File Name: sacred geometry philosophy and practice by robert lawlor .zip
For R. Diagrams by Melvyn Bernstein, A. Illustration on p. Any copy of ths book issued by the publisher as a. No part of this publication may bereproduced or transmitted in any form or by any means,electronic or mechanical, including photocopy, recordingor any other information storage and retrieval system,without prior permission in writing h m the publisher.
It seems that the same constituents in any other arrangement cannottransform the radiant energy of light into life substance.
In mythological thought,twelve most often occurs as the number of the universal mother of life, and so thistwelvefold symbol is precise even to the molecular level.
The specialization of cells in the body's tissue is determined in part by the spatialposition of each cell in relation to other cells in its region, as well as by an informationalimage of the totality to which it belongs. This spatial awareness on a cellularlevel may be thought of as the innate geometry of life. All our sense organs function in response to the geometrical or proportional -not quantitative - differences inherent in the stimuli they receive.
For example,when we smell a rose we are not responding to the chemical substances of its perfume,but instead to the geometry of their molecular construction. That is to say,any chemical substance that is bonded together in the same geometry as that of therose will smell as sweet.
Similarly, we do not hear simple quantitative differencesin sound wave frequencies, but rather the logarithmic, proportional differencesbetween frequencies, logarithmic expansion being the basis of the geometry ofspirals. Our visual sense differs from our sense of touch only because the nerves of theretina are not tuned to the same range of frequencies as are the nerves embedded inour skin.
If our tactile or haptic sensibilities were responsive to the same frequenciesas our eyes, then all material objects would be perceived to be as ethereal as projectionsoflight and shadow. Our different perceptual faculties such as sight, hearing,touch and smell are a result then of various proportioned reductions of one vastspectrum of vibratory frequencies. We can understand these proportional relationshipsas a sort of geometry of perception.
With our bodily organization into five or more separate perceptual thresholds,there is seemingly little in common between visual space, auditory space and hapticspace, and there seems to be even less connection between these physiological spacesand pure, abstract metric or geometric space, not to mention here the differingawareness of phychological space. Yet all these modes of spatial being converge inthe human mind-body. Within the human consciousness is the unique abilityto perceive the transparency between absolute, permanent relationships, containedin the insubstantial forms of a geometric order, and the transitory, changing formsof our actual world.
The content of our experience results from an immaterial,abstract, geometric architecture which is composed of harmonic waves of energy,nodes of relationality, melodic forms springing forth from the eternal realm ofgeometric proportion. From the apparent world tothe subatomic? Here we find twelvefold symmetry as the life-giver orwomb which transforms light into the basic spectrum of -organic substance.
This is recalled symbolically in thestained-glass window, which transforms light into thecolour spectrum. Geometry as a contemplative practice is personified by anelegant and refined woman, for geometry functions as anintuitive, synthesizing, creative yet exact activity of mindassociated with the feminine principle. But when these geometriclaws come to be applied in the technology of dailylife they are represented by the rational, masculine principle:contemplative geometry is transformed into practicalgeometry.
BELOW Pythagoras is credited with first establishingthe relationship between number ratios and soundfrequencies. He is shown here experimenting withbells, water-glasses, stretched cords, and varioussized pipes; his Hebrew counterpart, Jubal, usesweighted hammers on an anvil. The whole numberratios for determining the consonant sounds ina musical scale are either drawn from or aremultiples of the numbers in the two progressionsof the Lambda.
LBOVE Arithmetic is also personified as a woman, but not asgrand and noble in attire as Geometry , perhaps symbolicallyindicating that Geometry was considered as a higher order ofknowledge. On her thighs symbolizing the generative function are two geometric progressions.
The first series, 1, 2, 4,8, goes down the left thigh, associating the even numberswith the feminine, passive side of the body. The second1series, l,3, 9, 27, goes down the right thigh, associating theodd numbers with the masculine, active side, an associationwhich goes back to the Pythagoreans, who called the odd1numbers male and the even female. On the woman's left sitsPythagoras using an abacus system for computation.
In thisIsystem, number notation is still dependent upon spatialarrangement. Boethius sits on her right using Arabicnumerals in a modern,system of calculation in which numbernotation has become a separate, abstract system independentof its geometric origin. The ancient astronomers designatedthe movement and position of celestialbodies through angular notation.
Thevaried angular positions of the sun,moon, planets and stars were related tothe cyclic changes in the natural world,such as moon phases, seasons, tides,plant growth, human and animal fertility,etc. It was the angle which specifiedthe influences of celestial patternson earthly events. In this way we canappreciate the similar root of the wordsangle and angel.
