**CHAPTER FOUR: FOURS **

GROUP THEORY AND THE COSMIC CYCLE

Let's get down to real nuts and bolts.
In* With Hidden Noise*, four nuts and bolts hold together two brass
plates which compress between them the ball of twine; and within the dark
interior space thus generated shakes, rattles and rolls the secretly-inserted
object producing the hidden noise. They are brass plates, not copper; and
they are nuts and bolts, not screws. Sometimes precision is seen to matter,
sometimes it isn't. We must apply in our analysis of and writing about Duchamp's
art a scale of rigor that is necessarily flexible: on the one hand sensitive
to the artist's distinctive penchant for fineness and finesse, and on the
other hand accommodating the idiosyncracies of the his own sometimes immediate
and intimate, other times abstruse and recondite sense of style. There are
quirks: in a note from The Green Box Duchamp refers to the two brass plates
or plaques of *With Hidden Noise* as being made of copper ["cuivre"].

[ Sanouillet, Salt Seller, p. 32.]

A casual attitude in referring to the media or materials used in a piece may be indulged when it comes from the artist himself. Few, however, would suggest that writers knowingly ought to perpetuate sheer, non-"poetic" inaccuracies. The plates remain NOT copper, but brass. To be sure, there is a lot of copper in brass. But in most recipes for brass, as with the new poison pennies in America, there is also a lot of zinc. We would certainly not want to see a child or a pet die for a matter of imprecision worth just a few cents.

Scientists at the University of Georgia's College of Veterinary Medicine said pennies minted in 1982 and later are 96 per cent zinc, which if swallowed by animals or small children can react with stomach acid and cause serious illness or death. "Swallowing even a small number of recently minted pennies can have disastrous consequences, most notably a zinc-induced hemolytic crisis (destruction of red blood cells) that can be fatal," said Kenneth Latimer, associate professor of pathology.

[ San Francisco Chronicle (April 25, 1988). ]

Marcel Duchamp may have talked a good game of accuracy, but how accurate is accurate enough, or how precise is too precise? With his strong sense of Norman decorum, Duchamp was meticulous in his dress as in his manner of artistic expression. This was only to be expected from such a sophisticated sensibility, attracted to mechanical drawing and fascinated by the most arcane of intellectual speculations and revolutionary scientific theories of his day. So too, we should not be surprised that his attention strongly gravitated to the game of chess.

Duchamp must have appreciated the irony that he was known as the chess player among artists, and the artist among chess players. Yet his application of precisionist aesthetics was unpredictable, and he was often notoriously lazy about name and form. Of course, this allowed an expansive arena for his sense of humor as well as for his innovative pursuit of paradox in the overlappings of sense and sound, or image and function, supplying that sufficiently multivalent meaning from which Duchamp could weave his web of puns and jests. We described With Hidden Noise as presenting a GENERALLY cubic shape, occupying APPROXIMATELY a cubic space. Perhaps regarding the dimensions of the piece as 5 x 5 x 5 inches is too cavalier a generalization; but in this instance we have simply joined Duchamp himself, Robert Lebel, Richard Hamilton, Walter Hopps and many others. However, when Arturo Schwarz opted for excruciating exactitude, he was followed by Anne d'Harnoncourt and Kynaston Mc Shine. Self-evidently, these writers all have precision on their side, measuring their fractions with a fine focus, while our rounded-off reading--by which we simply sought to make a point about three figures five--begs indulgence with a sort of special-pleading that has become so typical of poets and aritsts it is often referred to as their own special "license."

Then what should it matter that almost all the writers on Duchamp --including Schwarz, d'Harnoncourt and McShine--refer to the four nuts and bolts of With Hidden Noise as "screws?" Who's counting, anyway?

