Horror Vacuii: The Desire to Fill Space       

Ben Nicholson  

 

Contents

 

Introduction  

Chapter 1.         Beans, Beans, Glorious Beans

Chapter 2.         String Lines    

Chapter 3.         Tiles

Chapter 4.         Spirals, Hooks & Meanders

Chapter 5.         Labyrinths

Chapter 6.         Drawing on the Beach

Chapter 7.         Taxonomy of Geometry

Chapter 8.         Numerology

Chapter I.  Beans, Beans Glorious Beans.  

 

A.  Beginnings

Surely there is no bigger conceit than the desire to know fundamental origins. To achieve knowledge of 'From whence they came' is a succulent prize; anybody who comes up with a good solution for a beginning of anything, particularly if it comes across as divinely inspired, is bound to go down in history.

Of the family of beginnings of things, the body makes the origin of counting engaged and comfortable. Repetitive practice is second nature to the body: drumming the fingers, putting one foot in front of the other, and the back and forth reciprocity of procreation is so basic to our very being, that these acts are surely to be amongst the origins of counting.  Current interest in ethno-mathematics has made us at ease with the idea that math has been alive and well for millennia in cultures other than our own; although it might come as a surprise to know humanity was engaged in empirical number structure in the form of geometric designs, recorded on pottery, 25,000 years ago.

Rather than hope to trace down the well spring of something that is too intrinsic to humanity to expect a single point of origin, a more healthy attitude towards the origin of number might be to follow Plato's lead made in the Timeaus, in which he suggested multiple beginnings, and to accept the structure of a foggy start in which no one knows exactly how to track down the instant when it all began.  Even if things are recorded, be they artifactual, textual, an allusion to a lost treatise, or oral tradition, we must still leave plenty of leeway for a tall tale to be told, one that does not quite fit into any of the paradigms of accepted veracity.

 

B.  Reading the Classics

Into this mix of illusions steps the quixotic Mr. Pythagoras, a man generations of school children know best by having been drilled with the school house mantra The square of the hypotenuse equals the sum of the squares of the other two sides, sung in a lilting chant that belies the fact that few have the foggiest idea of what it means.  The vision of a genial Mr. Pythagoras sitting on a stone in the Agora handing out boiled sweets to a mist of eager children at his knee, whilst they are busy drawing lines on slates, is wishful thinking.  Mr. Pythagoras had other things on his mind than a career clothed in tweed jackets and stumps of chalk.  Pythagoras was a power and a mystic: he headed up a esoteric sect which held onto a way of life, and a body of knowledge, that threatened or allied the leaders of the time.  So closely guarded was the inner core of this knowledge, that it could not be written down for fear of inadvertent disclosure to the wrong sort. 

Inevitably, stories leaked and a series of anecdotal tell-alls were written that passed through the quills of Aristotle, Theophratus, Plutarch, Iamblichus and others.  The texts are fragments of his thinking, yet they include a curious body of sentences that are memorable because of their singular non-conformance with what you would expect to hear from a revered mathematician: Pythagoras had a particular interest in beans.  The mystique continues to this day and typing in Pythagoras+beans into the Google search engine releases a flood of academically inclined and non-specific sites. The BBC Homepage History includes this on the life of Pythagoras:

Beans were equated with testicles, and therefore represented one's father; to eat beans was to eat the head of one's father. Furthermore they made one fart, which disturbed concentration [1] .

The Catholic Encyclopedia is a little more even handed on the subject, declaring:

The time-honored tradition that Pythagoras forbade his disciples to eat beans, for which various reasons, more or less ingenious, were assigned by ancient and medieval writers, has been upset by some recent writers, who understand the phrase, "Abstain from beans" (kyamon apechete), to refer to a measure of practical prudence, and not to a gastronomic principle. Beans, black and white, were, according to this interpretation, the means of voting in Magna Græcia, and "Abstain from beans" would, therefore, mean merely "Avoid politics"...

 

By assembling an improbable series of antique textual fragments, and making a close visual examination of the bean, a tale can be told to suggest the bean as the instrument of a grand conjunction of physiognomy, sexuality, the soul, reasoned proof, politics and numbers: a delightful, but equally improbable, vision of the world.

