HYPERSPACE

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..."That Cubists and other modernist artists should be inter-ested in higher-dimensional space is hardly surprising, for a pri- mary thrust of early twentieth-century art was to break with the tradition of perspective. If it turned out that physical space was not in fact three-dimensional, then the rules of linear perspective would simply be arbitrary. The possibility of higher-dimensional space thus served a powerful rhetorical function for the nascent modems. Recognizing this explicitly, Gleizes and Metzinger stated in Du Cubism that "If we wished to tie the painter's space to a particular geometry, we should have to refer it to the non- Euclidean scholars."17
The painter who most seriously took up this challenge was Marcel Duchamp. Originally associated with the Cubists, Duchamp soon spun off onto his own peripatetic paths. Like Malevich, his most famous work was also inspired by the fourth di-mension. The Bride Stripped Bare by Her Bachelors, Even, often known as The Large Glass, is surely one of the most pondered-over
works in the modem canon; and this time we have extensive notes by the artist detailing the process of genesis. Specifically, we know that in preparing for this work Duchamp embarked on a study of non-Euclidian and higher-dimensional geometries. The end result of these efforts was a complex work divided into two distinct halves: in the top half is the "Bride," and in the bottom half the "Bachelor Apparatus." According to Duchamp's notes the Bride is supposed to be a four-dimensional entity, while the bachelors are three- dimensional. Floating above her retinue, this higher-space spouse hovers enigmatically in a world of her own.

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With all this artistic, literary, and mystical speculation about a fourth dimension, what delicious synchronicity when the theory of relativity suddenly;enshrined the concept in physical reality. Einstein's revelation of the fourth dimension seemed to many hy-perspace enthusiasts a confirmation of what they had known all along. The common thread running between the worlds of rela-tivistic physics and that of the writers and artists was of course the new mathematics of non-Euclidian geometry. Ironically, many of the new-math pioneers had themselves been driven to their radi-cal geometries by a scientific interest in the structure of physical space. To these men, Gauss included, their fantastical new geo-metries had originally evolved as tools for helping them to better understand the nature of the concrete physical world. Thus while they are generally remembered today as mathematicians, along with Einstein these men ought also to be recognized as pioneers in the physics of space.
In fact, the whole development of non-Euclidian geometry that Gauss initiated emerged out of his work on the measurement of the earth. Given that the literal meaning of the word "geome-try" is "earth measurement," this was particularly apt. In its origi-nal incarnation, the science of geometry had emerged from ancient Egyptian surveying of the Nile Delta. This ancient (i.e., Euclidian) geometry had only dealt with flat space, such as the surface of this page. On a large scale, however, the surface of the earth is spherical, and hence curved. Thus a study of the earth's surface ultimately requires a geometry of curved surfaces. Gauss' seminal papers on curved-space geometry were inspired by his stint as scientific advisor to a geodetic survey of the region of Hanover. "Once again," says Max Jammer, "we see that histori-cally viewed, abstract theories of space owe their existence to the practice of geodetic work."18