If the plates were of wood, four long "screws" just might suffice to hold it together. But the structural logic of the piece--a ball of twine, with its dynamic resilience, resisting the compression of the two brass plates--requires bolts to hold it together at the four corners. Short of highly-specialized self-tapping metal screws, really any other screws just would not do at all. The term "screw" has a colloquial sexual connotation bordering on the vulgar; but so, too, does "nuts" and the suggestive yin/yang penetration of nuts and bolts. These risqué readings reflect Duchamp's playful, sly, rapscallion sensibility, but more to the point both the bolt and the screw are variations on the basic idea of a fastener. To draw the distinction finally, the screw is simpler and more general, while nuts and bolts are obviously more complex, being composed of two threaded, coupled elements; and a tapering screw, closely related to a drill bit, might be thought of as between the nail and the bolt. Yet, despite the cosmic sexuality of yin and yang, and their acknowledged antiquity in Taoist poetry and philosophical thought--interestingly enough,

the continuously winding screw-thread, male and female (as in bolt and nut), and the [worm gear], are the most outstanding examples of mechanical systems apparently unknown to Chinese engineers and artisans until the 17th century. (So foreign to the Chinese technical tradition was the screw that even in 1954, in an article on the bringing of modern agricultural machinery to remote districts, Ma Chi remarked that some of the peasants found difficulty in handling screws and bolts. On the other hand spiral forms in decorative art were quite common in traditional China, as e.g. on temple pillars, and on vessels such as the Buckley bowl.) On the history of the screw much has been written, and it is quite clear that the principle was very familiar in Hellenistic times. The reputed inventor was Archytas of Tarentum (fl. B.C. 365), and all the Alexandrians discuss apparatus involving worms and worm gearing...[as] screw presses used in the wine and oil industries, shown, for instance, in the wall paintings of Pompeii....The tapering wood screw appears in Gallo-Roman times....There was clearly no break in the European knowledge of screws; in the 13th century Villard de Honnecourt used them for raising weights, and the 15th century German engineering [manuscripts] frequently show them. About 1490 metal screws suddenly became common for the fastenings of armour.

A curious problem is raised by the fact that the only people other than Europeans to possess the continuous helical screws were the Eskimos. There has been much discussion as to whether this was an independent invention or due to cultural contact with Europeans, but the question is not yet solved.

[ Needham, Science and Civilisation in China, Volume IV, Physics and Physical Technology, Part 2 "Mechanical Engineering," Cambridge University Press (1965), pp. 119-120. For the Buckley bowl, "the most splendid piece of old Chinese glass," but of very uncertain date, see Volume IV, Part 1 "Physics," p. 104, Figure 295; and W. B. Honey, Glass; A Handbook and Guide to the Museum Collection, Victoria and Albert Museum, London (1946). ]

The Chinese did employ the principle of
the paddle wheel and zoetrope, the origin of which may go back to mechanisms
for wind-driven prayer-wheels of the Mongol-Tibetan culture (Needham, IV,
Part 2, p. 566); the "Glider" in the Large Glass relates to a
similar paddle wheel device. For the relationship of Duchamp to Eskimos,
of course, we have Rrose Sélavy's famous* mots exquis*, the
gouache *Project for the Rotary Demisphere*, (1924), with words written
by Man Ray:

Rrose Slavy et moi estimons les ecchymoses des Esquimaux aux mots exquis.

[ See, Schwarz, Complete Works, Cat. 281, pp. 491-492; and Rotary Demisphere (Precision Optics) (1925), Cat. 284, p. 493. Sanouillet records this text, but declines to translate it, Salt Seller, p. 106.]

It would seem simple enough, you might think, to distinguish between a screw and a bolt, although articulating their distinct efficient actions is a demanding verbal exercise. It might go something like this: the screw functions as a fastener by virtue of the friction (between the two surfaces) of the thread and the wood or other material into which it has been twisted. In contrast, the bolt is usually inserted through a drilled hole(s) to resist shear and-- although the friction between the surfaces of their female and male threads keeps the nut on the bolt--the bolt holds as a fastener owing to compression between the inside surfaces of nut and bolt head, respectively. The term screw also refers to the propeller of an airplane or a boat, and its use as a device for raising water is said to have been invented by the Greek Archimedes. More formally, the screw is a special case of the wedge--or, in terms of classroom physics, an inclined plane rotating about an axis (a helicoid). All these devices are inclined to work because matter describes a three-dimensional spiral when moving most easily through space.

