tu fiuide flBe$iltlteri
llniverte the [unfirtlctiru
tu fiuide flBegiltlteri
Iullrtru rtlng Ifl |jlllve rre Ihe ffiuthemuticul flrchetgpBr nfllnttlre, frt,rlld Icience
[|lrttnrr I ftttnn[tR HARPTFI N E W Y O R K
L O N D O N
T O R O N T O
S Y D N E Y
Tomy parents, SaIIyandLeonard, loae, for theirendless guidance, snd encouragement.
A universal beauty showed its face; T'he invisible deep-fraught significances, F1eresheltered behind form's insensible screen, t]ncovered to him their deathless harmony And the key to the wonder-book of common things. In their uniting law stood up revealed T'he multiple measures of the uplifting force, T'he lines of tl'reWorld-Geometer's technique, J'he enchantments that uphold the cosmic rveb And the magic underlving simple shapes. -Sri AurobitrdoGhose(7872-1950, Irrdinnspiritunl {uide, pttet) Number is the within of all things. -Attributctl to Ptlfltngorus(c. 580-500s.c., r nrrd rtutt Ircnutt i cinn) Greek Tililosopthe f'he earth is rude, silent, incomprehensibie at first, nature is incomprehensible at first, Be not discouraged, keep on, there are dir.ine things well errvelop'd, I swear to vou there are dirrine beings more beautiful than words can tell. -W nl t W hi tmm fl 819-1892,A meri cnnnoet) Flducation is the irrstruction of the intellect in the laws of Nature, under which name I include not merely things and their forces, but men and their ways; and the fashioning of the affections and of the r,t'ill into an earnest and loving desire to move in harmony with those laws. -Thomas Henry Huxley ( 1B25-1 Bg5, Ett glislthiologist )
hntellts ACKNOWLEDGMENTS
xt
GEOM€TRYAND THE QUEST FOR R E A L I T YB Y J O H N M I C H E L L
xiii
INTRODUCTION
xvii
1
MONAD
WHOLLY ONE
2
DYAD
IT TAKES TWO TO TANGO
2I
3
TRIAD
THREE.PARTI.{ARMONY
58
4
TETRAD
MOTHER SUBSTANCE
60
5
PENTAD
REGENERATION
96
6
HEXAD
STRUCTURE.FUNCTION.ORDER
178
7
HEPTAD
ENCI.IANTINGVIRGIN
221
8
OCTAD
PERIODIC RENEWAL
267
9
ENN€AD
THE HORIZON
5OI
DECAD
BEYOND NUMBER
323
NOW THAT YOU'VE CONSTRUCTED TIIE UNIVERSE
347
10
EPILOGUE
1
ix
Acknrwled I'd like to expressmy sincere appreciation and thanks to the good people who produced this book at HarperCollins, especiallymy hardworking editots Rick Ko! Alice Rosengard, and Eamon Dolan, and copy editor JeffSmith. I'd also like to thank Nancy Singer for helping celebrategeomefry in the way she designed the bool and Gail Silver for her insightful page layout. And many thanks to my astute agents, Katinka Matson and John Brockman. I especially wish to express my gratitude to John Michell, whose books, lectures, friendship, intelligence, humor, and vision have inspired me and manv others to explore the harmony in nunibers and shapes and *ho generously gave this book its main title and preface. The works of many peoplefrom the deeppast and immediate present have impassioned my interest in this subject, and I would like to acknowledge my debt to many including Keith Critchlow, John Anthony West, Robert Lawlor, Matili Ghyka, D. W. Thompsory T. A. Cook, ]ay Hambidge, M. C. Escher, R. Buckminster Fuller, David Fideler, Iean and Katherine LeMe6, Rupert Sheldrake, Rachel Fletcher, R. A. Schwaller de Lubicz, Sri Aurobindo, Pythagoras, and plato, as well as the artisans and architects of ancient Egypt and Greecewho knew how to make harmony visible. For years of deeply interesting conversatiory fun, friendship, love, and support I'd tike to offer warm thanks and love to my brother, Jeffrey,and my parents and to my friends who live in the canyons on that island off the mainiand, New York City, especially Naomi H. Cohen, fim pittman, Libby Reid, BarbaraCrane,Charlie and Evelyn Herzer,Mark Haj-
xii
ACKNOWLEDOHENTS
at seldis, and June Cobb of the SacredGeometry Library Se C"utn"at"f of St.lohn the Divine. To my friends embracedby the wide sky and natural order of Baker,Nevada' hearttelt ttlut*t fo. tfn" perpetual welcome and the tools with which to leam, i"ctudi"i naryn M. DeBit, Anita Hayden,-and Jim Dalton. I'd also hle tothank my. friends, fellow educators' and students for sharing a dozen lovely years in the green- ' of Gainesville,Florida. erv I'd liku to ."^umber my teachers,who tried as best they could to get me to like math, and my many students,in geometryivorkshops, from whom I have leamed so much' " Andi wish to eipress my wonder and thanks to nature' whose geometric jewelry adoms the world'
and fienmetry flealit thefluestfur lohn Michell
Sooner or later there comes a time in Me when you start thinking about Reality and where to find it. Somepeople tel l you there is no such thing, that the world has nothing permanent in it, and, as far as vou are concemed, consists merely of your fleeting experiences.Its framework, they say, is the random product of a nafural process,meaningless and undirected. Others believe that the world was made by a divine Creator,who continuesto guide its development.This soundsa more interesting idea, but, as skeptics point out, every religion and church that upholds it does so by faith alone. If you are naturally faithful and can accept without question the orthodoxy of your particular religion or system of beliefs, you will feel no need to inqufue further and this book will appear superfluous. It was written for those of us who lack or have lost the gift of simple faith, who need evidence for our beliefs. We cannot help being athacted by the religious view, that the world is a harmonious, divinely ordered creation in whidr, as Plato prornised the uninitiated, "things are taken care of far better than you could possibly believe." Yet superficially it is a place of confusion and chaos, where suffering is constant and the ungodly flourish.