Humans had long known that the surface of our planet is curved, but what about the space in which our globe is embedded? / Page 202 / Might space itself be curved? For Newton and his contemporaries there had been no mathematical alternatives to Euclidian space so they had simply assumed that this was-the correct model for phys-ical space. But after Gauss' work on curved surfaces, he began to wonder if the assumption of a Euclidian universe was justified. In the early nineteenth century -long before Einstein was born- Gauss actually tried to measure the curvature of physical space. He did this by the ingenious method of surveying a triangle formed by three mountaintops. In Euclidian, or flat space, three angles of a triangle must add up to 180 degrees, but if the space is curved the angles will add up to something else. (To more than 180 if the space is "positively" curved, like a sphere, and less than 180 if it is "negatively" curved, like a saddle.) Since Gauss failed to find any deviation from 180 degrees, he concluded that at least in the vicin- ity of the earth, space must be Euclidian.
The later Russian mathematician, Nikolai Ivanovich Lobachevsky, would try a similar experiment but on a much larger scale. Instead of mountains, Lobachevsky used distant stars, yet still he found no deviation from flat space. Both Gauss and Lobachevsky concluded, based on the evidence available to them, that our local area of the universe was Euclidian, but both realized there was no reason why this must be the case. As Gauss pre-sciently put it: "In some future life, perhaps, we may have other ideas about space which, at present, are inaccessible to us "19
While Gauss and Lobachevsky pioneered the idea of curved space, later in the nineteenth century a brilliant young mathe-matician named Bernhard Riemann even considered the pos- sibility that gravity was a by-product of curvature in higher- dimensional space. While there is no doubt that Einstein thought up this concept for himself, it is worth noting that the idea had already been imagined more than half a century before. The young man responsible for this astonishing insight was a disciple of Gauss, and he remains one of the most underrated / Page 203 / visionaries in modern science. Today Riemann is generally re-membered as a pure mathematician, but what really interested this pathologically shy Austrian was the problem of how physical forces arise. Decades before Einstein's birth, Riemann became convinced that the explanation for gravity must lie in the geometry of space.
Thinking about the problem of physical forces, Riemann imagined a world not unlike Abbott's Flatland, in which a race of two-dimensional creatures were living on a flat sheet of paper. Now what would happen, Riemann asked, if we crumpled the paper? Because the creatures' bodies are embedded in the paper, they would not be able to see the wrinkles - to them their world would still look perfectly flat. Yet Riemann realized that even if the space looked flat, it would no longer behave as if it were flat. He ar-gued that when the creatures tried to move about in their two - dimensional world they would feel a mysterious unseen "force" whenever they hit one of the wrinkles, and they would no longer be able to move in straight lines.
Extrapolating this idea to our three-dimensional universe, Riemann imagined that our three-dimensional space was also "crumpled" in an unseen fourth dimension. Like the two-dimensional beings of the paper universe, he reasoned that al-though we could not see such "wrinkles" in the space around us, we too would experience them as invisible forces. From this bril-liant insight, Riemann concluded that gravity was "caused by the crumpling of our three-dimensional universe in the unseen fourth dimension."20 Having outlined his basic theme, this shy genius set about developing a mathematical language in which to express these ideas. The result of his labors was the new geometry that Ein-stein would eventually use in his general theory of relativity. "In retrospect," says physicist Michio Kaku, "we now see how close Riemann came to discovering the theory of gravity 60 years before Einstein."2l In one way or other, speculations about the physics of Flatland have had profound consequences for us all.

