2. geometry PHILOSOPHY A N D PRACTICEwith 202 illustrations anddiagrams, 56 in two coloursThames Hudson 3. For R. A. Schwaller deLubicz and Lucie L r n yThis book originated in a series ofseminars held in New York Cityfor the Lindisfarne Association,Crestone, Colorado.Diagrams by Melvyn Bernstein, A.I.A.Illustrationon p.1: see p.53.Any copy of t h s book issued by the publisher asa .paperback is sold subject to the condition that it shallnot byway of trade or otherwise be lent, resold, hredout or otherwisecirculated without the publishersprior consent in any form ofbinding or cover otherthan that in which it is published andwithout a s i m i l vcondition includmg these words being imposedon asubsequent purchaser.Fiat published in the United Kingdom in1982 byThames & Hudson Ltd, 181A High Holbom,London W C l V7QXO 1982 Thames & Hudson Ltd, LondonReprinted 2002All RightsReserved. No part of this publication may bereproduced ortransmitted in any form or by any means,electronic or mechanical,including photocopy, recordingor any other information storage andretrieval system,without prior permission in writing h m thepublisher.ISBN 0-500-81030-3Printed and bound in China 4. ContentsIntroduction 4 The Practice of Geometry 7 Sacred Geometry :Metaphor of Universal Order16 The Primal Act: The Division ofUnity23 Workbook 1: The Square Cut by its Diagonal; J2 25-27Workbook 2 : The J3 and the Vesica Piscis 32-33 Workbook 3: The J536-37 IV AlternationWorkbook 4: Alternation 4 W 1 v Proportiorlandthe Golden Section Workbook 5 : The Golden Proportion 48-52 VIGnomonic Expanion the Creation of Spirals andWorkbook 6 : Gnomonicspirals 67-70*VII The Squaring of the Circle Workbook 7 : Squaringthe circle 7 4 7 9VIII Mediation: Geometry becomes MusicWorkbook 8: Geometry and Music 83-85 Anthropos I The Genesis of CosmicVolumes Workbook 9 : The Platonic Solids 98-1 02 BibliographySources of Illustrations 5. Introduction In science today we arewitnessing a general shift away from the assumption that thefundamental nature of matter can be considered from the point ofview of sub- stance (particles, quanta) to the concept that thefundamental nature of the material world is knowable only throughits underlying patterns of wave forms.Both our organs of perceptionand the phenomenal world we perceive seem to be best understood assystems of pure pattern, or as geometric structures of form andproportion. Therefore, when many ancient cultures chose to examinereality through the metaphors of geometry and music (music beingthe study o f the pro- portional laws of sound frequency), theywere already very close to the position of our most contemporaryscience.Professor Amstutz of the Mineralogical Institute at theUniversity of Heidelberg recently said :X-ray diffraction patterninberyl, indicating a patternedMatters latticed waves are spaced atintervals corresponding to the frets on a harparray of intervalssurround- or guitar with analogous sequences of overtones arisingfrom each fundamental.ing a central node much like The science ofmusical harmony is in these terms practically identical with thethepattern of partial over- science of symmetry in crystals.tonesaround a fundamentaltone. The point of view of modern force-fieldtheory and wave mechanics corresponds to the ancientgeometric-harmonic vision of universal order as being an interwovenconfiguration of wave patterns. Bertrand Russell, who began to seethe profound value of the musical and geometric base to what we nowcall Pythagorean mathe- matics and number theory, also supportedthis view in The Analysis o Matter: f What w e perceive as variousqualities of matter, he said, are actually differences inperiodicity.In biology, the fundamental role of geometry andproportion becomes even more evident when we consider that momentby moment, year by year, aeon by aeon, every atom of every moleculeof both living and inorganic substance is being changed andreplaced. Every one of us within the next five to seven years willhave a completely new body, down to the very last atom. Amid thisconstancy ofchange, where can we find the basis for all that whichappears to be consistent and stable?These geometric array pat-Biologically we may look to our ideas of genetic coding as thevehicle of replicationterns called kolams are drawn, andcontinuity, but this coding does not lie in the particular atoms(or carbon,with powdered chalk, byhydrogen, oxygen and nitrogen)ofwhich the gene substance, D N A , is composed;South Indian womenon thethese are all also subject to continual change andreplacement. Thus the carrier ofdoorstep each morning, tocontinuityis not only the molecular composition of the D N A , but also itshelixevoke the spirit of order andform. This form is responsiblefor the replicating power of the DNA. The helix,harmony into thehome. which is a special type from the group of regular spirals,results from sets of fixed geometric proportions, as we shall seein detail later on. These proportions can be understood to existapriori, without any material counterpart, as