xlv
AND THEOUESTFORREALIry OEO}IETRY
This is where we begin the quest for Reality' Lgokinq closelv at nature, the firsi insight we obtain is thai, behind the apparently endlessproli{eration of natural objects,there is a flr lesser number of apparently fixed tlpes' We see' for example,that through every generationcatsarecatsand are for catlike behavior. In the same way, every progrlmmed 'ro"""hu" the unique characteristics of a rosb and every oak leaf is definitely an oak leaf. No two specimens of these -are ever exactlythe same,but eachone is clearly a product of its formative i1pe. If it were not so, if animals and plants,simply inherited their progenitors' characteristics,the order of nature would soon dissolve into an infinite variety of creatures, undifferentiated by speciesand kinships' This observation, of one type with innumerable products, gives rise to the old philosophical problem o{ the One and the Manv. The problem is that, whereas the Many are visible and tangible ind can be examined at leisure, the One is never seen or sensed, and its very existence is only inferred tfuough the evident effect it has upon its products, the Many. Yet, paradoxically, the One is more truly real than the Many. In the visibte world of nature all is flux' Everything is either being bom or dying or moving between the two processes.Nothing ever achievesthe goal of perfection o. ti'," ttute of equilibrium that would allow it to be describedin essencb.The phenomenaof nature, said Plato, are always "becorning," never actually " are." Our fiue sensestell us that they are real, but the iniellect judges differently, reasoning that the One, which is constant, creative, and ever the sarne,is more entitled to be called real than its ever-fluctuating products. The search for Reality leads us inevitably toward the type, the enigmatic One that lies behind the obvious world of*re Many. Imrnediately we encounter difficulties. Being imperceptible and existing only as abstractions, tyPes cannot be apprehendedby the methods of physical science.A number of modem scientists, perceiving the influence of types in nature, have attempted to bring them within the range of empirical study. Rupert Sheldrake, author of A Nezu Science of Life and other works, has taken a bold step in that direction.In an earlier age,the Pythagoreansworked on systematizing the types by means of numerical formulae. Yet
SEOiETRY AND TH€ OUESTFOR REALITY
Plato, who wrote at length on the subject of constant types (referred to as "forms"), was carefully ambiguous in defining them and never made clear the means by which they influence the world of appearances. Plato did, however, give instructions on the procedue toward understanding the nature and function of the types. In the Republiche described the ascent of the mind through four different stages.It begins in Ignorance, when it does not even know that there is anything worth knowing. The next stage is Opinion, the stage in which TV chat-show particiDants are forever stuck. This is divided into two subcategories, Right Opinion and Wrong Opinion. Above that is the Ievel of Reason.By education and study, particularly in certain mind-sharpening subjects,the candidate is prepared for entry into the fourth stage, which is called Intelligence (nous). One can be prepared for it but with no guaranteeof success,for it is a level that one can only achieve on one's own, the level of heightened or true understanding, which is the mental level of an initiate. The studies that Plato specified as most effective in preparing the rnind for understanding are the so-calied mathematical subjects, consisting of number itself, music, geometry, and astronomy. These were the main studies of Plthagoras and his followers, who anticipated the realization of modem physics in proclaiming that all scales and departrnents of nature were linked by the same code of number. Geometry is the purest visible expression of number. In Platonic terms, the effect of its study is to lead the mind upward from Opinion onto the level of Reason,where its premises are rooted. It then provides the bridge or ladder by which the mind can achieve its highest level in the realm of pure Intelligence. Geometry is aiso the bridge between the One and the Many. When you draw one of its basic figures-a circle, say, or a triangle or regular polygon-you do not copy someone else's drawing; your model is the abstract ideal of a circle or triangle. It is the perfect form, the unchanging, unmanifest One. Below it are the Many-the expressions of that figure in design, art, and architecture. ln nature also the One circle gives rise to the Many, in the shapes and orbits of the planets, in the roundness of berries, nests, eyeballs, and the
xvi
GEO}IETRYAND THE OUESTFOR REALITY
cycles, of time. On every scale, every nafural pattern of growth or movement conforms inevitably to one or more of the simple geomerrictypes.The pentagoryfor example,Iies behind each specimen of the five-pelaled rose, the five_ fingered starfish, and many other living forms, whereas the sixfold, hexagonal gpe, as seen in the structure of snowflakes and crystal generally, pertains typically to inan_ imate nature. As soonasyou enter upon the world of sacred,symbolic, or phiJosophicaigeometry-from your first, thoughifuJcon_ struchon of a circle with the circumferencedivid;d into its natural six parts-your mind is opened to new influences that stirnulate and refine it. you begin to see,asnever before, the wonderfuily patlerned beauty-of Creation. you see true artistry, {ar above any human contrivance. T?risindeed is the very, sourceof art. By contact with it your aestheticsensesarc heightenedand set_uponthe firm bisis of truth. Beyond the oovlous pteasureot contemplatingthe works of nafure_the MTy-1" the delight that comes through the philosophical study of geometry, of moving toward-the prr".,." of th" One. Michael Schneideris an experiencedteacherand, as you areentitled to expect,a masterof his geometriccraft. No Lne less qualified could set out its basic piinciples so ctearty anJ simply. His much rarer assetis appieciation of the symbolic and,cosmologicals].'rnbolisminjrerent in geometryiThat is the best reasonf9r being interestedin the subject,and it is why. rhe.philosophers of ancient Gieece,Egypt, 1"_,t:i:." and other civilizations made geometry and numbei'the most important of their studies. The traditional science taught in their mystery schoolsis hardly known todav.Itis not available for study in any modem place of educition, and the.reis very little writing on the sublect.ln this book you will find something that cannot be obiained elsewhere, a complete introduction to the geometric code of nature, written and illustrated by the most perceptive of its modem investigators.
lntrndurt FANNINGANCIENTSPARKS
Thr*ort thing incomprehensible abouttheuniaerseis that it is comprehensible. -Albert
Einstein (1,879-1955)
Nothing in educationis soastonishing asthe amountof ignorance it accumulates in theform of inertfacts. -Henry Brooks Adams (183&-1918, American historian)
The universe may be a mystery, but it's not a secret.Each of us is capable of comprehending much more than we might realize. A vision of rnathematics different from that which we were taught at schoolholds an accessiblekey to a nearby world of wonder and beauty. In ancient Greecethe advanced students of the philosopher Pythagoras who were engaged in deep studies of natural science and self-understanding were called,mathematekoi,a'thosewho studied all.l' The word mathemasigntfied "leaming in general" and was the root of the Old English mathein,"to be aware," and the Old German munthen, "to awaken." Today,the word math has, for most people, constricted its scope to emphasize mundane measurement and mere manipulation of quantities. We've r;nwittingly traded wide-ranging vision for narrow expertise. It's a shame that children are exposed to numbers merely as quantities instead of qualities and characterswith distinct personalities relating to each other in various pattems. If only they coirld seenumbers and shapesas the ancierrtsdid, as s)zmbolsof principles availableto teachus about the naturai structure and processesof the universe and to give us perspective on human nature. Instead, "math education" for children demands rote memorization of procedures to get one "right answer" and pass innumerable "skill tests" to prove superficial mastery before moving on to the next isolated topic. Teacherscall this the "drill and kill" method. Even its terrninology informs us that this approach to math is full of problems. It's no wonder countless people are irmumerate. We've lost sight of the spiritual qualities of number and shapeby emphasizing brute quantity.