 

A gruesome tale indicates the lengths to which Pythagoreans were prepared to go to protect the secret of beans.  Iamblichus relates a story [2] of how a group of Pythagoreans were being hunted down in the countryside, and chose to die rather than run into a field of beans (Fig 1.1 A bean plant from a field of beans).  Just before the last survivor, Timycha, was to be tortured in order to pry out of her why beans were so important to Pythagoras, she chose to bite out her tongue and spit it at the tyrant Dionysius so that she could not divulge the secret of beans.  Pythagoreans maintained a formidable non-disclosure agreement.

 

               So (the Pythagoreans) would have escaped, and their pursuit would have been given up by Eurymene's soldiers, who were heavily armed, had their flight not led them up against a field sewn with beans, which were already flowering.  Unwilling to violate their principle not to touch beans, they stood still, and driven to desperation turned, and attacked their pursuers with stones and sticks, and what ever they found at hand, till they had wounded many, and slain some.  But [numbers told and] all the Pythagoreans were slain by the spearmen, as none of them would suffer himself to be taken captive, preferring death, according to the Pythagorean teachings.

As Eurymenes and his soldiers had been sent for the express purpose of taking some of the Pythagoreans alive to Dionysius, they were much crest-fallen; and... turned homewards.  But as they were returning they met two of the Pythagoreans who had lagged behind, Myllias the Crotian and his Lacedaemonian wife Timycha, who had not been able to keep up with the others, being in the sixth month of pregnancy.  These therefore the soldiers gladly made captive, and led to the tyrant with every precaution, so as to ensure their arrival alive.  On learning what had happened, the tyrant was very much disheartened, and said to the two Pythagoreans, "You shall obtain from me honors of unusual dignity if you shall be willing to reign in partnership with me."  All his offers, however, were by Myllias and Timycha rejected.  Then said he, "I will release you with a safe-guard if you will tell me one thing only."  On Myllias asking what he wished to learn, Dionysius replied:  "Tell me why your companions chose to die rather than tread on beans?"  But Myllias at once answered,  "My companions did indeed prefer death to treading on beans; but I had rather do that than tell you the reason."  Astonished at his answer, Dionysius ordered him forcibly removed, and Timycha tortured, for he thought that a pregnant woman, deprived of her husband, would weaken before the torments, and easily tell him all he wanted to know.  The heroic woman, however, with her teeth bit her tongue until it was separated, and spat it out at the tyrant, thus demonstrating that the offending member should be entirely cut off, even if her female nature, vanquished by the torments, should be compelled to disclose something that should be reserved in silence.

Iamblichus (c.250-c.325AD), The Life of Pythagoras, Chap. 31

 

The distinct physiognomy of the bean lends it qualities that go far beyond merely pleasing the eye: an Orphic fragment suggests parenticide.

Eating fava beans and gnawing on the heads of one's parents are one and the same.

Orphica, fr.291 Kern

Plutarch, Symposia II.3.635e (neither verified by B.E.N.)

 

Iamblichus announces the bean's other-worldly properties by declaring them to be sacred, along with certain other foods.

Food other than anima l w as by him also considered sacred, due to the nature of divine reason.  Thus his disciples were to abstain from mallows, because this plant is the first messenger and signal of sympathy of celestia l w ith terrestrial gods.  Moreover, the fish melanurus was interdicted because it was sacred to the terrestrial gods.  Likewise, the erythinus.  Beans were also interdicted, due to many causes, physical, psychic and sacred.

Iamblichus (c.250-c.325AD), The Life of Pythagoras, Chap. 24.

 

Pliny's Natural History includes a long description of the physical properties of the bean, as well as mentioning Pythagoras' condemnations against eating them as they contained the souls of the dead.  This might further explain why he refused to walk through a field of bean plants, for fear of disturbing the countless souls hanging from their stems. 