abstract, geometricrelationships. The architecture of bodily existence is determinedby an invisible, immaterial world of pure form and geometry.Modernbiology increasingly recognizes the importance of the form and thebonding relationships o f the few substances which comprise themolecular body of living organisms. Plants, for example, can carryout the process of photosynthesis only because the carbon,hydrogen, nitrogen and magnesium of the chlorophyll molecule arearranged in a complex twelvefold symmetrical pattern, rather like6. that of a daisy. It seems that the same constituents in anyother arrangement cannot --transform the radiant energy of lightinto life substance. In mythological thought,twelve most oftenoccurs as the number of the universal mother of life, and sothistwelvefold symbol is precise even to the molecular level. Thespecialization of cells in the bodys tissue is determined in partby the spatialposition of each cell in relation to other cells inits region, as well as by an informa-tional image of the totalityto which it belongs. This spatial awareness on a cellularlevel maybe thought of as the innate geometry of life. All our sense organsfunction in response to the geometrical or proportional -notquantitative - differences inherent in the stimuli they receive.For example,when w e smell a rose we are not responding to thechemical substances o f its per- %fume, but instead to the geometryof their molecular construction. That is to say,%*?.any chemicalsubstance that is bonded together in the same geometry as that oftherose will smell as sweet. Similarly, we do not hear simplequantitative differencesin sound wave frequencies, but rather thelogarithmic, proportional differences From the apparent worldtobetween frequencies, logarithmic expansion being the basis of thegeometry of the subatomic? all forms arespirals.only envelopes forgeometricpatterns, intervals and rela- O u r visual sense differsfrom our sense of touch only because the nerves ofthetionships.retina are not tuned to the same range of frequenciesas are the nerves embedded inour skin. If our tactile or hapticsensibilities were responsive to the same frequenciesas our eyes,then all material objects would be perceived to be as ethereal aspro-jections oflight and shadow. O u r different perceptualfaculties such as sight, hearing,touch and smell are a result thenof various proportioned reductions of one vastspectrum of vibratoryfrequencies. W e can understand these proportional relation-shipsas a sort of geometry of perception. With our bodily organizationinto five or more separate perceptual thresholds,there is seeminglylittle in common between visual space, auditory space andhapticspace, and there seems to be even less connection betweenthese physiological spacesand pure, abstract metric or geometricspace, not to mention here the differingawareness of phychologicalspace. Yet all these modes of spatial being converge inthe humanmind-body. Within the human consciousness is the unique abilitytoperceive the transparency between absolute, permanentrelationships, containedin the insubstantial forms of a geometricorder, and the transitory, changing formsof our actual world. Thecontent of our experience results from an immaterial,abstract,geometric architecture which is composed of harmonic waves ofenergy,nodes of relationality, melodic forms springing forth fromthe eternal realm ofgeometric proportion.Here we find twelvefoldsymmetry as the life-giver orwomb which transforms light into thebasic spectrum of -organic substance. This is recalled symbolicallyin thestained-glass window, which transforms light into thecolourspectrum. 7. I The Practice of Geometry What is God? He is length,width, height and depth.ST B E R N A RD O F CLAIRVAUX , OnConsideration Geometry means measure of the earth. In ancientEgypt, from which Greeceinherited this study, the Nile would floodits banks each year, covering the land andobliterating the orderlymarking of plot and farm areas. This yearly flood symbol-ized tothe Egyptian the cyclic return of the primal watery chaos, and whenthewaters receded the work began of redefining and re-establishingthe boundaries. This work was called geometry and was seen as are-establishment of the principleof order and law on earth. Eachyear the areas measured out would be somewhat different. The humanorder would shift and this was reflected in the ordering of theearth. The Temple astronomer might say that certain celestialconfigurations had changed so that the orientation or location of atemple had to be adjusted accord- ingly. So the laying of squaresupon the earth had, for the Egyptian, a metaphysical as well as aphysical and social dimension. This activity of measuring the earthbecame the basis for a science of natural law as it is embodied inthe archetypal forms of circle, square and triangle.Geometry is thestudy of spatial order through the measure and relationships offorms. Geometry and arithmetic, together with astronomy, thescience of temporal order through the observation of cyclicmovement, constituted the