Counting each nut and bolt component in With Hidden Noise--each piece of threaded metal--there are a total of eight pieces: four bolts and four nuts. Let us, however, reckon each one of the nut/bolt combi-nations as constituting, together, a single structural feature of the piece. In our analysis by-the-numbers, we are moving poetically from sixes to fives to fours, and so on, to the heart of the matter. In Duchamp's original piece, now in the Arensberg Collection in Philadelphia, the nuts are hexhead, i.e., six-sided, not four-sided. Counting from photographs, there seem to be about 64 threads on each bolt, and a whitish oxide has formed in the thread grooves. Similar problems of detail arise for anyone concerned with finer points of connoisseurship or who sets out to recreate the piece. The hemispheroidal- or "round-" headed bolts, furthermore, are threaded about two-thirds of the way along the (approximately) five-inch length of their shafts. Again, you might think such bolts would be easy to find, that one could walk into a hardware store and just pick them out of a bin; in the experience of the present author at least, and according to Ms. Tamara Blanken, who also has quested, this has proven to be not quite the case.

Within Marcel Duchamp's own *oeuvre*
was a piece thematically related to the four nuts and bolts of With Hidden
Noise, namely *Pulled at Four Pins* (1915), as it was actually inscribed,
in English. The French translation, *Tiré à quatre épignes*,
generates a nonsequitur double entendre: as a colloquial expression it means
something like "dressed to kill" or "dressed to the nines."
There may be another far-fetched pun at "weathercock," Schwarz's
translation of a term used by André Breton (possibly referring to
this piece), and the phrase "cock-of-the-walk" meaning a dandy
(Duchamp?) The sculpture was a Readymade: a sheetmetal ventillation vane
or louvre (or "louver"), executed in New York in the year Duchamp
first arrived (1915). He gave the piece to Louise Varèse, wife of
the modernist composer Edgar Varèse; but it has been lost, as have
so many of the other Readymades. In 1964, in Milan, Duchamp executed an
etching based on the original object,

the only extant representation of the 1915 Readymade, which was lost before any photographic record was made of it.

[ Schwarz, Duchamp, Cat. 373, p. 545, and Cat. 232, p. 454-455, inter-preting this Readymade's symbolism, and accounts of its disappearance. See also, D'Harnoncourt and McShine, Cat. 183, p. 314.]

Let us now explore some hopefully-illuminating though diverse analogs of fourness, continuing our analysis of With Hidden Noise by-the-numbers. The question of the Real--the way people today often want to have it answered--can be more or less determined by the physical sciences, once known as the realm of "natural philosophy." Basically, key components of the Primordial Soup of Life appear to have been four compounds: water, ammonia, methane, and carbon dioxide--although phosphorous and a few other elements are essential for life as we know it. The recipe for Primordial Soup contains these four chemical elements: hydrogen (Atomic Number 1), carbon (6), nitrogen (7), and oxygen (8). The cosmic building block and most abundant element in the universe, hydrogen, combines with the oxygen to form water (H2O), with nitrogen to form ammonia (NH3), and with carbon to form methane (CH4); and the oxygen and the carbon combine to form carbon dioxide (CO2). Recent research has established that the particular form of life we enjoy here on the Earth is crucially determined by the molecular structure of DNA (DeoxyriboNucleic Acid). It may be enough for our purposes simply to point out that this grand and complex molecule is composed of precisely four nucleic acid bases: namely, the two pyramidines (thymine and cytosine) and the two purines (adenine and guanine). These four nitrogenous bases, together with associated sugar and phosphate molecules, are known as nucleotides; and they are linked together in specific combinations by weak hydrogen bonds to form the double helix of DNA. Nucleic acid bases, identical to corresponding components of DNA, have been formed in the laboratory just by mixing up the four essential compounds mentioned above. Together with the quintessential phosphorus atom--and energy from, say, the odd bolt of lightning as symbolic of the Will of the Creator--this simple soup would be sufficient, in theory, for the generation of life. Of course, this would not yet make life "as we know it," in all its variety, with the usual choice of either soup or salad, to say nothing of DES[S]ERT.