 

We now come to the history of the leguminous plants, among which the place of honour must be awarded to the bean; indeed, some attempts have even been made to use it for bread. Bean meal is known as "lomentum;" and, as is the case with the meal of all leguminous plants, it adds considerably, when mixed with flour, to the weight of the bread. Beans are on sale at the present day for numerous purposes, and are employed for feeding cattle, and man more particularly. They are mixed, also, among most nations, with wheat, and panic (millet) more particularly, either whole or lightly broken. In our ancient ceremonials, too, bean pottage occupies its place in the religious services of the gods. Beans are mostly eaten together with other food, but it is generally thought that they dull the senses, and cause sleepless nights attended with dreams. Hence it is that the bean has been condemned by Pythagoras; though, according to some, the reason for this denunciation was the belief which he entertained that the souls of the dead are enclosed in the bean: it is for this reason, too, that beans are used in the funereal banquets of the Parentalia. According to Varro, it is for a similar cause that the Flamen abstains from eating beans: in addition to which, on the blossom of the bean, there are certain letters of ill omen to be found.

There are some peculiar religious usages connected with the bean. It is the custom to bring home from the harvest a bean by way of auspice, which, from that circumstance, has the name of "referiva." In sales by public auction, too, it is thought lucky to include a bean in the lot for sale. It is a fact, too, that the bean is the only one among all the grains that fills out at the increase of the moon, however much it may have been eaten away: it can never be thoroughly boiled in sea-water, or indeed any other water that is salt.

Pliny the Elder, The Natural History, Book XVIII ch. 30, 'The Leguminous Plants: The Bean', ed. John Bostock.

 

That the bean had some stature in the religious life of Greece is confirmed by Pausanius, who describes a Greek temple that was dedicated either to beans, or to the person who first sowed them.  He goes on to imply that because the bean was sacred, it had a different status than that of other plants.

Here is the grave of Theodectes of Phaselis, and also that of Mnesitheus. They say that he was a skilful physician and dedicated statues, among which is a representation of Iacchus. On the road stands a small temple called that of Cyamites (Cyamos means "bean"). I cannot state for certain whether he was the first to sow beans, or whether they gave this name to a hero because they may not attribute to Demeter the discovery of beans. Whoever has been initiated at Eleusis or has read what are called the Orphica knows what I mean.  

Pausanias , Description of Greece ,  1.37.1

 

To those Pheneatians who received her with hospitality into their homes the goddess gave all sorts of pulse save the bean only. There is a sacred story to explain why the bean in their eyes is an impure kind of pulse.

Pausanias, Description of Greece ,  8.15.1.

 

Iamblichus goes on to state that the bean is corruptive of our relationship to the gods and divine prophecy.  The context for this remark is most likely political, for voting with beans was standard practice at this time.

 

This, abstain from beans, advises us to beware of everything which is corruptive of our converse with the gods and divine prophecy.

Iamblichus, The Exhortation to Philosophy, Symbol 37.

 

Diogenes Laertus describes beans as having several qualities, both gastronomic and sacred. The flatulent aspect of beans has led some to link them with the presence of the devil in the body: the generosity of polytheism form's a relationship with just about anything, even farting.

His disciples were forbidden to eat beans because, as they are flatulent, they greatly partake of animal properties; and besides, the stomach is kept in much better order by avoiding them, and such abstinence makes the visions that appear in one's sleep gentle and free from agitation.

Diogenes Laertus (c.3^rd century), The Life of Pythagoras, Chap. 19.

 

Another selection of writings makes the case for considering the multivalent qualities of the bean, going so far as to think of it as the First Plant to emerge at the beginning of creation.  This text is a fine example of the power of metaphor, in which the physical properties of the bean are enough of a proof to suggest its sacred properties.

And it is said that Zaratas forbade men to eat beans because he said that at the beginning and composition of all things, when the earth was still a whole, the bean arose.  And he says that the proof of this is that if one chews a bean to a pulp and exposes it to the sun for a certain time (for the sun will affect it quickly), it gives off the odor of human seed.  And he says that there is another and clearer proof: if when a bean is in flower we were to take the bean and its flower, and putting it into a pitcher moisten it and then bury it in the earth, and after a few days dig it up again, we should see in the first place that it had the form of a womb, and examining it closely we should find the head of a child growing with it.

Hippol., Phil 2; Dox. 555, and The Doxographers, On the Pythagoreans.

 

Diogenes Laertus quotes Aristotle, from his lost treatise on Pythagoras, who gives other reasons why he required his pupils to abstain from eating beans.  He observes the structure of the plant, which leads one to believe that he is using the plant's physiognomy to indicate its diabolic intentions. The jointless plants, moving from ground to sky in one spiraling movement, are an apt metaphor for the relationship of heaven and hell.