major intellectual disciplines ofclassical education. The fourth element of this great fourfoldsyllabus,the Quadrivium, was the study ofharmony and music. Thelaws ofsimple harmonics were considered to be universals whichdefined the relationship and interchange between the temporalmovements and events of the heavens and the spatial order anddevelopment on earth.The implicit goal of this education was toenable the mind to become a channel through which the earth (thelevel of manifested form) could receive the abstract, cosmic lifeof the heavens. The practice of geometry was an approach to the wayin which the universe is ordered and sustained. Geometric diagramscan be con- templated as still moments revealing a continuous,timeless, universal action generally hidden from our sensoryperception. Thus a seemingly common mathe- matical activity canbecome a discipline for intellectual and spiritual insight.Platoconsidered geometry and number as the most reduced and essential,and therefore the ideal, philosophical language. But it is only byvirtue of functioning at a certain level of reality that geometryand number can become a vehicle for philosophic contemplation.Greek philosophy defined this notion of levels, so useful in ourthinking, distinguishing the typal and the archetypal. Followingthe indication given by the Egyptian wall reliefs, which are laidout in three registers, an upper, a middle and a lower, we candefine a third level, the ectypal, situated between the archetypaland the typal.T o see how these operate, let us take an example ofa tangible thing, such as the bridle of a horse. This bridle canhave a number of forms, materials, sizes, colours, uses, allofwhich are bridles. The bridle considered in this way, is typal;it is existing, diverse and variable. But on another level there isthe idea or form of the bridle, the guiding model of all bridles.This is an unmanifest, pure, formal idea and its level is ectypal.But yet above this there is the archetypal level which is that ofthe principle or power-activity. that is a process which theectypal formrand typal example of the bridle only represent. Thearchetypal is concerned with universal processes or dynamicpatterns which can be considered independently of any structure or8. Geometry as a contemplative practice is personified by anelegant and refined woman, for geometry functions as an intuitive,synthesizing, creative yet exact activity of mindassociated withthe feminine principle. But when these geo- metric laws come to beapplied in the technology of daily life they are represented by therational, masculine principle: contemplative geometry istransformed into practicalgeometry.BELOWPythagoras is credited withfirst establishingthe relationship between number ratios andsoundfrequencies. He is shown here experimenting with bells,water-glasses, stretched cords, and various sized pipes; his Hebrewcounterpart, Jubal, uses weighted hammers on an anvil. The wholenum-ber ratios for determining the consonant sounds in a musicalscale are either drawn from or aremultiples of the numbers in thetwo progressions of the Lambda. LBOVE Arithmetic is alsopersonified as a woman, but not asgrand and noble in attire asGeometry, perhaps symbolicallyindicating that Geometry wasconsidered as a higher order ofknowledge. O n her thighs(symbolizing the generative func-tion) are two geometricprogressions. The first series, 1, 2, 4,8, goes down the leftthigh, associating the even numberswith the feminine, passive sideof the body. The second1series, l , 3 , 9, 27, goes down the rightthigh, associating theodd numbers with the masculine, active side,an associationwhich goes back to the Pythagoreans, who called theoddnumbers male and the even female. The Greeks called these1twoseries the Lambda, and Plato in the Timaeus uses them todescribethe World Soul (see p. 83). O n the womans left sitsPythagorasusing an abacus system for computation. In thisIsystem, numbernotation is still dependent upon spatialarrangement. Boethius sitson her right using Arabicnumerals in a modern,system of calculationin which numbernotation has become a separate, abstract systemindependentof its geometric origin. 9. The ancient astronomersdesignatedthe movement and position of celestialbodies throughangular notation. Thevaried angular positions of the sun,moon,planets and stars were related tothe cyclic changes in the naturalworld,such as moon phases, seasons, tides,plant growth, human andanimal fer-tility, etc. It was the angle which speci-fied theinfluences of celestial patternson earthly events. (In this way wecanappreciate the similar root of the wordsangle and angel.) Todaythe newlyemerging science of heliobiology veri-fies that theangular position of themoon and planets does affecttheelectromagnetic and cosmic radiationswhich impact with theearth, and inturn these field fluctuations affect manybiologicalprocesses.material form. Modern thought has difficult access to theconcept of the archetypalbecause European languages require thatverbs or action words be associated withnouns. W