Essays, Page 2

  Abbott's oddity began with his repeated name, which a mathematical wit might see as A times A or A Squared, A[sup 2]. Abbott's protagonist is A Square, a much troubled spirit. Liberated into another character, Abbott seems to have broken out of his cover as a prim reverend, and poured out his feelings.

  The book has a curiously obsessive quality, which perhaps accounts for its uneasy reception. Reviewers termed it "soporific," "prolix,"" mortally tedious," "desperately facetious, "while others found it "clever," "fascinating," "never been equaled for clarity of thought," and "mind broadening," and they even likened it to Gulliver's Travels. This last comparison is just, because beneath the math drolleries lurks a penetrating satire of Victorian society.

  A Square's society is as constrained as were the prim Victorians. Women are not full figures but mere lines. Soldiers are triangles with sharp points, adept at stabbing. The more sides, the higher the status, so hexagons outrank squares, and the high priests are perfect circles.

  In a delicious irony, the upper classes are polygons with equal sides --but their views certainly do not embrace equality. Mathematicians term equal-sided figures "regular," and in nineteenth century terms, proper upper class polygons are of the regular sort.

  A Square learns that his view of the world is too narrow. There is a third dimension, grander and exciting. but his hidebound fellows cannot see it. This opening-out is the central imaginative event of the novel, Abbott echoing an emergent idea.

  In the late nineteenth century higher dimensions were fashionable. Mathematicians had laid the foundations for rigorous work in higher-dimensional space, and physicists were about to begin using four-dimensional spacetime. Twenty centuries after Euclid, the mathematician Bernhard Riemann took a great leap in 1854, liberating the idea of dimensions from our spatial senses. He argued that ever since Rene Descartes had described spaces with algebra, the path to discussing higher dimensions had been dear, but unwalked.

  Descartes' analytic geometry defined lines as things described by one set of coordinates, distances along one axis. A plane needed two independent coordinate sets, a solid took three. With coordinates one could map an object, defining it quantitatively: not "Chicago is over that hill." but "Chicago is fifteen miles that way." This appealed more to our logical capacity, and less to our sensory experience.

  Riemann described worlds of equal logical possibility, with dimensions ranging from one to infinity. They were not spatial in the ordinary sense. Instead, Riemann took dimension to refer to conceptual spaces, which he named manifolds.

  This wasn't merely a semantic change. Weather, for example, depends on several variables -- say, n -- like temperature, pressure, wind velocity, time of day, etc. One could represent the weather as a moving point in an n-dimensional space. A plausible model of everyday weather needs about a dozen variables, so to visualize it means seeing curves and surfaces in a twelve-dimensional world. No wonder we understand the motions of planets (which even Einstein only needed four dimensions to describe), but not the weather.

  Riemann revolutionized mathematics and his general ideas diffused into our culture. By 1880, C.H. Hinton had pressed the issue by building elaborate models to further his extra-dimensional intuition, he tried to explain ghosts as higher-dimensional apparitions. Pursuing the analogy, he wrote of a fourth-dimensional God from whom nothing could be hidden. The afterlife, then, allowed spirits to move along the time dimension, reliving and reassessing moments of life. Spirits from hyper-space were the subject of J.K.F. Zollner's 1878 Transcendental Physics, which envisioned them moving everywhere by short-cut loops through the fourth dimension.

  Mystics responded to the fashion by imagining that God, souls, angels and any other theological beings resided as literal beings of mass ("hypermatter") in four-space. This neatly explains why they can appear anywhere they like, and God can be everywhere simultaneously, the way we can look down on a Flatland and perceive it as a whole. Some found such transports of the imagination inspiring, while others thought them crass and far too literal. I am unaware of Abbott himself ever subscribing to such beliefs.

  Still, Abbott and his adventure-some Square longed for the strange. More than any other writer, Abbott coined the literary currency of dimensional metaphor. By having a point of view which is literally above it all, surveying the follies of a two-dimensional plane, Abbott can adroitly satirize the staid rigidities of his Victorian world. (Perhaps this is why he first published Flatland under a pseudonym.)