XYIII
This book concemsmathematics,but not the kind you were shown at school. The Roman goddessNumeria is said to have assistedin teaching each child to count' We must have had a misunderstanling becauseI grew uP on uneasy terms with the subject of irathematicJin schbol. I was intrigued b.y.the inJinity of numbers and could calculate mechanically if I had Bodily ,rrrcise,when for a given situation' doesno harm memorized the rule I was taught compulsory, intimidated me' I math art, and Although I liked science to the body;but knowledge remem6er my frustrated tears at age seven over the concePt tohichis acquiredunder of subtracting by borrowing {rom another column when it contained a zero.l dreaded math, its mindless memorizacompulsionobtainsno tion and its tedious paperwork' (Pify the teacherswho have holdon themind. to check it!) Math was dry and mechanical and had little rel-Plato (c. 428-348e.c.,Greek evanceto my wo d. If we had iooked at numbersto seehow they behave with each other in wonderful patiems I might mathematical philosopher) have liked math. Had I been shown how numbers and shapes relate to the world of nature I would have been ihdlled. hstead, I was dulled by math anxiety and pop ottizzes. Fortunately for me, when I was sixteen one of my teachers mentioned that mathematics can be found in nature: a shaped like a bee's cell and quartz Wrao* shouldbe six-sided snowflake is to make the cormection. How could I was stunned crystal. asa meansof cherished something as apparently irrelevant as mathematics be traaelingfrom youth to old related to something as wonderful as nature? The ordinary world opened up to rne and spoke the language of number age,for it is morelasting mathematics thanany otherpossession. and shape. No longer a foe, the dreaded what it wasn't tool. Nature and a a teacher becameat once -Bias of Priene (c.570B-c-, was made out to be, an antagonist to fear, conquer, and exploit, but a garden of wonder and a patient teacher worone of the SevenSagesof ancient Greece) thy of gteat respect. Over the years rny studies led me to seethat a profound understanding of number was prevalent in ancient times, more than is commonly acknowledged, and seamlessly wove mathematics, philosophy, ar! religiory myth, nature, science, technoloW, and everyday life. This book is a fanning of some little sparks of philosophy from deep antiquity to introduce ihe general reader to another view of mathematics, nafure, and ourselves that is our heritage and birthright. It requires that we liberate math from its Pigeon-
xix
CqlvinonaHobbes
by Bill Wotterson
'(EAl{. A\! 1tl6E EAUAnoN: IHls ul\{nE Bcd( ts N\! OFII{INGSTHATHA\IE AF€ L\KEMIRAC,LES, \C'\) lb IAG TTION\}USgSA\OiI\ITN E. Ak 2\
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dentally or by trial and error. The oldest construction on paper was taught by Euclid in hjs Elemmts (c. 300 r.c.). although much older Babylonianclay tabletsdescribingthe steps have been unearthed. Periodically throughout history the construction of the pentagon has been so revered that it was kept secretby var_ ious. societies who recognized its predominance in. nature and its potent effecton human natuie. They studied the.pen_ tad's principles in geometry and nature,and its psycholoe_ ical effect,deeply enough to seein its application t ,o*'i_ edge that could be misused.Around 50d i.c. the pentagram " star was the sign used by advanced members oi the rythagorean society to recognize one another, and one thou_ san_cl years later it was a guarded teachingtool only taughr orau_y, not written about,by the craftsguiJdswho in?used*its symbolism into the designs of Gothic cathedrals. It wasn,t until 1509when the monk Fra Lucapacioli, inventor of dou_ ble-entry bookkeeping and the mathematics teacher of Leonardoda Vinci, published De Dioina proportione, that the method of its constructionand unique geometricproperties was_revealedto artists and philosophG in public. out in the open again. Use tire geometer,stools, ard . ,,It's tortow these steps (seeillustrations on the next pages) ' to construct the pentagon and pattems that come from It. Some constructions of the pentagon you may come acrossare actualJyonly approximate, althoueh very close, but this method producesone that is perfectly froportioned.
REG€N€RATION
r05
Construct the pentagon. Follow these steps with the greatest precision and the sharpest pencil point.
itself, encompassing 1. Construct a z:esica piscis. and interpenetratlarge Draw a line connect- ing the four states of matter. ing the circles' centers and a vetical 3. Using the sarne line up the middle. compass opening, put its point at the 2. Put the point of the compass where center of the large the lines cross, and circle, and turn open it to the center another small circle, of one of the circles. making a small Turn a circle within aesicapiscis. tl,:'eaesicapiscisas you did to consuuct 4. Draw a vertical a square. The penta- line up the small gon represents the oesicapiscis.Find its "fifth essence,"life center where it
6. Put the compass point atop the small circle, and open the pencil to the point 5. Put the unmoving leg of your com- created on the circle's diameter. pass at the center of pisSwing the compass oesica Ihe small upward until it cis, and open its crossesthe small pencil to the point atop the small circle circle. This new point and the point within the large atop the circle are aesicapiscis.Swing the pentagon's first the compass downward until it crosses two comers.We the horizontal diam- have organized Monad by crystaleter (or complete Lizing the Pentad the arc to the full within it. circle).
crossesthe line betweencenters.