e therefore haveno linguistic forms with which to image a process oractivity thathas no material carrier. Ancient cultures symbolized these pure,eternalprocesses as gods, that is, powers or lines of actionthrough which Spirit is con-cretized into energy and matter. Thebridle, then, relates to archetypal activitythrough the function ofleverage; the principle that energies are controlled, specijedandrnodijied through the effects o angulation.f Thus we find thatoften the angle - which is fundamentally a relationship oftwonumbers - would have been used in ancient symbolism to designatea group of fixedIn ancient trigonometry anangle is a relationshipbe- relationships controlling interacting complexes or patterns.Thus the archetypes ortween two whole numbers.gods representdynamic functions forming links between the higher worlds ofIn thisexample the angle a t constant interaction and process and theactual world of particularized objects. Weleft is an expression ofthefind, for example, that a 60" angle has quite differentstructural and energeticratio 3 to 4, and with this properties froman angle o f 90" or of 45". Likewise, geometric optics revealsthatsystem spatial coordinatescan easily be put into rela-eachsubstance characteristically refracts light at its own particularangle, and it istionship with sound fre-this angle which gives usour most precise definition of the substance. Furthermore,quencies,such as the musical the angles in the bonding patterns of moleculesdetermine to a great extent thefourth (see p. 85). qualities of thesubstance. In the case of the bridle, this angulation or angularplay is manifested in therelation of the bit to the bridle strap,or between the bit and the bend of the horsesneck and jaw, bothcontrolled by the angulation between the forearm and thebiceps ofthe rider. From the level of the archetype or active Idea, theprinciple ofthe bridle can be applied metaphorically to manyregions of human experience. Forinstance, when St Paul describesthe process of self-discipline by which a higherintentionalityattempts to control the lower, animal nature, he says that whenonecan bridle the mouth he can then master the rest of his nature.But while at thearchetypal level this image can be metaphysicallyand poetically expansive, it alsofinds its exact, geometricalrepresentation in the angle. It is the precise angle of thearm inplay with the angle of the bridle that controls the energy of thehorse. Functioning then at the archetypal level, Geometry andNumber describe funda-mental, causal energies in their interwoven,eternal dance. It is this way of seeingthat stands behind theexpression of cosmological systems as geometric configura-tions.For example, the most revered of all Tantric diagrams, the SriYantra, imagesall the necessary functions active in the universethrough its nine interlockedtriangles. T o immerse oneself in sucha geometric diagram is to enter into a kind ofphilosophiccontemplation. 10. For Plato, Reality consisted o f pure essencesor archetypal Ideas, of which theThe Sri Yantra is drawnphenomena we perceive are only pale reflections. (The Greek work Idea is alsofrom nine triangles, fourtranslated as Form.) These Ideas cannot beperceived by the senses, but by purepointed downward and fivepointed upward, thus form-reason alone. Geometry was the languagerecommended by Plato as the clearest ing 42 (6 x 7) triangularmodelby which to describe this metaphysical realm.fragments around acentral triangle. There is probablyAnd do you not know that they[the geometers] make use of the visible formsno other set oftrianglesand talk about them, though they are not of them but ofthose things of whichwhich interlock with suchthey are a likeness,pursuing their inquiry for the sake o f the square as suchandintegrational perfection.the diagonal as such, and not for thesake of the image of it which they draw?And so on in all cases . .. What they really seek is to get sight o f those realitieswhichcan be seen only by the mind. PLATO, Republic, VII, 510 d, e.ThePlatonist sees our geometrical knowledge as innate in us, havingbeenacquired before birth when our souls were in contact with therealm of ideal being.All mathematical forms have a primarysubsistence in the soul; so that prior tothe sensible she containsself-motive numbers.; vital figures prior to such as areapparent;harmonic ratios prior to things harmonized ;and invisible circlespriorto the bodies that are moved in a circle.TH O M AS T A YLORPlato demonstrates this in the Meno where he has an untutoredservant boy solveby intuition the geometric problem of the doublingof the square. 