These fundamental chemical relationships
are clearly deeper and simpler than the complexities of macromolecular life;
but at an even deeper level, all physical existence may be constructed of
four different types of quarks: theoretical sub-atomic building blocks.
It may be that quarks, themselves, are made up of even Teenier constituents
in the game-like pursuit of Basic-Thing thinking wherein matter is conceived
as a sequence of nestled Russian *matryoshki* dolls. In its most fundamental
aspect, however, this analytical approach to being and existence--in both
theory and practice--is grounded upon the prior order of mathematics. As
we have noted above, mathematics is about the properties and relationships
of formal spaces, and particularly--at its deepest, most important levels--about
spaces not yet complex enough to admit of time, hence (to use precise language)
eternal. Its practice is traditionally considered to be not a science, but
an art.

By the time we have constructed spaces complex enough for the first order of time to begin--a mere oscillation, but with no duration --we also have the basis for counting with binary logic. Even so, this is not yet time as popularly conceived, with past, present and future, nor yet a time admitting the three Aristotelian poetic conventions of beginning, middle and end. Yet, from such a simple switching function (just this or that, on or off) other more complex spaces can be constructed, and with them, capabilities for representing both the recall of memory and the projection of plans, as today's computer logic so resplendently demonstrates. With such orders of complexity, we have more than sufficient logical resources for mapping the generation of the entire universe. Subsequently, we can perform operations in the familiar arithmetic of natural numbers such as those we use in the ordinary process of counting.

Within the class of things four, or things "in their fourness," suppose we direct our attention toward things that are NECESSARILY four, thereby hoping to reveal some essential properties of fourness itself. There are many fascinating things four in mathematics, such as the famous Four-Color Theorem in topology, that branch of mathematics dealing with properties of surfaces, with and without holes in them. For example, the question arises when coloring a map, say, like a map of modern Europe: How many colors are sufficient? If Germany is blue, France is yellow, and Italy is red, then what color is Switzerland? Obviously it must be green, or purple, or some fourth color, if its borders are to be distinguished from those of the adjacent, already-colored countries. Yet, as simple and as self-evident as this might appear, it is notoriously difficult to PROVE that four colors are sufficient for coloring ANY map that could be drawn on a plane surface. The problem turns out to be the same if the map is drawn on the surface of a sphere, which in the view of topologists is seen to have the same formal properties as a plane, those of a "genus zero" space, i.e., with no holes in it. Although a "proof" has been inferred by the number-crunching approach of recent computer attacks on the problem, other mathematicians with a cautious or skeptical eye and a rigorous respect for problems of "probability and scientific inference" are not prepared to be convinced by this kind of data.

[ See the recent work of G. Spencer-Brown; his early and seminal book on statistics--although it did not specifically treat the four-color problem later addressed elsewhere by Brown--bears the title:

Probability and Scientific Inference, Longmans, Green, London (1957).]

Spencer-Brown promised a solution for this problem in a paper which applies a second order calculus based on his Laws of Form.

[The] aim was to give the map in S1 [a plane surface or that of a sphere] a form that does not allow it more than four colors, and then to prove...that such a form will generate all maps....The nub of the proof lies in two theorems, the first essential to the method, and the second a crucial restatement of the four- color conjecture in a way that renders it susceptible to proof.

1. If a map is colorable with n colors, it is colorable with p primary colors, p being the least integer such that 2p>n.

2. Every standard map in a surface of unit connectivity can be expressed as a real construction of two sets of simple closed curves.