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Einstein's "discovery"-of a fourth dimension must surely rate as one of modem science's most amazing findings. With this dis-covery, man was now in a position (like the Square in Abbott's tale) to see his world from a new perspective. But as the Square said to Lord Sphere, why stop at four dimensions? With our vision thus expanded, might we too not "resolve that our ambition [should] soar" onward and upward to higher dimensions still? And since human beings are as naturally curious as Squares, indeed it was not long before someone began to dream about a fifth dimension. In the 1920s, a young Polish mathematician had the bright idea that if the force of gravity could be explained by the geometry of four-dimensional space, then perhaps he might be able to explain the electromagnetic force by the geometry of five- dimensional space. With this seeming science fiction fantasy begins one of the most curious episodes in the history of space.
If Riemann was a maverick in the history of science, Theodr Kaluza was decidedly an oddity. An obscure mathematician at the University of Konigsberg (in what is now Kaliningrad in the former Soviet Union), Kaluza was convinced that Einstein's approach to gravity could be expanded and enhanced. In particular, he wanted to apply Einstein's approach to the electromagnetic force-the force responsible for electricity, magnetism, and, light. Along with Riemann, in fact, Kaluza believed that electromagnetism must also be the result of curvature (or ripples) in a higher-dimensional space. But the problem Kaluza faced was that there did not seem to be any more dimensions left. With three of space and one of time, nature's stock seemed to be exhausted.
Yet Kaluza was not a man to be deterred by such prosaic objections. In an audacious move he simply rewrote Einstein's equations of general relativity in five dimensions. Lo and behold, when he did so it turned out that these five-dimensional equations contained within them the regular four-dimensional equations of relativity, plus an extra bit which turned out to be / Page 205 / precisely the equations of electromagnetism. In effect, Kaluza's five-dimensional theory consisted of two separate pieces that fit- ted together like a jigsaw puzzle-Einstein's theory of gravity and Maxwell's theory of.-.electromagnetism (the field equations of light).
Another way of understanding this "mathematical miracle," says physicist Paul Davies, is that "Kaluza showed that electro-magnetism is actually a form of gravity." Not the regular gravity of everyday physics, but "the gravity of an unseen [fifth] dimension of space."22 In 1919 Kaluza sent a paper on all this to Einstein. So stunned was the great physicist by the young Pole's radical addition of an extra dimension that like Lord Sphere in Abbott's Flatland, he was appalled. For two years Einstein apparently refused to an- swer Kaluza's letter. But the whole construction was so mathe-matically elegant he could not get it out of his mind, and finally in 1921 he became convinced of the importance of Kaluza's ideas and submitted the paper to a scientific journal.
Ironically, it was the very beauty of Kaluza's construction that so shook Einstein, and many other physicists. Was this five- dimensional space of Kaluza's "just a parlor trick? Or numerology? Or black magic?"23 It was all very well to propose that time was a fourth dimension (for that, after all, is a real aspect of our physical experience), but what on earth was this supposed fifth dimension? If Kaluza's equations were to be taken seriously-and not just as mathematical chicanery-then the awkward question arose: Where is this extra dimension? Why don't we see it?
To this query Kaluza had a disarmingly simple answer. He declared that the extra dimension is so small it escapes our normal attention. The reason we don't see, he said, it is because it is mi-croscopic. To understand this proposal, again it is helpful to resort to a lower-dimensional analogy. Imagine this time that you live on a line, what we might call Lineland, the one-dimensional sibling of Flatland. As a dot in this linear universe, you can travel up and / Page 206 / down your line, always remaining in a single dimension. Now suppose that one day a scientist in your Lineland announces she has discovered an extra dimension and that your universe is really two-dimensional. At first you think she is mad. Where is this other dimension? you ask. Why can't we see it? But then the scientist ex-plains that in fact you don't live on a line, but on a very thin hose. Each point of your line universe is not really a point, but a tiny circle, one so small that you never notited it. Taking this extra microscopic dimension into account, your world is not a line, but really a two-dimensional cylindrical surface.
This was the essence of Kaluza's explanation for his fifth di-mension. According to him, every point in our three dimensions of space is actually a tiny circle, so that in reality there are four di-mensions of space, plus one of time, making a total of five. In 1926 the Swedish physicist Oskar Klein made improvements to Kaluza's theory which enabled him to calculate the size of this tiny hidden dimension. According to Klein's calculations, it was no wonder we had not observed the extra direction because it is ab-solutely minute. Its circumference was just 10-32 centimeters-a hundred billion billion (102°) times smaller than the nucleus of an atom!
So small was Kaluza's dimension that even if we ourselves were the size of atoms we would still not notice it. Yet this tiny di-mension could be responsible for all electromagnetic radiation: light, radio waves, X rays, microwaves, infrared, and ultraviolet. A powerful punch indeed for something so small. Unfortunately, the Kaluza-Klein dimension was so small there was no way of measuring it directly. Even our largest accelerators today still cannot measure things on such a minute scale. So what then are
we to make of Kaluza's vision? Is this fifth dimension physically real? Or is it just an elegant mathematical fiction?
Kaluza himself insisted that the beauty of his theory could not "amount to the mere alluring play of a capricious accident."24