11. For the human spirit caught within a spinninguniverse in an ever confusing flow of events, circumstance andinner turmoil, to seek truth has always been to seek theinvariable, whether it is called Ideas, Forms, Archetypes, Numberso r Gods. T o enter a temple constructed wholly of invariablegeometric proportions is to enter an abode of eternal truth. ThomasTaylor says, Geometry enables its votary, like a bridge, to passover the obscurity of material nature, as over some dark sea to theluminous regions ofperfect reality. Yet this is by no means anautomatic happening that occurs just by picking up a geometry book.As Plato says, the souls fire must gradually be rekindled by theeffort: You amuse me, you w h o seem worried that I imposeimpractical studies upon you. It does not only reside with mediocreminds, but all men have difficulty in persuading themselves that itis through these studies, as if with instruments, that one purifiesthe eye of the soul, and that one causes a new fire to burn in thisorgan which was obscured and as though extinguished by the shadowsof the other sciences, an organ whose conservation is moreimportant than ten thousand eyes, since it is by it alone that w econtemplate the truth. Republic, VII, 527 d, e (as quoted by Theonof Smyrna (2nd c. AD) in his Mathematics Useful for UnderstandingPlato)Geometry deals with pure form, and philosophical geometryre-enacts the un- folding of each form out of a preceding one. Itis a way by which the essential creative mystery is renderedvisible. T h e passage ffom creation to procreation, from theunmanifest, pure, formal idea to the here-below, the world thatspins out from that original divine stroke, can be mapped out bygeometry, and experi- enced through the practicc of geometry: thisis the purpose of the Workbook sections of this book.Inseparablefrom 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 wasnot thinking of numbers in the ordinary, enumera- tive sense. Inaddition to simple quantity, numbers on the ideal level arepossessed of quality, so that twoness, threeness o r fourness, forexample, are not merely composed of 2, 3, o r 4 units, but arewholes or unities in themselves, each having related powers. Two,for instance, is seen as the original essence from which the powerof duality proceeds and derives its reality.R.A. Schwaller deLubicz gives an analogy by which this universal and archetypalsense of Number can be understood. A revolving sphere presents uswith the notion of an axis. W e think of this axis as an ideal orimaginary line through the sphere. ItThe twelfth-centuryarchi-tecture of the CistercianOrder achieves its visualbeautythrough designswhich conform to the pro-portional system ofmusicalharmony. Many of the abbeychurches of this periodwereacoustic resonators trans-forming a human choir intocelestialmusic. St Bernard ofClairvaux, who inspired thisarchitecture, saidof theirdesign, There must be nodecoration, only proportion. 12.Christ is shown using com- passes to re-enact the crea-tion of theuniverse from thechaos of the primal state.This icon can also beunder-stood as an image of indi-vidual self-creation; forhere, asin many medievalimages of Christ, Tantricsymbolism is evident.Christholds the compass with hishand across the vital centrecalledthe heart chakra, andfrom this centre he organizesthe turmoil ofthe vital ener-gies contained in the lowerchakras which areindicatedon the body by centres a t thenavel and genitals.Geometryis symbolized here in boththe individual and universalsenseas an instrumentthrough which the higherarchetypal realmtransmitsorder and harmony to thevital and energetic worlds.has noobjective existence, yet w e cannot help but be convinced of itsreality; andto determine anything about the sphere, such as itsinclination o r its speed of rotationwe must refer to thisimaginary axis. Number in the enumerative sense correspondsto themeasures and movements of the outer surface of the sphere, whiletheuniversal aspect of Number is analogous to the immobile,unmanifest, functionalprinciple of its axis. Let us shift ouranalogy to the two-dimensional plane. If w e take a circle andasquare and give the value 1 to the diameter of the circle and alsoto the side o f thesquare, then the diagonal of the square willalways be (and this is an invariable law)an incommensurable,irrational number. It is said that such a number can becarried outto an infinite number of decimal places without ever arriving ataresolution. In the case o f the diagonal o f the square, thisdecimal is 1.4142 . . . andis called the square ro,ot of 2, or J2.With the circle, if w e give the diameter thevalue I, thecircumference will also always be of the incommensurabletype,3.14159 . . . which w e know by the Greek symbol 71, pi. 13.The principle remains the same in the inversion: if we give thefixed, rationalvalue of 1 to the diagonal of the square and to thecircumference of the circle, thenthe side of the square and theradius of the circle will become of the incommensur-able irrationaltype: 1/J2 and lln.It is exactly at this point that quantifiedmathematics and geometry go theirseparate ways, because numericallywe can never know exactly the diagonal of thesquare nor thecircumference of the circle. Yes, we can round-off after acertainnumber of decimal places, and treat these cut off numberslike any other number,but we can never reduce them to a quantity.In geometry, however, the diagonaland the circumference, whenconsidered in the context of formal