Today the newlyemerging science of heliobiology verifiesthat the angular position of themoon and planets does affect theelectromagnetic and cosmic radiationswhich impact with the earth, and inturn these field fluctuations affect manybiological processes.
In ancient trigonometry anangle is a relationship betweentwo whole numbers. In this example the angle atleft is an expression of theratio 3 to 4, and with thissystem spatial coordinatescan easily be put into relationshipwith sound frequencies,such as the musicalfourth see p.
Modern thought has difficult access to the concept of the archetypalbecause European languages require that verbs or action words be associated withnouns. We therefore have no linguistic forms with which to image a process oractivity that has no material carrier. Ancient cultures symbolized these pure, eternalprocesses as gods, that is, powers or lines of action through which Spirit is concretizedinto energy and matter.
The bridle, then, relates to archetypal activitythrough the function of leverage; the principle that energies are controlled, specijedand rnodijied through the effects of angulation. Thus we find that often the angle - which is fundamentally a relationship of twonumbers - would have been used in ancient symbolism to designate a group of fixedrelationships controlling interacting complexes or patterns.
Thus the archetypes orgods represent dynamic functions forming links between the higher worlds ofconstant interaction and process and the actual world of particularized objects.
Wefind, for example, that a 60" angle has quite different structural and energeticproperties from an angle of 90" or of 45". Likewise, geometric optics reveals thateach substance characteristically refracts light at its own particular angle, and it isthis angle which gives us our most precise definition of the substance. Furthermore,the angles in the bonding patterns of molecules determine to a great extent thequalities of the substance.
In the case of the bridle, this angulation or angular play is manifested in therelation of the bit to the bridle strap, or between the bit and the bend of the horse'sneck and jaw, both controlled by the angulation between the forearm and thebiceps of the rider.
From the level of the archetype or active Idea, the principle ofthe bridle can be applied metaphorically to many regions of human experience. Forinstance, when St Paul describes the process of self-discipline by which a higherintentionality attempts to control the lower, 'animal' nature, he says that when onecan bridle the mouth he can then master the rest of his nature. But while at thearchetypal level this image can be metaphysically and poetically expansive, it alsofinds its exact, geometrical representation in the angle.
It is the precise angle of thearm in play with the angle of the bridle that controls the energy of the horse. Functioning then at the archetypal level, Geometry and Number describe fundamental,causal energies in their interwoven, eternal dance. It is this way of seeingthat stands behind the expression of cosmological systems as geometric configurations. For example, the most revered of all Tantric diagrams, the Sri Yantra, imagesall the necessary functions active in the universe through its nine interlockedtriangles.
To immerse oneself in such a geometric diagram is to enter into a kind ofphilosophic contemplation. For Plato, Reality consisted of pure essences or archetypal Ideas, of which thephenomena we perceive are only pale reflections. The Greek work 'Idea' is alsotranslated as 'Form'.
These Ideas cannot be perceived by the senses, but by purereason alone. Geometry was the language recommended by Plato as the clearestmodel by which to describe this metaphysical realm. And do you not know that they [the geometers] make use of the visible formsand talk about them, though they are not of them but of those things of whichthey are a likeness, pursuing their inquiry for the sake of the square as such andthe diagonal as such, and not for the sake of the image of it which they draw?
And so on in all cases. What they really seek is to get sight of those realitieswhich can be seen only by the mind. The Platonist sees our geometrical knowledge as innate in us, having beenacquired before birth when our souls were in contact with the realm of ideal being. All mathematical forms have a primary subsistence in the soul; so that prior tothe sensible she contains self-motive numbers.
The Sri Yantra is drawnfrom nine triangles, fourpointed downward and fivepointed upward, thus forming42 6 x 7 triangularfragments around a centraltriangle. There is probablyno other set of triangleswhich interlock with suchintegrational perfection.
For the human spirit caught within a spinning universe in an ever confusing flowof events, circumstance and inner turmoil, to seek truth has always been to seek theinvariable, whether it is called Ideas, Forms, Archetypes, Numbers or Gods. Toenter a temple constructed wholly of invariable geometric proportions is to enteran abode of eternal truth.
Thomas Taylor says, ' Geometry enables its votary, likea bridge, to pass over the obscurity of material nature, as over some dark sea to theluminous regions ofperfect reality.
As Plato says, the soul's fire mustgradually be rekindled by the effort:You amuse me, you who seem worried that I impose impractical studies uponyou. It does not only reside with mediocre minds, but all men have difficulty inpersuading themselves that it is through these studies, as if with instruments, thatone purifies the eye of the soul, and that one causes a new fire to burn in thisorgan which was obscured and as though extinguished by the shadows of theother sciences, an organ whose conservation is more important than ten thousandeyes, since it is by it alone that we contemplate the truth.