TH€ UNIV€RS€ GUIDETO CONSTPUCTINO A BEGTNNER'S
106
7. Using the same compassoPening, walk the compass around the circle to find each new Point until all five Points are identified. Your pencil's final steP should end precisely at the toP where it began. (Helptul Tip: This construction requiresgreat preci sion. If your com-
pass was off even slightly at any stage, the difference will be magnified and the pentagon loPsided. To double' check, walk the compass around the circle in the opPosite direction. If there's any differencebetween the marks made in each directiory the actual points of the penta-
gon are found halfway between them.) 8. Connect the five points to make the pentagon visible. Welcomethe emissary of living structures,the geometric flag of life itself. 9. Corurect alternating points to
reveal the pentagram star. 10. A varietY of pentagonal forms can be constructed by this method, eachof which expressesthe Pentad's principles. Practice constructing thern. Save one as a template for producing others quickly
@ o o @f f i * *
R€OEN€RATION
BUILD A DODECAHEDRON (Greek for "two The fifth Platonic volurne is the dodecahedron plus ten faces"). Like the four other volumes, it expressesthe principle of equality in all directions, having twelve identical regular pentagonal faces,thirty equal edges, and twenty vertices rneeting at three-comer joints. It is closestof the five forms to having the volume of the Monad's perfect sphere. As "quintessence" it isn't aboaet}jrefour elements, the configured statesof matter, but encompassesthem, infusing the force of life and excellencethroughout their structures. Even nonliving crystalline minerals take part in the cycles of life when they're ingested by plants, animals, and humans, thereby becoming incorporated into our bodies. Like the dodecahedrory life's cycles encompassall the universe in its processes. Today, we encciunter dodecahedra, if at all, as those twelve-month desk-calendar paperweights and as special dice. The ancient mathematical philosophers revered the dodecahedron for its geometric properties and s).'rnbolism. Like a pomegranate ready to burst forth its seeds, the dodecahedron representsthe archetype of life and fecundity made visible. It's a three-dimensional pentagonal web on which life expressesits fullness. The best way to appreciatea dodecahedronis to build one. Unlike triangles, squares, and hexagons, pentagons cannot fill flat spacewithout leaving gaps. But twelve pentagons will enclosethree-dimensional space as the Quintessenceenclosesthe elements.
107
108
TH€ UNIV€RS€ A SEOINNER'S OUIDETO CONSTRUCTING
Sorrotrcsaidthat,from aboTre, theEarthlookslike oneof thosetweloe-patched luthernballs.
Scientists have recently constructed dodecahedral molecules using carbon and various metal atoms that join to form larger molecules. Recently, a third form of pure carbon beyond graphite and diamond was discovered and formally named "Buckminsterfullerenes," or 'lBuckyballs," after the mathematiciary architect inventor, and visionary R. Buckminster Fuller, who used similar shapesfor his geodesic domes. A Buckyball consists of sixty carbon atoms,fiae arowrd each corner of a dodecahedron.
Dodecahedral skeletons of microscopic radiolaria.
r09 Hold and tum the dodecahedron to appreciate the beauty of its lines and surfaces. Get quiet within yourself and be aware of feelings this fifth Platonic volume evokes.
WHOLES IN NATURE'SFABRIC Avoid extremes-keep the Golden Mean.
-lleobulusofLindus (sixthcmturyB.c., oneoftheoSeam ,,y;6ffi
Nature expressesthe cosmic phitoroptry tt geometry. "orrgh A deep look into the star's unique geometry will take us even closer to the core of the archetype of the Pentad. There, perhaps, we can discover the source of its power and the secretof its appearancein nature and its psychological effect on us. A clue is found in the ability of vegetation and certain creafures to regenerate another whole like themselves from a part, not only from their seeds. For example, when put near soil, the leaves of some plants send out roots and grow whole new plants. Flatworms and other creatures with prirnitive nervous systems also regenerate new creatures from their parts. Most familiar is the ability of starfish to regrow a lost leg, while the lost leg grows a whole new body. We can seethe principle of regeneratiory the whole in the part, in many places. The veins in a leaf reveal the branching pattem of the whole leafless tree. Each ridge of a fem's leaf models at once the whole leaf and the whole plant. Each seedon a dandelion'shead mimics the whole head in miniature. Il/ho hasn't noticed that the tips of broccoli and cauliflower resembleminiature trees? Look at the top of a fulI broccoli or cauliflower head while squinting and you can see the overall pentagonal groupings. Five major clumps contain constellations of smaller and smaller pentagonal structu-res:each clump is made of five smaller clumps, and so on. This is the rhythm by which vegetation grows, a pulsing fivefold rhythm. Each successively smaller branch models the others and the whole. This whole-in-the-part principle is the basis of bonsai, the Japaneseart of growing miniature trees and bushes from the branches of a large tree. A tree's leaves swirl around their
The veins of a leaf, dandelion seeds,and branches of broccoli model the whole plant in eachpart.
THE,UNIVERS€ A B€GINNER'SGUTDETO CONSTRUCTIN9
twies in the samepattern that twigs swirl around branches' asb"ranchesaroun-dthe boughs, asboughs emergefrom the tmnk. Each taken by iiself is a model of the whole' The next time you can dwell with a bush, look for this structural selfsyrtttttetry. Distinguish each level, and build your vision l-Ut yori can seJ a[ the structure comprehended as the whole. Cognizance of this harmony in nature and rnathematics attunes us to harmony at our owrl core. Today,we limit the word "symmetry" to mean a simPle rnirror image, such as our two hands. But the ancient Greek word symmetriallterallymeant "alike measure,"a term pelfectly zuited to these expressions of self-similarity on different scales. This fascinating characteristic of living forms curiously corresponds with the geometric property of the PenBFam to repl-icateforever smaller and larger versions of itself' The star ieplicates and inverts itself in its own central pentagory whictr in tum has its central Pentago& in an endlessly diminishing series of inverting stars. Flip the centrai pentagram outward and it transforms at each of the star's five points into a whole pentagram' These become nuclei for smaller stars to replicate within, in its center and around its points, multiplying endlessly. The Pentad holds the principles of the geometry of regeneration' It is
111
nature's way of producing endless variety using a single self-reproducingscheme. Pentagonsperpetuate their own image in endless detail. We are dranm into them as into the eyes of a lover, as we see the whole by looking at the part, Know one star and know the whole. This self-similar accord, repeating the same shape on different scales,is the basis of {ractal mathematics, which is at the heart of modern chaostheory. For example, the microscopic bumps on a grain of sand at the seashoreresemble an aerial view of the whole beach'sshoreline.Each small part of a turbulent system tums out to be a model of the whole and can be described mathematically by self-replicating formulas. Even in chaos we find the signature of cosmos. This part-as-wholeoccursin thelcientific art of producing holograms. Unlike ordinary photographic negatives, in which a piece from the comer holds only a small part of the picture, any piece cut from a hologram contains the information of the whole scenein miniafure. Explore the Pentogonb Regenerotion Construct a pentagon or trace the five comers of any illustration, and connect them to enclose a pentagram star. Then do theseactivities: Doodle pentagonsand pentagrams.Explore pentagons on your own. Draw them freehand, with each hand, as a continuous line. Extend lines and
The pentagram exhibits selfstnilarity in endless variations. Explore its regenerating geometry on your own by subdividing each part into smaller pentagrams.