 

A state of purity is brought about by purifications, washings and lustrations, by a man purifying himself from.......animals that lay eggs, from beans, and from other things that are prohibited by those who have charge of the mysteries in the temples.

In his treatise On the Pythagoreans, Aristotle says that Pythagoras' reason for demanding abstention from beans on the part of his disciples, was that either they resemble genitals, or because they are like the gates of hell, [...] they are the only plants without parts (joints/hinges), or because they dry up other plants, or because they are representative of universal nature, or because they are used in elections in oligarchal governments.

Diogenes Laertus (c.3^rd century), The Life of Pythagoras, Chap. 19.

 

It has already been mentioned that beans are linked to politics: there are several mentions of how they were used, some even being specific to an actual event. 

 

Beans should be scorned, because they are used in voting, inasmuch as the Pythagoreans selected office-holders by appointment.  To rule should be an object of desire, for it is better to be a bull for a day, than for all one's life to be an ox. 

Iamblichus (c.250-c.325AD), The Life of Pythagoras, Chap. 35.

 

And since it was a hard task for him to restrain the Athenians in their impatience of delay and eagerness to fight, he separated his whole force into eight divisions, had them draw lots, and allowed the division which got the white bean to feast and take their ease, while the others did the fighting. (5.71)

Plutarch, Lives, Pericles, chap. 27, sec. 2.

 

The ancient Greeks and Romans made use of the beans in gathering the votes of the people, and for the election of magistrates.  A white bean signified absolution, and a black bean condemnation. Beans had a mysterious use in the lemuralia and parentalia, where the master of the family, after washing his hands three times, threw black beans over his head nine times, reiterating the words "I redeem myself and my family by these beans."

Encyclopedia Britannica, 11^th edition, Bean, vol III, p. 572-573.

 

St. Jerome , who lived between 340-420, during the same century as Iamblicus, picks up on the strain of psycho-sexual attributes of the bean in his warning against nuns having anything to do with beans.  "Once cooked, the bean becomes a plump, yet firm signal to a woman's lips, heart, womb, that she is prepared to become with child," and so the cycle continues, creating love and children everywhere."  "(He) forbade beans to the nuns in his charge on the grounds that 'they tickle the genitals."

Texts extolling the nature of beans have been no secret to more recent writers, some of whom fancied themselves as Pythagoreans.  The satirist Jonathan Swift describes a bedroom scene in his poem Strephon and Chloe, evoking Pythagoras, the Sage of Samos:

Now, Ponder well ye Parents dear; [115]
Forbid your Daughters guzzling Beer;
And make them ev'ry Afternoon
Forbear their Tea, or drink it soon;
That, e'er to Bed they venture up,
They may discharge it ev'ry Sup; [120]
If not; they must in evil Plight
Be often forc'd to rise at Night,
Keep them to wholsome Food confin'd,
Nor let them taste what causes Wind;
('Tis this the Sage of Samos means, [125]
Forbidding his Disciples Beans ²¹ )
O, think what Evils must ensue;
Miss Moll the Jade will burn it blue:
And when she once has got the Art,
She cannot help it for her Heart; [130]
But, out it flies, even when she meets
Her Bridegroom in the Wedding-Sheets.

 

            In writing On Walden Pond, Thoreau refers to the discourse on Pythagoras and beans,  and no doubt enjoyed the irony of his having planted a bean field, whilst claiming to be a Pythagorean, but he may well have been right to have suspected that the Pythagoreans were bean farmers, but for purposes other than have been described here. 

 

Not that I wanted beans to eat, for I am by nature a Pythagorean, so far as beans are concerned, whether they mean porridge or voting, and exchanged them for rice; but, perchance, as some must work in the fields if only for the sake of tropes and expression, to serve a parable-maker one day.

Henry David Thoreau, On Walden Pond , Chap. 7, The Bean Field.

 

C.  Bean Anatomy

 

It is only fair that modern science should be given a chance to weigh in on the discussion, for it may shed some 'non-metaphoric' light onto the matter why beans had such a compelling interest to the ancients; but first, a definition of beans.  A clue to the complexity of categorizing the plant comes from the names given to the four great Roman families who were named after varieties of the legume: Fabius (fava); Lentulus (lentil); Piso (pea); and Cicero (chick pea).  The classification of the plant kingdom situates all these legumes into the family of fabaceae, the pea family, which breathe, have flowers, and bear seeds that are split into two parts (dicotyledons). The fabaceae family is subdivided into many genus, among which are the above four, and they in turn are divided into species.