  "Irregulars" are cruelly executed, for example. Do they stand for foreigners? Gypsies? Cripples? We are left to fill in some blanks, but the overall shape of the plot is clear -- flights of fancy are punished, and A Square does not finish happily.

  At a deeper level, the book harks toward deep scientific issues, and the difficulty of comprehending a physical reality beyond our immediate senses. This is the great theme of modem physics. The worlds of relativity and the quantum are beyond the rough-and-ready ideas we chimpanzees have built into us, from our distant ancestors' experience at throwing stones and poking sticks on African plains.

  Still deeper, in this fanciful narrative the good Reverend tries to speak indirectly of intense spiritual experience. The trip into the higher realm of three dimensions is a fine metaphor for a mystical encounter.

  The thrust of the deceptively simple narrative is to make us examine our basic assumptions. After all, our visual perceptions of the world are two-dimensional patterns, yet we somehow know how to see three-dimensionality. One knows instantly the difference between a ball and a fiat disk by their shading in available light. Objects move in front of each other, like a woman walking by a wall. We automatically discount a possible interpretation -- that the woman has somehow dissolved the wall for an instant as she passes. Instead, we see her in her three-dimensionality. The eye has learned the world's geometry and discards any other scheme.

  A Square learns this lesson early as he first visits Lineland in a dream. The only distinction the natives can have is in their length. They see each other as points, since they move along the same universal straight line. They estimate how far away others are by their acute sense of hearing picking up the difference between a bass left voice and a tenor right; the time lag in arrival tells the distance. The king is longest, men next, then boys are stubby lines. Women are mere points, of lower status. Their views of each other are partial and instinctive. They never dream of how narrowly they see their world.

  This sets the stage for A Square's conceptual blowout when a Sphere visits him and yanks him up into the hallucinogenic universe of three dimensions. Its realities are surrealistic. A Square straggles to fathom what for us is instinctive.

  The reality of three dimensions we take for granted, but for us, what is the reality of two dimensions? Would flatlanders have physical presence in our world -- that is, could we perceive a two-dimensional universe embedded in our own? Could we yank them up into our world?

  Flatlanders could be as immaterial as shadows, mere patterns in our view. If an isosceles triangle soldier cut your throat it would not hurt. Abbott did not consider this in his first edition, but in the second he says that A Square eventually believes that flatlanders have a small but real height in our universe. A Square discusses this with the ruler of Flatland:

  * I tried to prove to him that he was "high," as well as long and broad, although he did not know it. But what was his reply? "You say I am 'high'; measure my 'highness' and I will believe you." What could I do? I met his challenge!

  If flatlanders were even quite thick, they would not be able to tell, if in that direction they had no ability to move or did not vary. Height as a concept would lie beyond their knowable range. Or if they did vary in height, but could not directly see this, they might ascribe the differences to qualitative features like charisma or character or "presence." There would be rather mysterious forces at work in their world, the Platonic shadows of a higher, finer reality.

  If a flatlander soldier of genuine physical
thickness attacked, it would cut us like a knife. Otherwise, it could not impinge upon us. We would remain oblivious to all events in the lesser dimensions.

  In a sense, a truly two-dimensional flatlander faces a similar problem if it tries to digest food. A simple alimentary canal from stem to stem of, say, a circle would bisect it. To keep itself intact, a circle would have to digest by enclosing whatever it used for food in pockets, opening one and passing food to the next like a series of locks in a canal, until eventually it excreted at the far end.

  This is typical of the problems engaged by thinking in another dimension. Not until 1910 did artists respond to non-Euclidean spaces, with Cubism and its theories. Mute image and poetic metaphor, they said, were ways of perceiving what scientists could only describe in abstractions and analogies.

  They were right, and many, including Picasso and Braque, struggled with the problem. Looking downward at lower dimensions is easy. Looking up strains us.