112
A BEGINNEE'S GTI'I]!
I belieaethegeometrit proportionseroedthe Crmtor asan idm whenHe introduced thecontinuous generationof similar objectsfrom similar objects. -Johannes Kepler
Intuition is theclear conception of thewholeat once, -Johann Lavater (1741,-1801, >wss poet, mystic, and philosopher)
,6
connect points to seehow the star regeneratesitself endlessly into smaller and larger moiels of the whole. The pentagon is extreriely rich in enJlessly lascinating self-replicatinggeomefry that is accessi_ o1" y,rh u.."13.ppencil point. Discover its galaxies ot stars, all identical in shape, differing on[z in size. 1. Decomposeand fold a penragon. 2. Geometricallyconstructa pentagonand connect rrs comers to inscribea pentagram star. 3. Connect the comers of its central pentagon to draw a smaller but inverted pentagram. 4. Extend the lines of this inner pentagram out_ * they meet the sides'of the"largepentaI^1-.9 gon (see*r, iltustration on page 111).Use this as a tem_ plate to makeotherpentagrams. Use the straigLrtedgeto creasesharply all *re ,. rmes both ways. Feeland experiment *iih the p"n_ tagon's tendency to fold various ways. Get all ire points to_meetat the top and let them sprine Ua;k; thls shou.ldremind you of a flower bud ope-ning. the star tnto pieces.Construct another sJar, and,Lut cut the whole into piecesalong the lines. Re_ arrangethem to make smaller and larger starsand pentagons. Notice their interrelationsiips. parts added together match exactly with larger ones. . Construct dodecahedra. ilepeat the"corrst u"-
apentason. cut u srituro.,jrn"
li:1,of 1"::p::ys srde eachof the small outer triangles(seeitius_ puUt113)potathesidesipward,anJ
Iif,"" 9"small strp.each triangle under (or over) the piece next to it. Clue or tape it there as a tab, corurectine me whole into a ,,basket,,resemblingan open flower. Two of_thesebasketsjoined fice_to'_face will rorm a dodecahedron.Or twelve of these baskets can be joined back-to-backby their sides to _rke u targerdodecahedron.With a needleand thread vou can sew stuaightlines in three dimensions betwJen
fi3
the points and comer folds of the pentagonal "basket" to reveal unexpected and beautiful geometric relationships within it.
m r\,,,n
r-v/t\-4
e#
Discover Pentogonot Setf-SUmfnetrgin Noturol Forms Each life form below is enclosed within a pentagon. Use a straightedge to connect its five corners to inscribe a pentagram star. Further decompose the pentagons and notice the remarkable detail with which the creature exDressesits archetypaldesign,and how eachsmall part is structured as the whole. Some examples have been completed to different degreesto show you how it's done.
14
Radiolarian
Starfish
Starfish mouth
Stin$ay
REGEN€RATION
THERABBITRIDDLE It is worihwhile for anyoneto havebehind him a few generationsof honest,hard-working ancestry (1893-1'960, noaelist) -lohn PhiltipsMarquafld American The ways in which numbers behave and interact reveal the most eisential workings of their archetype. A look at the actual numbers and prbcessesinvolved in pentagonal selfs1'mrnetry will take ui as close as we can get to understanding the piinciples of the Pentad. The key to this regenerative pentagonal giometry can be {ound in a particular. selfg"r,"ritit g ttirmbe. ieries known as the Fibonacci (fib-ohNAH-chee) sequence. Until the twelfth century, EwoPean commerce,banking, and measurement relied upon the clumsy system of Roman numerals, a vestige of the ancient Roman Ernpire, to calculate with. Try multiplying or even adding two Roman numerals together to seehow difficult it is even to begin. But in the year 1202a book, LiberAbaci(TheBookOf Computation), was published by a mathematician and merchant, Leonardo of Piia. This booi convinced Europe to convet to the numerals we use today, zerc through nine, known by their origin and Dath to the West as the Hindu-Arabic numerals. Because his father was nicknamed Bonaccio ("man of good cheer")' Leonardo was known by the Latin for sonof Bonaccio,"hlius Bonaccio,"contractedto "Fibonacci." Of interest to us from his book is an ancient number puzzle about breeding rabbits. Although there's no evidence that Fibonacci suspected it this Puzzle holds the key to selfreplicating growth in geometry and nature. The "rabbit riddle" essentially goes like this, The date is January 1 and a pair of newbom rabbits, a male and female, are in a pen. How many pairs of rabbits will there be one year later if newbom rabbits take exactly one month to mature, at which time they immediately mate, gestate for another month, then give birth to another pair like themselves, and mate every month thereafter? Every newborn pair repeats this pattem of rnonthly maturing, mating, gestating, and breeding other identical pairs, which likewise do the same. (We assume an ideal situation: no pair dies or deviates from the pattern.) A visual sketch of their family tueeis tJrebest way to sta*.
fl5
Tfi€ UNIV€RsE ABEGINN€R'SGUIDETO CONSTRUCTING
116
We soon become bogged down in the drawing of s1'rnbols representingrabbiipairs. But Fibonacciwanted us to notrce ilLenumbelsrepresenting their growth. A count of the newborn, mature and total rabbit pairs each month produces an interesting pattem, which recurs for each pair' \44ren we notice its secret-that each two consecutive numbers add together to produce the next number-we can solve the puizle easily by continuing this pattem to the next -Tanuarv1. fftit same pattem would recur if someone decided to soread a rumor in a crowd' If he thinks about it for thirty seconds and then tells it to someone who hasn't heard it every thirty secondsthereafter, and if everyone who hears it waits thirty secondsand then also passesit along every thirty secondi, the numbers of knowers, tellers, and hearers will increase by terms of the Fibonacci sequence. Other phe-
o
Newbom pair
O
Mature pair
Rule 1 I;r one month each newborn pair becomesmatu(e.