Today there are many varieties of bean, most of which are fabaceae phaseolus (pea family, bean genus), introduced to Europe from central America in the 1500's. 

(Fig.1.2 Side views of selected fabaceae) (Fig.1.3 End views of selected fabaceae)The bean referred to by Pythagoras and later writers is thought to be fabaceae vicia faba (pea family, vetch genus, horsebean species) commonly known as the fava bean, or broad bean.  It is understood to be the only bean available in Greek antiquity, having been cultivated across Europe and the Middle east for millennia.  A difference in identity between the European and American bean seeds is that the hilum (umbilical 'tummy button' scar) of vicia is placed at the base of the seed, and at the side in phaseolus. 

The Greek and Latin text fragments mentioned above have been largely dismissed in academia because they are scatological, mildly pornographic and have overtones of religion, a deadly mixture for the latter day scholar.  But the opaque remarks are so intriguing that it is worth taking each phrase at face value, and examining a bean in its own light. (Fig.1.4  Bean anatomy of fabaceae phaseolus) If a bean is described as having genitalia, is it apparent to the naked eye?  A close inspection of a bean is best done with a magnifying lens: araraatificial forms of magnification were known in antiquity, but the simplest magnification system can be set up by coaxing a tear to puddle upon your iris whilst the head is bent down.  The eye becomes a very effective magnifying lens with a focal length of about one inch, good as a X8 photographer's loupe, but it does depend on remembering how to cry.

After reading the classics, looking at a magnified bean is a startling sight, verging on the unprintable. (Fig.1.5  Bean anatomy of fabaceae vicia faba) The underside of  fabaceae vicia faba, aptly known as the Broad bean, has four dominant aspects.  In the middle is a wide opening (hilum) surrounded by engorged, plump lips.  At one end of the opening is a pinprick hole (micropyle), beyond which is a triangular perforation which bursts open when the root finds its way out of the shell.  At the other end of the hilum is the raphe, a biological and medical term used to designate a line of closure in the folds and swellings of both the bean and the scrotal and penile areas.  The side view of the broad bean is equally suggestive; it appears as a powerfully erect membrum virile attached to a pair of drawn up testicles.  Aristotle's remarks about the bean's likeness to genitals is confirmed: the bean comes with its own penis, urethratic pee hole, vulva, raphe and pair of testicles, perfectly crafted into a whole.

When the bean is stripped of its skin (testa), the form intensifies.  The resemblance of a little penis (radicle) is found beneath the bulge of the shell, complete with a wimpley little tip.  To the left and right of it are protrusions and, at its top, is a flap attached to one of the buttock-like forms (cotyledons) which works its way across to the other cotyledon, like a trouser fly. With the testa stripped away, the vulvate opening is gone too, replaced with a sinuous curve of an opening.  Color has been drawn out of the bean, it has an ivory whiteness, marked by form and light alone. 

When placed in a bowl of water and left to its own devices, the hard skin (testa) of the bean begins to soften from fluid passing inwards through the tiny hole (micropyle), and the membrum (radicle) within begins to stir (Fig.1.6 Bean episiotomy).  So powerful is the arousal to life that the skin tightens up to form a codpiece until it splits open with a powerful rip that makes a horizontal gash, a self-made episiotomy, announcing the call to life.  With time, the little root, replete with tip, begins to fill itself out and stir, getting bigger and longer each day, until it makes its way out of the rent skin between the buttocks.  From one aspect the radicule is a penis and from another a clitoris, transexually ambiguous to the end. 

This first conception has happened without coition; the unruly membrum never entered the cavity, it went off on its own route to the center of the earth, down towards Tartarus, through the Gates of Hell, no doubt to be met by Pneuma Akathartos, the evil spirit of fou l w inds.  The devil is in the details.