[ Letter from G. Spencer-Brown, dated 17 December 1976, originally addressed to the editors of Nature. ]

Spencer-Brown promised a book containing his complete proof of the four-color theorem, but the book never appeared. On one occasion when a Brownian alter-ego, Mr. James Keys, in lecture, outlined this most ingenious proof, he introduced heuristically a four-part metaphor from the world of letters (rather than numbers), that concerned the Platonic (or Platonistic) entities: the Real, the True, the Beautiful, and the Good. The idea was that if we had any three of them, we would automatically have the fourth; and this supplied a useful clue for devising the mathematical proof. For our discourse, we might also note that these four qualities are coupled like nuts and bolts in the everyday, dialectic mind: real and imaginary, true and false, good and bad, ugly and beautiful.

When we compare documents written in very precise language--such as mathematics, computer programs or musical scores--we find that ordinary words in the American-English language can sometimes acquire slightly different, though quite specific meanings. Accuracy and precision are esential to injunctions in particular; hence the heightened attention given to them in formal languages where they are so characteristically employed. For, while all of us occasionally employ injunctions, such as "Please pass the salt," the greater part of ordinary communication uses descriptive language. Naturally, there are several other formal categories of discourse as well. Sailors, for example, traditionally salt their language with expletives. And very young children fondly employ interrogatives, once they have learned from us the Big Game: pretending they do not already know everything they REALLY need to know, then asking adults interminable questions. However, an important part of the training for both sailors and small children involves learning to recognize (and to heed) injunctions, i.e. orders or commands, and to distinguish them from descriptions.

The special nature of injunctive language quite simply arises from the circumstance that it must be precise if one expects to have done what is being enjoined. This is the root wisdom of the Confucian counsel to call things by their proper names.

And there was to be no equivocation about what virtue, peace and justice really were. Basing themselves upon certain passages in the Analects, later...Confucians developed a doctrine of the "rectification of names" (chêng ming), i.e. the precise definition of actions and relations. (This might be described as the determination to call a spade a spade, no matter what powerful influences might be desirous of having it called something else)....The nicety of the distinctions made in the rectification of names may be seen from the fact that in the traditional text of the Chhun Chhiu, of the thirty-six acts of regicide there recorded some are qualified as shih (murder, implying the guilt of the assassin), while others are termed sha (killing, implying that the act was legally justified). Legally justified because Confucian teaching also contained the democratic idea that the prince (and later, the emperor) derived his power primarily from the will of the people, expressing Heaven's will or mandate....Thus that "right of rebellion against unchristian princes" which so exercised the minds of the 16th and 17th century theologians of Europe had already been laid down two thousand years before by the Confucian school.

[ Needham, Science and Civilisation, II, pp. 9-10. ]

Injunctions in the written law may be positive or negative (prohibitions) such as those requiring the President of the United States to uphold and defend the Constitution, and to observe those limita-ions on government (negatively) enjoined by the Bill of Rights. Among the uses of injunctive language, that of the cookbook comes reasonably close to ordinary communication. Still, one must learn what "separate two eggs," or what a "pinch of baking powder," or what "shave some ice" all really mean. Otherwise, a quite literal--and syntactically perfectly valid--interpretation could be (mis)construed, like that leading to one of the Three Stooges's funniest routines: Curly (or was it Shemp?) lathers up a block of ice, strops a straight razor, etc. Because, if we do not follow every respective injunction quite faithfully and in the proper sequence, we cannot reasonably expect, in the end, Providence to favor our enterprise. And so it is--or, in the United States of America, so it is supposed be--even for presidents.