AD in hisMathematics Useful for Understanding Plato Geometry deals with pure form, and philosophical geometry re-enacts the unfoldingof each form out of a preceding one. It is a way by which the essentialcreative mystery is rendered visible. The passage ffom creation to procreation,from the unmanifest, pure, formal idea to the 'here-below', the world that spinsout from that original divine stroke, can be mapped out by geometry, and experiencedthrough the practicc of geometry: this is the purpose of the 'Workbook'sections of this book.
Inseparable from this process is the concept of Number, and, as we shall see, forthe Pythagorean, Number and Form at the ideal level were one. But number in thiscontext must be understood in a special way. When Pythagoras said, 'All is arrangedaccording to Number', he was not thinking of numbers in the ordinary, enumerativesense.
In addition to simple quantity, numbers on the ideal level are possessedof quality, so that 'twoness', 'threeness' or 'fourness', for example, are not merelycomposed of 2, 3, or 4 units, but are wholes or unities in themselves, each havingrelated powers.
Schwaller de Lubicz gives an analogy by which this universal and archetypalsense of Number can be understood. A revolving sphere presents us with the notionof an axis. We think of this axis as an ideal or imaginary line through the sphere. ItThe twelfth-century architectureof the CistercianOrder achieves its visualbeauty through designswhich conform to the proportionalsystem of musicalharmony.
Many of the abbeychurches of this period wereacoustic resonators transforminga human choir intocelestial music. St Bernard ofClairvaux, who inspired thisarchitecture, said of theirdesign, 'There must be no'decoration, only proportion.
Number in the enumerative sense correspondsto the measures and movements of the outer surface of the sphere, while theuniversal aspect of Number is analogous to the immobile, unmanifest, functionalprinciple of its axis.
Let us shift our analogy to the two-dimensional plane. If we take a circle and asquare and give the value 1 to the diameter of the circle and also to the side of thesquare, 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 becarried out to an infinite number of decimal places without ever arriving at aresolution. In the case of the diagonal of the square, this decimal is 1. With the circle, if we give the diameter thevalue I, the circumference will also always be of the incommensurable type,3.
Christ is shown using compassesto re-enact the creationof the universe from thechaos of the primal state. This icon can also be understoodas an image of individualself-creation ; forhere, as in many medievalimages of Christ, Tantricsymbolism is evident. Christholds the compass with hishand across the vital centrecalled the heart chakra, andfrom this centre he organizesthe turmoil of the vital energiescontained in the lowerchakras which are indicatedon the body by centres at thenavel and genitals.
Bookings : Roger Green, info fengshuiseminars. Robert Lawlor is the translator of R. Schwaller de Lubicz's masterwork — The Temple of Man - the culmination of his exhaustive year study at the great temple of Amun-Mut-Khonsu at Luxor, which is revealed to be an architectural encyclopedia of humanity and the universe. A ground breaking book which introduces the mythological properties assigned to geometric forms, and covers the Golden Section, gnomonic spirals, music, and the squaring of the circle. The thinkers of ancient Egypt, Greece and India recognized that numbers governed much of what they saw in their world and hence provided an approach to its divine creator.
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He is length, width, height and depth. The autho r covers the theories of the ancient geometers, and involves the reader in pract ical work doing geometric constructions using only a pencil, compass, straight-e dge and graph paper. These simple exercises lead us from first principles to a g rasp of the dynamics of logarithmic spirals and the Golden Proportion. The nine workbooks include Golden Section, creation of spirals, squaring the circle, geom etry and music, and the Platonic Solids. The book has illustrations and diag rams with 56 in color.
Constructing the Universe Activity Book Vols. Bradford Hansen-Smith , author: Wholemovement.
They lived mostly within their own memories, making only brief visits to the real world. But from time to time they came out with unexpected things. He had settled Jerry in front of the TV, and shortly he would ring Nilla to arrange a time to pick her up, but first he wanted to see the sunset. The sun had lost its heat much earlier in the evening, but it was still dazzling as it hovered just above the water line in the west, bright and golden. It was only when it had slipped halfway below the horizon that it lost its glow, staining the clouds dotted over the mainland dark red, like blood-filled arteries.
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Carol Martin Watts, Donald J. Journal of the Society of Architectural Historians 1 September ; 46 3 : — The Garden Houses complex at Ostia is an urban development consisting of apartments, shops, and gardens, planned and executed around A. Extensive on-site study and measurements of the major formal elements and their spatial relationships provide evidence for a recurrent geometrical pattern which is found at successive scales from urban planning to painting and mosaics. This particular geometric operation is based upon concentric regulating squares subdivided in such a manner as to perpetuate itself ad infinitum.