relationship(diagonal to side;circumference to diameter), are absolutely knowable,self-evidentrealities: 1 : J2 and 1:n. Number is considered as aformal relationship, and this typeof numerical relationship iscalled a function. The square root of 2 is the functionalnumber ofa square. Pi is the functional number of a circle. Philosophicgeometry-and consequently sacred art and architecture - is verymuch concerned with theseirrational functions, for the simplereason that they demonstrate graphically a levelof experience whichis universal and invariable. The irrational functions (which wewill consider rather as supra-rational) are akey opening a door toa higher reality of Number. They demonstrate that Numberis aboveall a relationship; and no matter what quantities are applied tothe side andto the diameter the relationship will remaininvariable, for in essence this functionalaspect of Number isneither large nor small, neither infinite nor finite : it isuniversal.Thus within the concept of Number there is a definite,finite, particularizing powerand also a universal synthesizingpower. One may be called the exoteric or externalaspect of number,the other the esoteric or inner, functional aspect. Let us look atthe first four primary numbers in this spirit. The number O NE canof course define a quantity; as, for example, one apple.But in itsother sense, it perfectly represents the principle of absoluteunity, and assuch has often been used as the symbol to representGod. As a statement of form itcan in one sense represent a point -it has been called the pointal number, the binduor seed in theHindu mandala - or in another sense it can represent the perfectcircle. T w o is a quantity, but symbolically it represents, as wehave already seen, theprinciple of Duality, the power ofmultiplicity. At the same time it has its formalsense in therepresentation of a line, in that two points define a line. T HREEis a quantity, but as a principle it represents the Trinity, avital conceptwhich we will meet again later. Its formal sense isthat of the triangle, which isformed from three points. With threea qualitative transition is made from the pure,abstract elements ofpoint and line to the tangible, measurable state which is calledasurface. In India the triangle was called the Mother, for it is themembrane or birthchannel through which all the transcendent powersof unity and its initial divisioninto polarity must pass in orderto enter into the manifest realm of surface. Thetriangle acts asthe mother of form. But three is yet only a principle of creation,forming the passage between thetranscendent and the manifestrealms, whereas FOUR represents at last the first bornthing, theworld of Nature, because it is the product of the procreativeprocess,that is of multiplication: 2 x 2 = 4. As a form, four isthe square, and representsmaterialization. The universality ofNumber can be seen in another, more physical context. Welearn frommodern physics that from gravity to electromagnetism, light, heat,andeven in what we think of as solid matter itself, the entireperceptible universe iscomposed of vibrations, perceived by us aswave phenomena. Waves are puretemporal patterns, that is dynamicconfigurations composed of amplitude, intervaland frequency, andthey can be defined and understood by us only through Number.Thusour whole universe is reducible to Number. Every living bodyphysically 14. : This Japanese Zen calli-graphic drawingbeautifullyshows creation through thesimple progression fromtheUnity of the circle, throughthe triangle, to the manifestform ofthe square.I. .vibrates, all elemental or inanimate matter vibratesmolecularly or atomically, andevery vibrating body emits a sound.The study of sound, as the ancients intuited,provides a key to theunderstanding of the universe. Weve noted already that the ancientsgave considerable attention to the study ofmusical harmony inrelation with the study of mathematics and geometry. Theorigin ofthis tradition is generally associated with Pythagoras (560-490 BC)andhis school, yet Pythagoras may be considered as a window throughwhich we canglimpse the quality of the intellectual world of anolder, eastern and mideasterntradition. For this line of thinking,the sounding of the octave (an octave is for ex-ample twosuccessive Dos on a musical scale) was the most significant momentofall contemplation. It represented the beginning and goal ofcreation. What happenswhen we sound the perfect octave? There is animmediate, simultaneous coincidingof understanding which hasoccurred on several levels of being. Without any inter-vention ofthought or concept or image, we immediately recognize therecurrenceof the initial tone in the form of the octave. It is thesame note, yet it is different; itis the completion of a cycle, aspiral from seed to new seed. This timeless, instan-taneousrecognition (more accurate than any visual recognition) isuniversal amonghumans. But something else has happened as well. Aguitarist sounds a string. He nextdepresses this string with hisfinger exactly at its midpoint. He sounds