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He firmly believed in the reality of his fourth spatial dimension. He knew his tiny dimension could not be tested directly, so he de-cided instead to conduct an experiment of his own to test the gen-eral correspondence between theory and reality. The test case he chose was not anything from the realm of physics, but the art of swimming. As someone who could not swim, Kaluza decided he would learn all he could about the theory of swimming and when he had mastered that then he would test this theoretical framework against the reality of the sea. Giving himself over to the,project, he diligently studied all aspects of the aquatic art until finally he felt he was ready. Now, trunks in hand, the young Pole escorted his family to the seaside for the crucial test. With no prior experience, in front of the assembled Kaluza clan, Theodr hurled himself into the waves. . . and 10 and behold he could swim! Theory had been born out by practice in the real world. Could the tiny dimension also be there in the real world?"

 

4	REAL	36	18	9
5	WORLD	72	27	9
7	REALITY	90	36	9

"Unfortunately, if in Kaluza's own mind the swimming ex-periment supported a general correspondence between theory and reality, few others were willing to embrace the idea of an unseen and unmeasurable fifth dimension. Sadly, after an initial flurry of interest, the physics community turned away. Yet the startling el-egance of Kaluza's equations raised an uneasy question: How many dimensions of space are there really in the world around us?
As happens so often in the history of science, it was not in fact a new question. As long ago as the second century, Ptolemy had considered the matter and had argued that no more than three di-mensions are permitted in nature. Kant also had argued that three dimensions are inevitable. In this he could call upon the support a good deal of hard science. For instance it is well known that gravity and the electromagnetic force both obey "inverse square laws" - the strength of the force drops off according to the square of the distance. As early as 1747, "Kant recognized the deep con-nection between this law and the three-dimensionality of space."25

BREWER'S

 DICTIONARY OF PHRASE AND FABLE

Ivor H Evans

1985

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"Nimbus (Lat., a cloud). In Christian art a HALO of light placed round the head of an eminent personage. There are three forms: (1) Vesica piscis, or fish form (cp. ICHTHYS), used in representations of Christ and occasionally of the Virgin Mary, extending round the whole figure; (2) a circular halo; (3) radiated like a star or sun. The enrichments are: (1) for our Lord, a CROSS; (2) for the Virgin, a circlet of stars; (3) for ANGELS, a circlet of small rays, and an outer circle of quatrefoils; (4) the same for SAINTS and martyrs, but with the name often inscribed round the circumference; (5) for the Deity the rays diverge in a triangular direction. Nimbi / Page 786 / of a square form signify that the persons so represented were living when they were painted.
The nimbus was used by heathen nations long before painters Introduced it into sacred pictures of saints, the TRINITY, and the Virgin Mary. PROSER. PINE was represented with a nimbus; the Roman EMPERORS were also decorated in the same manner because they were divi.  Cpo AUREOLE."

"Nimrod. Any daring or outstanding hun-ter; from the "mighty hunter before the Lord" (Gen. x, 9 , which the TARGUM says means a "sinful hunting of the sons of men". Pope says of him, he was "a mighty hunter, and his prey was man" (Windsor Forest, 62); so also Milton inter-prets the phrase (Paradise Lost, XII, 24, etc.).
The legend is that the tomb of Nimrod still exists in Damascus, and that no dew ever falls upon it, even though all its sur-roundings are saturated..."

Nine. Nine, FIVE, THREE are mystical num-mbers-the DIAPASON, diapente, and dia-trion of the Greeks. Nine consists of a trinity of trinities. According to the Pythagoreans man is a full chord, or eight notes, and deity comes next. Three, being the TRINITY, represents a perfect unity; twice three is the perfect dual; and thrice three is the perfect plural. This explains why nine is a mystical number.
From ancient times the number nine has been held of particular significance. DEUCALION'S ark was tossed about for nine days when it stranded on the top of Mount PARNASSUS. There were nine MUSES, nine Gallicenae or virgin priest-esses of the ancient Gallic ORACLE; and Lars Porsena swore by nine gods.
NIOBE'S children lay nine days in their blood before they were buried; the HYDRA had nine heads; at the Lemuria, held by the Romans on 9, 11 , and 13 May, persons haunted threw black beans over their heads, pronouncing nine times the words: "Avaunt, ye spectres, from this house!" and the EXORCISM was complete (see Ovid's Fasti).
There were nine rivers of HELL, or, according to some accounts, the STYX en-compassed the infernal regions in nine circles; and Milton makes the gates of HELL "thrice three-fold", "three folds were brass, three iron, three of adaman-tine rock". They had nine folds, nine plates, and nine linings (Paradise Lost, II, 645).
VULCAN, when kicked from OLYMPUS, was nine days falling to the island of LEM- NOS; and when the fallen ANGELS were cast out of HEAVEN Milton says "Nine days they fell" (Paradise Lost, VI, 871).
In the early Ptolemaic system of astronomy, before the PRIMUM MOBILE was added, there were nine SPHERES; hence Milton, in his Arcades, speaks of

The celestial siren's harmony
That sat upon the nine enfolded spheres.