Rule 2 Each mature pair breeds a newborn pair and mates again.
Follow the two "rules" of growth to find how many pairs of rabbits there will be on January L. Notice, in the fashion of self-similarity, that if you break off arry "branch" it resemblesthe whole family tree.
'qrqqer ;o slrudgg7aqllllr araql 1 l,renwf u6 :raa,rsuy
January February
Total Pairs 1 0 1 0 1 1
March
1 . 1 . 2
Anril
t
z
J
May
2
3
5
June
3
5
8
Iuly
5
August
8
8 13
1
3 21.
117
nomena from handshakes in a crowded room to robotbuilding robots would do the same. The Fibonacci sequenceactually begins with two terms, zero and unity, nothing and everything, the Unknowable and the manifest Monad. Theseare the first two terms. Their sum, another unity, is the thfud term. To find eachnext term, just add the two latest terms together. This processproduces the endlessseries0, 1,,1,,2, 3, 5, 8, 13,21,,34, 55, 89,1,M,233, 377.610. ... Look at this series.At first glance we seea chain of numbers. But look beyond the visible numbers to the self-accumulating process by which they grow. The series grows by accruing terms that come fuorr.within itself, from its immediate past, taking nothing from outside the sequencefor its growth. Each term may be traced back to its begirming as in the Monad, which itself arose from the incompre""ity hensible mystery of zero. This principle of ongoing growth-from-within is the essenceof the Pentad's principle of regeneration and the pulsing rhyttr-ms of natural growth and dissolution. It appears in plants, music, seashells, spiral galaxies, the human body, and everything associatedwith the fiveness in naflfe.
Genealogy of the drone (male)honeybee.Some irsects and microscopic life expeience pafihenogmesis ("virgin birth") whereby unfertilized females give birth to males. But fertilized females always give birth to other females. This pattem maintains the d1'namic balance of the hive. The family tree of any bee branches in the accumulative Fibonacci growth rhythm. And each branch resembles the whole family history.
fl8
THE UNIVERSE A BEOINNER'sOUIO€TO CONSTRUCITNO
(('('( The average number of petals on each type of flower in a field will be Fibonacci numbers, as are the numbers of pine needles accumulating as clusters in different Pine sPecies.
x#,*
Sneezewort (Achillea ptarmica). Eachbranc}:. lengthens through time and then reproduces another like itself, which repeats the Pattem of Fibonacci branching.
€X€r
ss'\
# Petals 2
3 5 The piano keyboard is a metaphor of accumulating vibration and so is structured by terms of the Fibonacci sequence.The thirteen-note chromatic musical octave consists of eight white keys (whole tones) and five black keys (sharps and flats) arranged in groups of threes and twos making one full octave.
34
55 89
Flower Enchanter's nightshade kis, lilies, trillium Al1 edible fruits, delphinium, larkspurs, buttercups, colurnbines, milkwort Other delphiniums, lesser celandine, some daisies, field senecio Globe flower, ragwort, "double" delphiniums, mayweed, corn marigoid, chamornile Heleniums, asters,chicory doronicum, some hawkbits, many wildflowers Common daisies, plantains, gaillardias, pyrethrums, hawkbits, hawkweeds Michaelmasdaisies Michaelmasdaisies
fl9
REG€N€RAIION
THEOOLDENMEAN
great t'uro Geometryhas
oneis the treasures: Write out the first few terms of the Fibonacci sequence' ft".""aer eachto makeit a fraction,andundemeath Theoremof PYthagoras;the Jt"* " Fibonaccisequenceshiftedback oneterm' write the other,thediztisionof a line 55 34 into extremeandmean 34 2l ratio.Thefirst we maY of compareto a measure Use a calculator to divide and convert each fraction into gold;thesecondwe maY decimal form. to closer namea preciousjeutel. A graph of the results shows each termgetting
6 + + ?t g € #
an iaei ot f.ef903398875. ' . or rounded off to 1'518or even 1.62. Notice how ii begins crudely, pulsing far over then
-Johannes KePler
The Fibonacci sequencehas fascinating number ProPerties. For examPle, its Part resemblesthe whole. Use a calculator to divide the 1.6180339.. twelfth term, 89, into uniry The result is an endless decimal that, broken into parts, replicates the entire Fibonaccisequence:1/89 = ... 0.011235955056179 = the sum of
1.63 1.65 t63 162 161 1.53 t.51 1.56 1.55 ),53 t52 l.5l 1.50
Historical names of the Golden Mean: thesection Plato theextremeandffieanntio EucIid aurm sectio(goldm section) Romans the diaine proportion Luca Pacioli
ChristopherClavius Godlikeproportion fohannes Kepler
thediainesection
JohannF.Lorerrtz J.Leslie Adolf Zeising Mark Barr
theffiedialsection thegoldm cut
thecontinueddioision
o (phi)
.0 .01 .001 .0002 .00003 .000005 .0000008 .00000013 .000000021 .0000000034 .0000000005s .000000000089 .0000000000144 .00000000000233 .00000000000032 +...