The word REAL, in law, comes from the Latin word res, which has the same meaning as the Old English word THING, although THING also meant--as in Old Norse, from whence it came--an assembly. Further study of the lexical history and evolution of the word REAL reveals how REAL estate became confounded with ROYAL estate, often enough through the power of the Spanish silver coin, the real, the name for which is derived from rex, Latin for RULER or king. These lexical distinctions can be seen more clearly when we examine the Indo-European roots for the respective words, both of which are written the same way: R-E-A-L. In the sense of thingness, and in REAL estate, it derives from the root rei(3), from which we also derive cognates at REBUS, REIFY and REPUBLIC. On the other hand, in the opinion of Professor Calvert Watkins and the staff of the American Heritage Dictionary, REAL as ROYAL or rex (i.e. king, not president) comes from the basic root reg(1), meaning right, just, correct and straight, yielding cognates that include RULER (in both senses), the Spanish real, ERGO, and REALM, as well as RECKLESS and RECTUM.

In group theory, a branch of mathematics concerned with the most deeply generalized properties of systems and relationships, we may discover a profound basis for drawing distinctions between the real and the imaginary. For the record (without going into formal details just now), the word REAL in this insistently, unforgivingly strict, injunctive language refers to that part of a "complex number" which is not "imaginary." Our ordinary sense of language can be misleading here, because an "imaginary" number, strictly speaking, is every bit as real--or as unreal, if you rather--as a so-called real number. The existence of imaginary numbers had been implied since Renaissance mathematicians encountered an apparent paradox in trying to solve the quadratic equation X squared plus one equals zero. Approached with the standard techniques of high school algebra, this yields: X = plus or minus the square root of minus one. Although X is apparently some form of unity, it can be neither plus one (+1) nor minus one (-1) since both +1 and -1, when squared, give the product + 1, whereas the equation demands that the value of X squared be -1.

Of course we could take the position adopted in the Renaissance and earlier that quadratics leading to complex roots have no solution...[for] when mathematics gets to negative, irrational, and complex numbers, it departs [progressively and] radically from concepts which are intuitively familiar and readily under-standable. Ultimately one must acquire the more sophisticated view of mathematics....[which] rises above simple intuitions and employs concepts which are non-intuitive, or formal if you like, and yet these concepts are useful.

A pursuit of this whole matter really should involve the question: What is really intuitive? When we say that whole numbers and fractions are intuitive I believe we mean intuitive for people with certain experiences which are common to modern Western man. But there are and were primitive societies which never got to understand large numbers or operations with fractions. These societies would balk at the statement that large whole numbers and fractions are intuitive [i.e. self-evident]. Hence perhaps what we should recognize is that intuition is the product of experience and education and one can reach a level where the complex number is intuitively acceptable.

[ Morris Kline, Professor at the Courant Institute of Mathematics at New York University, quoted by Arthur Paul, "The Possible Meaning of Imaginary Numbers," in Charles Musés and Arthur M. Young, editors,

Consciousness and Reality: The Human Pivot Point, Outerbridge and Lazard, New York (1972), pp. 182, 194-5.]

That which is intuitively REAL in the world-view
of Buddhism, or from the similar but even earlier perspectives of Hindu
belief, is deeply conditioned by the ontological doctrine of maya: the world
as illusion. Indeed, the ephemeral qualities of experience, the transient
nature of life itself, and the impermanence of things are lessons the world
provides, daily, for awakening the self-awareness of all people; and who
is to say that this perception of what constitutes the "really real"
is not as intuitive as some judgment about the hardness of the rock against
which one stubs a toe? Charles Musés, the principal editor of *Consciousness
and Reality*, comments on Professor Kline's probe of the limitations
to Western ethnocentric intuition, which

implies that by a more significant and meaningful educational process, a society could produce, by teaching important ideas early and clearly enough, what less enlightened and cultivated societies would call a very high percentage of genius. Such a development would demand a very clear understanding of the fact that all algebras possess geometric and topological meaning and non-arbitrary representation. This understanding would then prepare the way for isomorphic models of profound ideas--thus conveying those ideas and their relationships into young minds with minimal burdens of terminology and verbiage. The most intelligent children then would not be intellectually deformed as so many now are into facile word jugglers. Rather they would be able to develop the ability to grasp ideas themselves and to think genuinely with imagination joined to logic.