In Scandinavian mythology there were nine earths, HEL being the goddess of the ninth; there were nine worlds in NIFL-HElM, and ODIN'S ring dropped eight other rings every ninth night.
In folk-lore nine appears frequently.
The ABRACADABRA was worn nine days, and then flung into a river; in order to see the FAIRIES one is directed to put "nine grains of wheat on a four-leaved clover"; nine knots are made on black wool as a charm for a sprained ankle; if a servant fmds nine green peas in a peascod, she lays it on the lintel of the kitchen door, and the fIrst man that enters is to be her cavalier; to see nine magpies is most un-lucky; a cat has nine lives (see also CAT O'NINE TAILS); and the nine of Diamonds is known as the CURSE OF SCOTLAND.
The weird sisters in Shakespeare's Macbeth (I, ill) sang, as they danced round the cauldron, "Thrice to thine, and thrice to mine, and thrice again to make up nine"; and then declared "the charm wound up"; and we drink a Three- limes-three to those most highly hon-oured.
Leases are sometimes granted for 999 years, that is three times three-three-three.

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Many run for 99 years, the dual of a trinity of trinities.
See also the NINE POINTS OF THE LAW below, and the NINE WORTHIES under WORTHIES. There are nine orders of angels; in HERALDRY there are nine marks of cadency and nine different crowns recognized.
Dressed up to the nines. See DRESSED. Nine days' Queen. Lady Jane Grey. She was proclaimed queen in London on 10 July 1553; Queen Mary was proclaimed in London on 19 July.
Nine days' wonder. Something that causes a great sensation for a few days, and then passes into the LIMBO of things forgotten. An old proverb is: "A wonder lasts nine days, and then the puppy's eyes are open", alluding to dogs, which like cats, are born blind. As much as to say, the eyes of the public are blind in aston-ishment for nine days, but then their eyes are open, and they see too much to won-der any longer.

King: You'd think it strange if I should marry her. Gloster: That would be ten days' wonder, at the least. Clar.: That's a day longer than a wonder lasts.
SHAKESPEARE: Henry VI, Pt. III, III, ii.

The Nine First Fridays. In the ROMAN CATHOLIC CHURCH the special observ-ance of the first FRIDAY in each of nine consecutive months, marked by receiv-ing the EUCHARIST. The practice derives from St. Mary Alacoque (see SACRED
HEART under HEART), who held that Christ told her that special grace would be granted to those fulfilling this observ-ance.
Nine Men's Morris. See under MORRIS. Nine-tail bruiser. Prison slang for the CAT-O'-NINE-TAILS.
Nine tailors make a man. See TAILOR.
Nine times out of ten. Far more often] than not.
Possession is nine points of the law. It is every advantage a person can have short of actual right. The "nine points of the law" have been given as: (1) a good deal of money; (2) a good deal of patience; (3) a good cause; (4) a good lawyer; (5) a good ]
counsel; (6) good witnesses; (7) a good jury; (8) a good judge; and (9) good luck. To look nine ways. To squint.
Ninepence. Commendation Nine-pence. See COMMENDATION.
Nice as ninepence. A corruption of "Nice as nine-pins". In the game of nine- pins, the "men" are set in three rows with the utmost exactitude or nicety.
Nimble as ninepence. Silver ninepences were common till the year 1696, when all Unmilled coin was called in. These nine- pences were very pliable or "nimble", and, being bent, were given as love tokens, the usual formula of presentation being To my love, from my love. There is an old proverb, A nimble ninepence is bet-ter than a slow shilling.
Ninepence to a shilling. An old rustic phrase in the West of England meaning that the person referred to is deficient in common sense or intelligence.
Right as ninepence. Perfectly well; in perfect condition.
Ninus. Son of Belus, husband of SEMI-RAMIS, and the reputed builder of Nineveh. It is at his tomb that the lovers meet in the PYRAMUS and This be trav-esty:

Pyr.: Wilt thou at Ninny's tomb meet me straight- way?
This.: 'Tide life, 'tide death, I come without delay.
SHAKESPEARE: Midsummer Night's Dream, V, i.

Niobe (ni' o be). The personification of maternal sorrow. According to Greek legend, Niobe, the daughter of TANTA-LUS and wife of AMPHION, King of THEBES, was the mother of fourteen chil-dren, and taunted LATONA because she had but two)-APOLLO and DIANA. Lato-na commanded her children to avenge the insult and they consequently de-stroyed Niobe's sons and daughters. Niobe, inconsolable, wept herself to death, and was changed into a stone, from which ran water, "Like Niobe, all tears" (SHAKESPEARE: Hamlet, I, ii).
The Niobe of Nations. So BYRON styles ROME, the "lone mother of dead empires", with "broken thrones and temples"; a "chaos of ruins"; a "desert where we steer stumbling o'er recollec-tions" (Childe Harold, iv, 79).

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