  Visualizing the fourth dimension preoccupied both mists and geometers. A cube in 4D is called a tesseract. One way to think of it is to open a cubical cardboard box and look in. By perspective, you see the far end as a square. Diagonals (the cube edges) lead to the outer "comers" of a larger square -- the cube face you're looking through. Now go to a 4D analogy. A hypercube is one small cube, sitting in the middle of a large cube, connected to it by diagonals. Or rather, that is how it would look to us, lowly 3D folk.

  Cutting a hypercube in the right way allows one to unfold it and reform it into a 3D pattern of eight cubes, just as a 3D cube can be made up of six squares. One choice looks like a sort of 3D cross. Salvador Dali used this as a crucifix in his 1954 painting Christus Hypercubus. Not only does the hypercube suggest the presence of a higher reality; Dali deals with the problem of projecting into lower dimensions. On the floor beneath the suspended hypercube, and the crucified Christ, is a checkerboard pattern -- except directly below the hypercube. There, the hypercube's shadow forms a square cross. (Shadows are the only 2D things in our world; they have no thickness.) Comparing this simple cross with the reality of the hypercube which casts the shadow, we contemplate that our world is perhaps a pallid shadow of a higher reality, an implicit mystical message.

  Robert Heinlein gave this a twist with "And He Built a Crooked House," in which a house built to this pattern folds back up, during an earthquake, into a true hypercube, trapping the inhabitants in four dimensions. Much panic ensues.

  Rudy Rucker, mathematician and science fiction author, has taken A Square and Flatland into myriad fresh adventures. I met Rucker in the 1980s and found him much like his fictional narrators, inventive and wild, with a cerebral spin on the world, a place he found only apparently commonplace. His The Sex Sphere (1983) satirizes dimensional intrusions, many short stories develop ideas only latent in Flatland, and his short story "Message Found in a Copy of Flatland" details how a figure much like Rucker himself returns to Abbott's old haunts and finds the actual portal into that world in the basement of a Pakistani restaurant. He finds that the triangular soldiers can indeed cut intruders from higher dimensions, and flatlanders are tasty when he gets hungry. As a sendup of the original it is pointed and funny.

  In science fiction there have been many stories about creatures from the fourth dimension invading ours, generally with horrific results. Greg Bear's "Tangents" describes luring 4D beings into our space using sound. While we puzzle over whether an unseen fourth dimension exists, modem physics has used the idea in the Riemannian manner, to expand our conceptual underpinnings. Riemann saw a mathematical theme of conceptual spaces, not merely geometrical ones. Physics has taken this idea and run with it.

  Abbott's solving the problem of flatlander physical reality by adding a tiny height to them was strikingly prescient. Some of the latest quantum field theories of cosmology begin with extra dimensions beyond three, and then "roll up" the extras so that they are unobservably small --perhaps a billion billion billion times more tiny than an atom. Thus we are living in a universe only apparently spatially three-dimensional; infinitesimal but real dimensions lurk all about us. In some models there actually are eighteen dimensions in all!

  Even worse, this rolling up occurs by what I call "wantum mechanics" --we want it, so it must happen. We know no mechanism which could achieve this, but without it we would end up with unworkable universes which could not support life. For example, in such field theories with more than three dimensions, which do not roll up, there could be no stable atoms, and thus no matter more complex than particles. Further, only in odd-numbered dimensions can waves propagate sharply, so 3D is favored over 2D. In this view, we live not only in the best of all possible worlds, but the only possible one.

  How did this surrealistically bizarre idea come about? From considering the form and symmetries of abstruse equations. In such chilly realms, beauty is often our only guide. The embarrassment of dimensions in some theories arises from a clarity in starting with a theory which looks appealing, then hiding the extra dimensions from actually acting in our physical world. This may seem an odd way to proceed, but it has a history.

  The greatest fundamental problem of physics in our time has been to unite the two great fundamental theories of the century, general relativity and quantum mechanics, into a whole, unified view of the world. In cosmology, where gravity dominates all forces, general relativity rules. In the realm of the atom, quantum processes call the tune.