Returning to science for a minute, the membrum is of course no more than a radicule attached at its hilt to the dicotyledons of the bean.  The root runs downwards into the earth in order to get enough purchase to push the plumule, a pair of perfectly formed embryonic leaves, into the light so that they can get life-giving sustenance from the sun. The radicule and plumile gradually draw out of the dicotyledons the nourishment needed, until they are sucked dry of energy and eventually drop off the stalk.  Thus the bean stalk moves in two directions, to moist earth, via the hypocotyl, and the warm light, via the epicotyl.  The bean has no arms, legs or trunk for it is all genitals; the bean is universal nature, as Aristotle is supposed to have remarked, one end of it going to hell and the other up to heaven.

 

D  The Sexual Sacred

A pair of themes intertwine in the Pythagorean reflections on beans: their sexuality and their sacredness, a topic quite familiar in Classical culture.

            The physiognomy of the bean shows it to be an overt instrument of sexual metaphor, whilst having the rare ability to be all things to all people.  It suggests masculine & feminine as well as hermaphrodite & androgyny.  It is both active and passive; the roles surreptitiously changing between the time when the bean is dry and made wet with water.   Beans appear to be both homosexual & heterosexual and bisexual & introsexual to a degree that is beyond notions of promiscuity or political correctedness, ever flexible and open to change.  Multitudinous traits of sexuality are latent in every human being, most of them laid low in favor of one grudgingly declared, the whole apparatus presided over by the agitated moral cause of religious authority.  With all this to contend with, could there possibly be a straight bean?  Probably not. 

This rampant ambiguity is reminiscent of the uncontrollable dilli-dalliance of the Olympian Zeus, who unabashedly ran the full gamut of sexual impropriety.  The pan-sexuality divined from the appearance of beans is of an ardor and tenderness so strong that humanity is left standing in the dust. Is this the quality that is being referred to by the ancients, when they speak of the Sacred Bean, a notion of singularity beyond gender, represented by the all inclusive Monad within which everything is bound up? The concept of the Sacred Bean conjures up a nightmarish scenario of a Pythagorean initiate, kneeing in front of Pythagoras, receiving a milky white, wafer sized, split fava bean between the lips, a uncommon host of the pre-Christian era.  A loaf of bread and a mess of beans may share more in common than at first thought.

Flipping back to nature once more, the disturbing truth is that plants are notoriously immoral by current standards of behavior.  The majority of flowering plants are both male and female at one and the same time and quite happy to mate with themselves if nothing else comes along.  Admittedly, a few are gender sexed, the prickly holly bush being one example, but generally the male's germ floats freely whilst the female aspect passively lies in waiting.  A flower's female pistil maintains a presence for the male stamen to shake off its fugitive pollen: between the two of them they create the generative seed.  Our own view of nature's biological ways confirm the immoral imperative alluded to by the ancients: Creation and the Gods are doing it together in strange ways, and it is unwise for the likes of humans to challenge them, for fear of retribution.

 

E.  Number from Counting

            The nature of the orgasm, that pocket sized metaphor for the astronomer's Big Bang, still intrigues humankind as much as it ever did.  At the macro/level, we constantly hunt for the singular moment of life's beginning. Adjacent to the entries on beans, farting, sex and religion in the writings on Pythagoras, are passages about the quality of number.  This is not a coincidental juxtaposition, for number is at the essence of generation. The proofs of Euclid , and the cryptic language of contemporary mathematics, make it difficult to access the unfolding beauty of number: however the bean offers up a very accessible perspective when it is observed directly.      

            If a single bean is stuck into warm moist earth the tiny desiccated root and leaves spring into life, pushing apart the two halves of the bean; from one bean comes two parts, and with the passing of the season (and all the tricky circumstance of the plant's life) the vine becomes covered in pods, each of which is filled with a fixed number of beans. (Fig.1.7  Bean pods with beans.) A pod can hold one bean, two, three, four, five, six or seven beans; in Chicago , the Scarlet Runner is reluctant to produce more than seven beans to a pod.  That one bean can generate a fixed number of pods, with a set of 1, 2, 3, 4, 5, 6 or 7 beans in each pod, cannot fail to have been noticed by Pythagoras.  The hermaphrodite bean, all genitals and no head, has completed a measured cycle; it has spawned from a single bean sequences of fixed numbers, lending credibility to the act of Creation through quantifiable materiality:  Number asserts Creation.

 

F  A Philosopher Stalks the Beans.