We have found by actual pedagogy that the analytic geometry of hypernumber spaces, orbits and forms, together with studies and models in the geometry of three and more dimensions, provide the readiest and the most attractive and powerful means of promoting such an educational development. Education in this sense is actually the acceleration of human evolution. That acceleration in turn is a noetic process since man's special characteristic is his quality of mind. Things urgently needed often come to pass. Let it be so for such a program for advancement of insight for our children who, without considerably deeper understanding than the human past has shown, will scarcely be able to negotiate the crises--ecological, political and psychological--now facing the human race and...due to climax.

[ Muses and Young, Consciousness and Reality, p. 195. ]

The significance of complex numbers and the possibility of higher orders of imaginary, hypernumbers were dramatically realized in a genuinely visionary experience, meticulously recorded by the Irish mathematician, William Rowan Hamilton, in 1843. On the occasion of seeing, in a gleaming flash of insight, the nature of what he was to call "quarternions," Hamilton is said to have cut the key equation into a post on the bridge over the River Liffey in the same Phoenix Park in Dublin later to become a famed locale in James Joyce's Finnegans Wake. In that literary context William Rowan has to share mention along with two other Hamiltons, all three being tucked into an ollave's prolix riddle (the answer to which, by the way, is Finn Mac Cool!) The play on (the three) Hamiltons with himmeltones suggests the music of the spheres (Himmel = heaven in German). The mathematician Hamilton, for the last few decades of his life, shut himself up in his room to pursue his abstract speculations uninterruptedly. And so, sadly, Sir William obtains only one oblique accolade in the Wake:

and wholed a lifetime by his ain fireside wondering was it hebrew set to himmeltones or the quicksilversong of qwarternions

[ James Joyce, Finnegans Wake, the Viking Press, New York (1939; sixth printing, 1955), p. 137-138. ]

Thus, the first hypernumber, i, leads us to a sublime manifestation of fourness: not just four things because people have chosen to group them by fours, nor because four is habitual or handy, nor for reasons of poetry or magic, but rather an epiphany of fourness that MUST be and can ONLY be four...always was and ever will be. These are the four members of the group defined as the power series of i, where i stands for the "imaginary" value, the value represented in numerical notation as the square root of minus one. We must remember that the terms "real" and "imaginary" are wholly conventional from this technical perspective, and that there is no reason whatsoever to bestow upon the so-called real numbers any more or less actual reality than that which the so-called imaginary numbers do equally deserve.

In "Calvin and Hobbes," the comic strip by Bill Watterson (for January 6, 1988), six-year-old Calvin asks, "Here's another math problem I can't figure out. What's 9 + 4?" Hobbes (his stuffed tiger who comes to life whenever grownups aren't around) quizzically inspects the text, "Ooh, that's a tricky one. You have to use calculus and imaginary numbers for this." Calvin looks freaked, "IMAGINARY NUMBERS?!" But Hobbes explains, "You know, eleventeen, thirty-twelve, and all those. It's a little confusing at first." In the last panel, looking miffed, Calvin asks, "How did YOU learn all this? You've never even gone to school!" Hobbes replies with a superior air, "Instinct. Tigers are born with it." Current readers having imaginative instincts are invited to add their own, such as lebbenty-lebben times umpteen plus one is prime. In any case, given the imaginary value--which we can represent either symbolically as i or numerically as the square root of minus one--the power series is constructed by listing, in order, the values for i, when it is raised by successive exponents:

We begin with i to the zero power, which is easy when we remember that any number raised to the power of zero has the value of unity; therefore, i to the zero equals plus one.

Next is i to the first power, which again is easy because any number raised to the first power is equal to itself; therefore, i to the first power equals the square root of minus one, to give this value a numerical expression.

Then comes i to the power of two; that is i squared, or i times i. When we multiply the square root of minus one times the square root of minus one, we are left by definition (oof square root) with the value minus one; therefore, i squared equals minus one.