  They do not blend. General relativity is a "classical" theory in that it views matter as particles, with no quantum uncertainties built in. Similarly, quantum mechanics cannot include gravity in a "natural" way.

  Here "natural" means in a fashion which does not violate our sense of how equations should look, their beauty. Aesthetic considerations are very important in science, not just in physics, and they are the kernel of many theories. The quantum theorist Paul Dirac was asked at Moscow University his philosophy of physics, and after a moment's thought wrote on the blackboard, "Physical laws should have mathematical beauty." The sentence has been preserved on the board to this day.

  One can capture a theorist's imagination better with a "pretty" idea than with a practical one. There have even been quite attractive mathematical cosmologies which begin with a two-dimensional, expanding universe, and later jump to 3D, for unexplained reasons.

  Einstein wove space and time together to produce the first true theory of the entire cosmos. He had first examined a spacetime which is "flat," that is, untroubled by curves and twists in the axes which determine coordinates. This was his 1905 special theory of relativity. He drew upon ideas which Abbott had already used.

  The Eminent British journal Nature published in 1920 a comparison of Abbott's prophetic theme:

  * (Dr. Abbott) asks the reader, who has consciousness of the third dimension, to imagine a sphere descending upon the plane of Flatland and passing through it. How will the inhabitants regard this phenomenon? . . . Their experience will be that of a circular obstacle gradually expanding or growing, and then contracting, and they will attribute to growth in time what the external observer in three dimensions assigns to motion in the third dimension. Transfer this analogy to a movement of the fourth dimension through three-dimensional space. Assume the past and future of the universe to be all depicted in four-dimensional space and visible to any being who has consciousness of the fourth dimension. If there is motion of our three-dimensional space relative to the fourth dimension, all the changes we experience and assign to the flow of time will be due is reply to this movement, the whole of the future as well as the part always existing in the fourth dimension.

  In special relativity, distance in spacetime is not the simple result we know from rectangular geometry. In the ordinary Euclidean geometry everyone learns in school, if "d" means a small change and the coordinates of space are called x, y and z, then we find a small length (ds) in our space by adding the squares Of each length, so that

  * (ds)[sup 2] = (dx)[sup 2] + (dy)[sup 2] + (dz[sup 2]<
br />
  The symbol "d" really stands for differential, so this is a differential equation.

  Contrast special relativity, in which a small distance in space-time adds a length given by dt, a small change in time, multiplied by the speed of light, c:

  * (ds)[sup 2] = (dx)[sup 2] + (dy)[sup 2] + (dz)[sup 2] center dot (cdt)[sup 2]

  The trick is that the extra length (cdt) is subtracted, not added. This simple difference leads to a whole restructuring of the basic geometry. The mathematician Minkowski showed this some years after Einstein formulated special relativity.

  A thicket of confusions lurks here. Reflect that the total small (or differential, in mathematical language) length is (ds), found by taking the square root of the above equation. But if (cdt) is greater than the positive (first three) terms, then (ds) is an imaginary number! What can this mean? Physically, it means the rules for moving in this four-dimensional (4D) space are complex and contrary to our 3D intuitions. Different kinds of curves are called "spacelike" and "timelike," because they have very different physical properties.

  Einstein was fond of saying that he viewed the world as 4D, with people existing in it simultaneously. This meant that in 4D the whole life of a person (their "world-line") was on view. Life was eternal, in a sense --a cosmic distancing available mostly to mathematicians and lovers of abstraction.

  Einstein's was the first major scientific use of time as an added dimension, though literature had gotten there first. By 1895 the widespread use of dimensional imagery led H.G. Wells to depict time as just another axis of a space-like cosmos, so that one could move forward and back along it. In a sense Wells's use domesticated the fourth dimension, relieving it of genuinely jarring strangeness, and ignoring the possibility of time paradox, too.