A gardener and a philosopher look at beans quite differently.  The gardener opts for a balance of circumstance that will optimize the quality and quantity of beans, whilst the philosopher inquires into the nature of the bean and where it is situated in the larger scheme of things [3] .  For the philosopher, the beginning of an abstract concept is formed upon the clues suggested by workaday nature: if a bean begins as one and ends as many, it would seem logical that latent within the bean are other numbers.  How then is it possible to get at them?  After laying a pile of beans on a table and inspecting them, (Fig. 1.8  Pile of fava beans) one notices how very different they are from each other, they can be big/small, fat/thin, lumpy/smooth, speckley/plain, tapered/bulbous, ugly/beautiful; the more they are inspected the more differences they seem to hold.

If beans are laid out top to tail in a line, the big and the small average themselves out to suggest a mean length of a bean; despite their differences they have achieved a standardized unit. (Fig. 1.9 Set number of different beans measures out the same length) The bean retains its individualistic singularity and difference, yet when put together becomes a quantifiable and measurable length.  As a device for counting and measuring, a bean is more animated than a pebble; beans create numbers, pebbles confirm them.

A single bean is now selected to discover its quality.  If it is stared at, poked at, and wondered at, curiosity will inevitably lead to its being cut open. The bean naturally cleaves into two halves; what was once one whole is now two, this action shows that two is latent in one.  To extend this logic from the physical to the conceptual, if one became two, two could become four, and four could become eight.     If the bean's physique naturally leads from one to two, is there any reason why the same principle of doubling cannot be extended conceptually, and the bean become tripled: if you can double something, then why not triple it, or set it to the power of any number for that matter.  And, if a bean thereby becomes three beans it follows that three could be tripled to become nine and, in turn, nine could become twenty seven.  (Fig.10: A single bean is doubled, tripled and empowered) That this single bundle of creation includes a pair of fleshy halves presents the idea that the one (the Monad) and the two (the Dyad) are difficult to cleave apart from each other. Necessarily male number and female number are present at one and the same time. In his Introduction to Arithmetic, written in 100AD, Nicomachus of Gerasa speaks about the inseparable nature of one and two, he suggests that number is better off beginning at three, the conjunction of the male Monad and the female Dyad.

            If beans are laid out as two limbs, according to the two generative acts described above, there comes the desire to do to the pile of three beans what was done to the single bean; to double it to make six beans, and to double that to make twelve.  It becomes apparent that six is also the sum of two beans that have been tripled; the qualities that emerged out of a single bean are now confirming themselves, forming a new kind of whole.

 

G  Geometry Joins the Arithmetic Fray

            If the little piles of beans are joined with lines, a simple triangular grid forms which can be overlain with lines which project further associations. New qualities emerge: if the pile of two beans is added to the pile of three beans they add up to five beans.  If the adjacent numbers are added on the lines parallel to it below, (6+9=15 and 18+27=45) it will be seen that all adjacent numbers situated on lines parallel to the line formed by 2+3 are factored by 5 (15=3×5, 45=9×5). 

            If these two piles of beans joined by a line are associated with the number five, then other piles of beans may have similar qualities: fig. 8 shows that every pair of adjacent numbers situated on a line composes a prime number or is factored by a prime number.  In a triangular structure, whose limbs are formed by 1, 2, 4, 8 and 1, 3, 9, 27, the factors of adjacent numbers can compose the primes 2, 3, 5, 7, 11, 13, 17, 19, & 29.  Only 23 is absent, but will reemerge if the grid is extended.  Thus, within one bean, the monad, a universe of numbers emerges, primes and all.

            This triangular grid of numbers, generated from the monad bean, has another quality which demonstrates the measure of infinity.  If the grid is generated outwards and a line is drawn in space, that line will be parallel to one that passes through two points on the extended grid: thus it will snag a prime or be factored by a prime.  Thus, any straight line drawn at any angle on a two dimensional grid is identified with a prime number, thus the concept of random direction is inextricably linked to the prime number.

            The dicot bean has spun a perfect yarn about itself, with qualities of coincidence which are as immaculate as the concept of creation.  Number is the match for the pair of desiccated leaves and root nestled between the flanks of the bean, but the majesty of number and growth remains outside of human comprehension.

