came from or to Orkney. For whatever reason, the computer generated flower pic.

It does contain all Freemasonry' s sacred symbols. A computer generated version of the complete flower I thought I'd add this picture as you will recognize the pattern pici4.

Pattern of snake's skin The complete flower of life does have another use. If you overlay a map after getting the correct scale with the complete flower of life, all sacred sites, standing stones etc, will sit at the centre of six points.

This picture gives you idea how the grid looks pic. I believe the flower is what the square and compass really symbolize. Is this geometry the origin of the "Star of David"!? Crop circles found within the complete flower http: Reprinted with Permission Is this freemasonry?He next depresses this string with his finger exactly at its midpoint. The tools of u geometry and number represent the means to attain know- ledge of both external and internal space and time. The visible growth of the plant, its proliferation into specificity, depends upon the'root for stability and nutrition.

When we cross a vertical with a horizontal giving these line-movements equal units of length, say 4 for example, we see that this crossing generates a square surface: F When we further limit ourselves to three terms, that is when we lift ourselves or Let us add to the side the diagonal, and to the diagonal, let us add two sides, because what the side can do two times the diagonal can do once.

This brings us to the problem of whether the sides of the square are curved or straight: Ancient mathematics had no decimal system with which the numerical equiv- alency of the incommensurable square root of 2 1. The hieroglyphic pith of a vibrating string sign for the mouth o is the same sign used to write the name of the supreme being, have a flattened, vesical RZ who, as creator, is known as Atum-RG.