the half-string.Thefrequency of vibrations produced is double that given by the wholestring,and the tone is raised by one octave. The string length hasbeen divided by two, and the number of vibrations per second hasbeen multiplied by two : 112 has created its mirror opposite, 211.Thus in this moment an abstract, mathematical event ispreciselylinked with a physical, sensory perception; our direct, intuitionalresponseto this phenomenon of sound (the octave) coincides with itsconcrete, measured definition. Hence we experience in this auditoryperception a simultaneous interwovenness of interior with exterior,and we can generalize this response to invoke the possi- bility ofa merger of intuitional and material realms, the realms of art andscience, of time and space. There may be another such moment in thecreated world, but the Pythagoreans did not know of it, nor do we.This is the essential spirit of the perception of Harmony, and forthe Pythagoreans it was the only true supernatural moment: atangible experience of the simultaneity of opposites. It wasconsidered to be true Magic, an omnipresent and authentic mystery.15. It was by means of geometry that the Pythagoreans poisedthemselves at this unique transition where heard vibration becomesseen form; and their geometry, as w e shall see, explores therelationships of musical harmony. Although interwoven in function,our two major intellectual senses, sight and hearing, use ourintelligence in t w o completely different ways. For example, withour optic intelligence, in order to form a thought we make an imagein our mind. Hearing, o n the other hand, uses the mind in animmediate, unimaged response whose action is expansive, evok- ing aresponse 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 toforget that it is also involved when the reason perceives invariantrelationships. Therefore when we place the auditory capacity at thecentre of our sensory experience w e can become aware that it ispossible to listen to a colour, or to a movement. This intellectualcapacity is quite different from the visual, analytical andsequential one we normally employ. It is this capacity, which isassociated with the right hemisphere o f the brain, that recognizespatterns in space, o r wholes o f any kind. It can perceiveopposites in simultaneity and grasp functions which to the analyticfaculty appear irrational. It is in fact the perfect complement o fthe left hemisphere, visual, analytic capacity, for it absorbsspatial and simul- taneous orders while the left rational facultyis ,best suited t o grasp temporal, sequential organization. T h eesoteric, functional aspect of Number, for instance, , would beapprehended through the right hemisphere faculty, while theexoteric, enumerative aspect of Number is apprehended by theleft.This innate intellectual quality resembles very closely whatthe Greeks called Pure Reason, o r what in India was called theheart-mind. The ancient Egyptians had a beautiful name for it, theIntelligence of the Heart, and to achieve this quality ofunderstanding was lifes implicit goal. The practice of Geometry,while also utilizing the analytic faculty, uses and cultivates thisaudial, intuitive aspect of mind. For example, one experiences thefact of geometric growth through the image of the square with itsdiagonal which forms the side of a second square. This is anunreasoned certainty absorbed by the mind from the actualexperience of executing the drawing. The logic is contained withinthe lines o n paper, which cannot be drawn in any other way.Asgeometers, equipped only with compasses and straight-edge, we enterthe two-dimensional world o f the representation of form. A link isforged between the most concrete (form and measure) and the mostabstract realms of thought. By seeking the invariable relationshipsby which forms are governed and inter- connected w e bringourselves into resonance with universal order. B y re-enacting thegenesis o f these forms w e seek to know the principles ofevolution. And by thus raising our o w n patterns o f thought tothese archetypal levels, w e invite the force of these levels topenetrate our mind and thinking. O u r intuition is enlivened, andperhaps, as Plato says, the souls eye might be purified and kindledafresh for it is by it alone that w e contemplate the truth.t --,c- -.. . %. -. - V"Numbers are the sources of form and energy inthe world.They are dynamic and active even among themselves . ..almost human in their capacity for mutual influence. (TheonofSmyrna.) Numbers, in the Pythagorean view, can beandrogynous orsexual, procreators or progeny, active orpassive, heterogeneous orpromiscuous, generous or miserly,undefined or individualized. Theyhave their attractions,repulsions, families, friends; they makemarriage contracts.They are in fact the very elements of nature.The tools ofu geometry and number represent the means to attainknow-ledge of both external and internal space and time.Theseinstruments, once used by architects and philosophers, be-
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