 

H.  Beans, Bottoms, Measure and the Golden Mean

            It is easy to be lulled by the narcotic logic of number regarding the generative properties of beans, but why was so much ink spilled by the ancients on the subject of flatulence?  The Pythagoreans are reputed to have sworn their allegiance by the sign of the pentacle [4] , no doubt because the pentagonal star is shot through with the Golden Mean.  One of Leonardo da Vinci's more curious drawings, currently held by Her Majesty The Queen in Windsor Castle , is of the external anal sphincter composed of a five part muscle, which he draws with the point towards the rear of the torso [5] . (Fig.1.11: Leonardo's Sphincter Muscles.)  Who would have guessed that the anus is such a tricky piece of geometry, and that when we do our morning ablutions, yesterday passes through a pentagonal die formed by the Golden Mean.  Leonardo worked closely with Fra Luca Pacioli to produce a series of drawings of Platonic and Archimedean Solids, which included the twelve pentagons that compose a dodecahedron.  Pacioli revealed the construction of the pentagon in De Divina Proportione  in 1509.  In a string of associations, connections and translations into continuous relationships composed of sex, Creation, farts, sphincter muscles, Leonardo & Fra Pacioli, (more sex), De divina proportione, number and earthy measurement, the bean has become the instrument that pulls it all together. 

Has the compound nature of the sacred bean revealed itself in corporeal glory, glimpsing at a pure moment of light as it lets off a passing wind?  Does that which is created by the gods automatically have body?  Do architects provide the first measurement that connects the body to the cosmos, thereby tripping themselves at the first hurdle?  What is measurement? Is it the only thing that the Gods forgot to manage, humankind snatching it up fearful that it might be taken away from them?  And where is measure within void?  Architecture is ready to make a place that is beyond measure and materiality, where anthropocentricity gets cracked once and for all time, severing the corporeal shackle.         

 

I.  A Natural Reflex..

There is something especially pleasurable about engaging in a historical inquiry on a subject for which all prime documents are missing, and which is composed of second hand hearsay.  It confronts the iffy status of architectural para-theorists, living in an underworld of poorly constructed arguments, without either proof or truth, yet with a certain poetry that eludes the Doctors.  Perhaps these sorts of inquiries, and the spaces that they invoke, are the new frontier of debate, a deliberately loose sort of terrain, in which the story is better than the facts that become walled around it.  By not looking for what is deemed to be defunct, or what is supposedly new, consider a realm in which all exists, waiting to be found, without provability, a story land beyond history.  Pythagoras' beans certainly conjure up such a terrain.

In this aftermath litter of quasi-history the question still remains why beans were forbidden by Pythagoras to be eaten.  To eat a plate of beans that look like, generate like, smell like, and then act out every aspect of nature, number, scatology and sexuality, which only divinity had any hope of pulling back into singularity, would not be especially conducive to a strong appetite.  Would you want to eat beans after knowing this much conjecture about them?  But seen in this light, beans do have an especially satisfying capacity to bring together the more primal qualities of the body and the more tricky aspects of the intellect.  Perhaps beans are the mean between that which desires order and that which gives it as it comes, grunts and all.  The beauty of the bean is that it has qualities of measure and chaos, the one thriving upon the other's inadequacy. Measure is not an isolated concept religated to mathematics for it needs to be coupled with the anecdotal to form a culturally inspired metaphor.  When the mathematical and the mythical are brought together, they create implied spaces that lie forever outside of comprehension, and yet somehow they come to be given form.

 

Bon appetite!

Illustrations: Chapter 1, Beans  

 

 

 (Fig 1.1 A bean plant from a field of beans)

 

(Fig.1.2 Side views of selected fabaceae)

(Fig.1.3 End views of selected fabaceae)

(Fig.1.4  Bean anatomy of fabaceae phaseolus)

 

(Fig.1.5  Bean anatomy of fabaceae vicia faba)

 

(Fig.1.6 Bean episiotomy).

 

(Fig.1.7  Bean pods with beans.)  

 

(Fig. 1.8  Pile of fava beans)  

 

(Fig. 1.9 Set number of different beans measures out the same length)

 

(Fig.10: A single bean is doubled, tripled and empowered)  

(Fig.1.11: Leonardo's Sphincter Muscles.)