T+tEUNIVERSE ABEGINN€R'SOUID€TO CONSTRUCTING
120
under then over the ideal, getting closer and closer on-its wav toward an inJinite at which it will never arrive' We Lir" coa,theDiuine *uich us its decimals get longer and longer, like a thirsty Proportionis alwaYs taproot reachingfor the infinite that beckonsit onward' I he similarto itself. farther out you go in the Fibonacci sequencethe more precise the result becomes as it hones into the ideal' -Fra Luca Pacioli The ideal it approacheshas had manyrlames over the centuries,often eipressing highest regard for it with terms including "golden," "divine" and even "Codlike'" lt is the prouerbiil i'qolden mean," the ideal balance of life' hresentlyit isil'rnbolized by the Greekletter Phi (O),named in this century to honor the Greek sculptor Phidias, who used it to proportion his designs, ranging from the Parthenonto his famous statueof Zeus. . We tend to Actually @is not a number but a rclationship Eorn o, thefinite focus on the visible numbers, but they rePresent t}:.eaccu' an infinite series, mulatiaeproicusthat manifests them. We could actually start encloses with any two numbers, not just zero and one, adding each And in theunlimited consec;ive pair to get the next term, yet the ideal limit will limitsappear, always approach IooI'raq1m;due pear no^ aroJag
091
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135
R€OENERATION
You'll need stiff cardboard (index-card material will work)/ a cutting tool, a needle and thread. Consfruct three identical golden rectangles. Cut slits aiong their centers as shown, long enough to slip another rectangle'ssmaller side through. The third rectangle, whose slit extends to one edge, slips around both to createthe three-dimensional frame. You miy wish to run tape aiong the ioints to strengthen the frame. Using a needle and thread you can sew through five corners to reveal the pentagram star (see iilustration) that sharesthe (Dratio with the golden rectangles. By cormecting with thread only the midpoints of short sides, you can outline the edges of an octahedron. And by connecting every comer, you can make the twenty hiangular facesof an icosahedron. Wiih ingenuity, you can also build the remaining Platonicvolumes on this frame. Discover O in Ctossic Art, Crofts, ond Architecture From their deep study of geometry and nature the ancient mathematicalphilosophersinfluenced artists, craftspeople, and architects. Since ancient times the (D proportion has
Regutatinglines. . . are ...aspfingboardandnot . . . theY a straightjacket satisfytheartist'ssense. . . nnd conferon theworka qualityof rhythm. -Le Corbusier (1887-19 65, Swiss architect, city planner, and writer)
I(nowledgeofa basiclaw giaesafeeling of sureness theartist to whichenables put into rmlizationdreams rahichotherwisewould in haoebeendissipated uncertainty. -Jay Hambidge
This Babylonlan stela depicting an initiate being led by priests into the presenceof the sun god contains many (D relationships.
134
A BEOINNER'S OUIDETO CONsTRUCTING TH€ UNIV€RS€
This sym-rtrycannotbe usedunconsciously althoughmanyof its shapes areapproximated by designersof great natizte ability whosesenseofform is highly dezteloped. -Jay Hambidge (1867-L924, American architect, geometer, educator,and author)
Art withoutknowledge is nothing! -Jean Mignot (Gothic master builder from Paris, addressing architects in Milan, 1398)
The west facade of the Greek Parthenon fiis within a golden rectangle. The United Nations Building in New York was designedas three golden rectangles, like three Parthenonsstackedupon each other, appropriate to the mission of promoting world harmony.
This circular Greek libation pan achievesvisual and geometric balance with its handle by the golden mean. Togethei they fit within the frame of a golden rectangledivided into a squire and smaller golden rectangle.
135
R€GEN€RATION
The west facade o{ the Cathe-
dral of Notre Dame in Paris is rich in golden mean relationships. Use the golden mean calipers to verifY them and find more.
shown up in artistic cornposition, it's simplest expression being a golden rectangle.The most complex designsemploy the diecompositionof the pentagon Somedesignsare higruy complex, such as those underlying Codric cathedrals ano They are comprehensiveand Dain'tinesof the Renaissance. iutirfulig, evm uplifting, due to the harmony inherent in *," n6o*1trv of self-replication to which our eye responds' Lse the golden mean calipers to verify and explore the O proportiois in the following cornpositions' Discover Pentogonot Composition in Art Of alJpeoples,the ancientEglptians were the most brilliant at integrating geometry, symbolism, art, myth, ancl lar*us" fi theii works. For example,their hieroglyph for the luoi. A. "underworld" within ourselves,is a star within a circle. So whenevet they created art that related to the duat ttey composed it on a large pentagonal grid decomposed into smiller pentagons and pentagrams. There arg many examples of the O proportion in Eglptian art and architecture.
is not, S acredarchitecture to see asour timechooses it, a "free"art, deaeloPed and from "feelings" butit is an " sentiment," art strictlYtiedbYand deoeloped from thelawsof geometry. -Fredrik
MacodY Lr':nd
lT hegotde,proportionl is a scaleofproPortions thebad whichmakes and dfficult [to produce] thegoodeasy. -Aibert
Einsiein
136
A B€GINNER'SGUID€TO CONSTRUCTING THEUNIV€RSE
The gold mask of the sarcophagus of Tut-AnkhAmon, which contained his muruny, was appropriately designed using the selfsimilar geometry of the pentagon, sl.rnbol of regeneration and rebirth.
The hieroglyph of the daat (Eglptian for "underworld") underlies the geometry of Egyptian art concerning journeys ihrough the underworld.
Pentagonal symmehy allows each small part to be a whole in itself while harmonizing with other parts and the greater whole. The visual effect is pleasing,balanced,even beautiful, and conveys the star's feelings of excellence, power, authority, life, and humarmess. Use the golden mean calipers to find the @relationships harmonizing the elementswithin eachcomposition. With the publication in 1509of Fra LucJ pacioli,s De Draina Proportiane,which rnarveled at the geometric relationships within the pentagory artists regained accessto the star's conskuction and to its value for organizing spaceaesthetically in harmonic composition. As tlorne,Jdirector of antiquities, Raphael (1,483-1520)was responsible for draw_ ing newly unearthed art, crafts,and archiiecture.He noticed
137
Salvador Dali was a student of ancient design ald used pentagonal syrnrneky in many of his paintings.
The good,of course,is alwaysbeautifuI, andthe beautifulnez;er lacks proportion. -Plato
Jewelry of Tut-Ankh-Amon from Thebes.The rectangular pec_ toral symbolizes the creation of the universe by the sin above the -watersof chaos,while the nearly hiangular pectoral represents rne Duth ot the sun and moon.Both were designedusinq pentas_ onal symmetry. Note how the sun in one and ihe moon i"'tt"" " other are at the center of five-pointed stars, forming the glyDh of the d df, the "underworld,, path to rebirth and ."gjrr"rrEo.,.