And i to the third power is the same as multiplying i squared times i. Now minus one (the value we just obtained for i squared) times i is the same as minus i; therefore, i to the third power equals minus the square root of minus one.

Once again, i to the fourth power is the same as multiplying the value of i squared times i squared. If we multiply minus one times minus one, we get plus one. Recall that this was the value of i to the zero, which also equaled plus one. So with i to the fourth, we are back where we started with i to the power of zero.

Yet again, if we go on with this series we have the subsequent values of plus i, minus one, minus i, plus one, plus i, minus one, minus i, plus one, and so forth.

GROUP THEORY AND THE COSMIC CYCLE

So we see that the power series of i is cyclic, with a two-fold symmetry: the one-phase alternation between real and imaginary values, and the two-phase alternation between plus and minus sign. There must be these four members of this group, and there can only be these four members of this group. Certain fourness! In fact, this same group was cited by way of example in a chapter on "The Group Concept," by the late Adrain Professor, Cassius J. Keyser of Columbia University:

For an example of a group that is finite and Abelian it is sufficient to take the system S whose [class] C is composed of the four numbers, 1, -1, i, and -i, where i is the square root of minus one, and whose rule of combination is multiplication; you notice that the identical element is 1, and that 1 and -1 are each its own reciprocal and that i and -i are each the other's reciprocal.

[

Cassius J. Keyser, Mathematical Philosophy: A Study of Fate and Freedom, New York (1922). A chapter, "The Group Concept" illustrates group theory in the "small library" edited by James R. Newman,The World of Mathematics, Volume III, Part 9, "The Supreme Art of Abstraction: Group Theory," Simon & Schuster, New York (1956), pp. 1540 ff. ]

Introducing Professor Keyser's essay, editor James R. Newman comments:

Group theory has to do with the invariants of groups of transformations. One studies the properties of an object, the features of a problem unaffected by changes of condition. The more drastic the changes, the fewer the invariants. What better way to get at the fundamentals of structure than by successive transformations to strip away the secondary properties. It is a method analogous to that used by the archaeologist who clears away hills to get at cities [Schliemann at Troy!], digs into houses to uncover ornaments, utensils and potsherds, tunnels into tombs to find sarcophagi, the winding sheets they hold and the mummies within [Howard Carter in the tomb of King Tutankhamen!].

...The theory is a supreme example of the art of mathematical abstraction. It is concerned only with the fine filigree of underlying relationships; it is the most powerful instrument yet invented for illuminating structure.

[ Newman, The World of Mathematics, III, pp. 1536, 1534. ]

Characteristically, Professor Keyser, who had broad interests in mathematics, philosophy and history, concludes this essay by drawing associations between the mathematical idea of a cyclic group--first technically formulated by the brilliant young mathematician Evariste Galois in 1830--and notions of the "Cosmic Cycle," also known as the "Platonic Year," corresponding with the Hindu idea of Kalpas and other ancient systems, and with ideas of the Phoenix and the sacred round.

The fact that so precise a formation of the mathematical concept of group is of so recent date is all the more curious because an idea closely resembling that of group has haunted the minds of a long line of thinkers and is found stalking like a ghost in the midst of philosophic speculation from remote antiquity down even to Herbert Spencer. I refer to those worldwide, age-long, philosophic speculations which, because of their peculiar views of the universe, may be fitly called the Philosophy of the Cosmic Cycle or Cosmic Year. This philosophy...of the cosmic cycle regards all the changes of which the universe is capable as constituting an immense indeed but finite and closed system of transformations, which follow each other in definite succession, like the spokes of a gigantic revolving wheel, until all possible changes have occurred in the lapse of a long but finite period of time--called a cosmic cycle or cosmic year--whereupon everything is repeated precisely, and so on and on without end. This philosophy...has lost its vogue; but, if the philosophy be true, will regain it, for if true, it belongs to the cosmic cycle and hence will recur.

[ Keyser, in Newman, The World of Mathematics, III, pp. 1554-1555. ]

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