Beauty dothofitself persuade theeyesof men withoutanorctor. -William
Shakespeare
t38
THE UNNEESE ABEGINNER'SOUIDETO CONSTRUCTTNG
aside, But let usleaoe Glaucon,theproportion betweenlines. . . or we shallhaoeourfill of many timesthenumberof discussions zaehadbefore. -Socrates (470-3998.c., Greekphilosopher)
that the ancient Roman works emulated the Greek and Egyptian symrnetries. From this observation and his many teachers,Raphael learned to use the pentagon's compositional harmony which he, like many artists of the Renaissance,applied to quite a few of his paintings. His Dispute ooer the Holy Sacramentuses the regenerating pentagram inscribed in the circle. The subdividing pentagons and pentagrams provide a frame of regulating lines along which both the figures and architectureare aligned.
o Our final look at the principles of the Pentad draws us into the most widespread shape in nature-the spiral. Spirals are deeply rooted in the architecture of the universe; they are found in every size and substance. We're aiways intimate with spirals yet rarely notice them. Sometimes we miss them due to familiarity, as in water whirling down the tub's drain and in the shape of our ears. Sometimes we miss them becauseof their obscurity,as in the spi-
E€OEN€RATION
fal "stafucase"of leaves whirling around a stem' Sometimes we miss them because of their size, or distance, hufflcanes or galaxies. And sometimes we miss them because of their invlsibility, as in the shape of the wind and waves of emotion. \iVhat makes spirals so prevalent in cosmic design? They are the purest expression of moving energy' Wherever energy is left to -o:rre on its own it resolves into spirals' The univ"eise moves and transforms in spirals, never straight lines. Spirals show up as the paths of moving atoms and atmospireres,in molecules and rninerals, in the forms ol flowing water, and in the bodies of plants, animals, humans, and the greater bodies of outer space.A universal integrity of soirals r.rnitesall creation' Like the pentagramstar,the spiral expressesthe geometry of self-similarity.At fust the spiral doesn't apgeal to b.e pentagonal.but wherever you seea star you will find a spiial rotLd within it. \ y'henwe feel its hlpnotic lure while gazing into a spiral we are responding to the Pentad's siren call into the self-sirnilar infinite. The spiral's role in nature is transformation. Similarly, in myth and religion it is the path of spiritual and mystical transformation. You may be surprised at all the spiral processesyou already know abou!.now you may leam to look at plants, fruit, vegetables, and other familiar natural forms differently. ihree principles of nature's spiral will reveal to us its secretsof universal construction. @ Spirals grow by self-accumulation. (6; Every spiral has a "calm eYe." resolve into spiral balance. @ Clashing opposites
UNRAVELINOSPIRALS Growthbg Setf-occumulotion Although mathematiciansdistinguish many speciesof spirals, they all have in common the fact that they wrap around a fixed point at a changing distance.
159
A B€GINN€R'SGUIDETO CONSTRUCTINO fHE UNIV€RSE
140
r
It is perhapsa more fortunate destiny to haztea tastefor collectingshells than to beborna millionaire. -Robert Louis Stevenson (1850-1894, Scottishessayist, novelist, and poet)
We can begin to understand spirals by doodling them. Draw some spirals with your normal writing hand. Then draw spirals with the opposite hand. Then read on. Did your spirals spin in the same or opposite directions? Did they flow outward or inward? Did the distance between coils stay the same,widery or narrow? Adding to the definition of these shapes,we can say that different spirais grow by curling around their points at different rates. The two main types of spirals we're most often exposed to are called the Archimedian spiral and the golden spiral. Named after the Greek mathematician and scientist Archimedes,who describedit in his book On Spirals(c. 225 B.c.),an Archimedian spiral is one whose distancefrom the point grows at a fixed rate.The distancebetweensuccessive coils is always the same.We encounter the Archimedian spiral as a coil of rope, clock springs,record grooves,and a ro11 of paper towels. The helix is a three-dimensionalversion of the Archimedian spiral, found in bolts and coil-springs, in candy-cane stripes and barber poles, and in the double helix of the DNA molecule, our genetic heritage. But the spiral most commonly found in nature's public manuscript is another f1pe, the golden spiral. Unlike the Archimedian spiral, the distance between the golden spiral's coils keeps increasing, growing wider as it moves away from the source or natlower as it moves toward it. Called by many n:unes through history, it was apparently weil known in ancient times as evidenced in prehistoric art and worldwide ornament. This is nature's spiral of seashellsand ram's horns, of our ears and fists, oi waterv whirlpools and star-spangled galaxies. It grows from within itself and increases according to the Fibonacci processof accumulation. A simple experiment will reveal the important difference in the ways these two kinds of spirals growFotd the Archimedion ond Gotden Spirots Cut two ships of papeL On one, mark off distancesat one inch (or one centimeter, or one finger width, or any unit you may choose),at two inches, tfuee inches, four inches, and so
1I+1
R€CEN€RAIION
1 1 2
3
4
5
6
ilfr-=illl // tl llllu-/ll
ArchirnediansPiral
tr=1\ llt l a)
u-=_= Golden spiral
Nautilus shell
Red cabbage
\Atrirlpool
on. In other words, make the counting sequencevisible. Fold a right angle at each mark so that the striP tums around to become an Archimedian spiral. The distance between its coils remains constant, always that of a single unit. Its accumulation comes from outside itself, always adding an external "one" to the previous term. On the second strip, mark off units measured by the . . . inches.\l/henyou Fibonaccisequence1,L,2,3,5,8,13,21, that the distance between you'll see and fold the strip crease golden spiral found This is the increaseS. its coils continually in nature. Like the {amous Fibonacci nunber series, the golden spiral grows from within itselt';nothing is absorbed from outside. It is the physical representation of self-accumulation. Its appeatancetells us that the Fibonacci sequence
Clock spring
Paper towels
THEUNIV€RS€ GUIDETO CONSTRUCTINO A BEGINNER'S
142
and @ undertie an internal dynamic balance at work.
hatmony,
excellence'
and
o The Swiss mathematician Jakob Bernoulli (1654-1705), Datriarch of a family of distinguished mathematicjansand icientists, devoted a great deil of study to this particular spiral. He discovered its self-accumulating,self-reproducine nature and gave the spiraI a motto (perhapsthe only one asiociatedwitliu geo*ut i. shape):EademmutatoresurSoAlthough changed, I arise again the same' He. ryas .so impresied with the properties of the golden spiral that he reouestedthat the shape and its motto be carved onto his tomb. Un-fortunately, the stonemason mistakenly carved an Archimedian spiral. Perhaps he didn't realize the important difference between them, or perhaps he just didn't know how to geometrically construct the spiral Bemoulli had reouested,Here's how it's done:
6\ )
145 RE6ENERATION
unfurt NotureS Gotden Spirot from o Gotden Rectongte are in a
