Gregory Benford - Essays and Short Stories Part 15

They were right, and many, including Pica.s.so 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 a.n.a.logy. 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 s.e.x 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 bas.e.m.e.nt 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 s.p.a.ce 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 s.p.a.ces, 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 un.o.bservably 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 embarra.s.sment 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 "cla.s.sical" 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 s.p.a.ce and time together to produce the first true theory of the entire cosmos. He had first examined a s.p.a.cetime 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 pa.s.sing 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 a.s.signs to motion in the third dimension. Transfer this a.n.a.logy to a movement of the fourth dimension through three-dimensional s.p.a.ce. a.s.sume the past and future of the universe to be all depicted in four-dimensional s.p.a.ce and visible to any being who has consciousness of the fourth dimension. If there is motion of our three-dimensional s.p.a.ce relative to the fourth dimension, all the changes we experience and a.s.sign 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 s.p.a.cetime 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 s.p.a.ce are called x, y and z, then we find a small length (ds) in our s.p.a.ce by adding the squares Of each length, so that * (ds)[sup 2] = (dx)[sup 2] + (dy)[sup 2] + (dz[sup 2]

The symbol "d" really stands for differential, so this is a differential equation.

Contrast special relativity, in which a small distance in s.p.a.ce-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) s.p.a.ce are complex and contrary to our 3D intuitions.

Different kinds of curves are called "s.p.a.celike" 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 s.p.a.ce-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.

Einstein's theory contrasts strongly with visions such as Wells' in The Time Machine, which treats motion along the (dt) axis as very much like taking a train to the future, then back. In Einstein's geometry, only portions of the s.p.a.ce can be reached at all without violating causality (the "light cone" within which two points can be connected by a single beam of light). Paradoxes can abound.

Logical twists have inspired many science fiction stories. The issues are quite real; we have no solid theory which includes time in a satisfying manner, along with quantum mechanics, as a truly integrated fourth dimension. I spent a great deal of s.p.a.ce in my novel Timescape wrestling with how to make this intuitively clear, but the struggle to think in four dimensions is perhaps beyond realistic fiction; perhaps it is more properly the ground of metaphor.

Physicists began envisioning higher dimensions because they got a simpler dynamic picture, at the price of apparent complication. More dimensions to deal with certainly strains the imagination, and is at first glance an unintuitive way to think. But they can lead to beauties which only a mathematician can love, abstruse elegances. Thus Einstein, in his 1916 theory of general relativity, invoked the simplicity that objects move in "geodesies" -undisturbed paths, the equivalent of a straight line in Euclidean, rectangular geometry, or a great circle on a sphere -in a four-dimensional s.p.a.ce-time. The clarity of a single type of curve, in return for the complication of a higher dimension.

Einstein's general relativity said that matter curved the four-dimensional s.p.a.cetime, an effect we see as gravity. Thus he replaced a cla.s.sical idea, force, with a modem geometrical view, curvature of a 4D world. This led to a cosmology of the entire universe which was expanding and therefore pointed implicitly backward to an origin.

Einstein did not in fact like this feature of his theory, and in his first investigations of his own marvelously beautiful equations fixed up the solution until it was static, without beginning or end. His authority was so profound that his bias might have held for ages, but Edmund Hubble showed within a decade that the universe was expanding.

Even so, the concept of a beginning land perhaps an end) may be an artifact of our persistent 3D views.

Implicitly, s.p.a.ce and time separate in the Einstein universe. They are connected, but can be defined as ideas that stand alone.

The essence of talking about dimensions is that they can be separately described. But this may not be so. At least, not in the beginning.

Even Edwin Abbott did not foretell that in the hands of cosmologists like Stephen Hawking and James Hartle, time and s.p.a.ce would blend. Though the universe remains 4D, definitions blur.

Following the universe back to its origins leads inevitably to an early instant when intense energies led to the breakdown of the very ideas of s.p.a.ce and time. Quantum mechanics tells us that as we proceed to earlier and earlier instants, something peculiar begins to happen. Time begins to turn into s.p.a.ce. The origin of everything is in s.p.a.cetime, and the "quantum foam" of that primordial event is not separable into our familiar distances and seconds.

What is the shape of this s.p.a.cetime? Theory permits a promiscuously infinite choice. Our usual view would be that s.p.a.ce is one set of coordinates, and time another. But quantum uncertainty erupts through these intuitive definitions.

Begin with an image of a remorselessly shrinking s.p.a.ce governed by a backward marching time, like a cone racing downward to a sharp point. Time is the length along the axis, s.p.a.ce the circular area of a sidewise slice. Customarily, we think of the apex as the beginning of things, where time starts and s.p.a.ce is of zero extent.

Now round off the cone's apex to a curve. There, length and duration smear. This rounded end permits no special time when things began. To see this, imagine the cone tilted. This model universe could be conceptually tilted this way or that, with no unique inclination of the cone seeming to be preferred. Now the "earliest" event is not at the center of the rounded end. It is some spot elsewhere on the rounded nub, a place where s.p.a.ce and time blend. No particular spot is special.

Another way to say this is that in 4D, time and s.p.a.ce emerge gradually from an earlier essence for which we have no name. They are ideas we now find quite handy, but they were not forever fundamental.

In the primordial Big Bang, there is no dear boundary between s.p.a.ce and time. Rather than an image of an explosion, perhaps we should call this event the Great Emergence. There we are outside the conceptual s.p.a.ce of precisely known s.p.a.ce and well defined time. Yet there are still only four dimensions -- just not sharp ones.

Einstein's cosmology thus begins with a time that is limited in the past, but has no boundary as such.

Neither does s.p.a.ce. As Stephen Hawking remarked, "The boundary condition of the universe is that it has no boundary."

Perhaps Edwin Abbott would not like the theological ramifications of these ideas. He was of the straitlaced Church of England. (The American version is the Episcopal faith, which happens to be my own. As an boy I was an acolyte, charged with lighting candles and carrying forth the sacraments of holy communion, in red and white robes. The robes were intolerably hot in our Atlanta church, and once I fainted and collapsed in service -- overcome by the heat, not the ideas. I'm told it provoked a stir.) However, it is notable that members of that faith had a decided dimensionally imaginative bent, at least in the nineteenth century; Lewis Carroll and H.G. Wells come to mind.

No doubt, psychologically the sharp-cone cosmological picture, with its initial singular point suggests the idea of a unique Creator who sets the whole thing going. How? Physics has no mechanism. For now, it merely describes.

Here lurks a conceptual gap, for we have no model which tells us a mechanism for making universes, much less one in which such basics as s.p.a.ce and time are illusions. We need a "G.o.d of the gaps" to explain how the original, defining event happened. These new theories seem to bridge this gap in a fashion, but at the price of abandoning still more of our basic intuitions.

Much of G.o.d's essence comes from our perceived necessity for a creator, since there was a creation.

But if there is no sharp beginning, perhaps we need no sharp, clear creator. Without a singular origin in time, or in s.p.a.ce for that matter, is there any need to appeal to a supernatural act of creation?

But does this mean we can regard the universe as entirely self-consistent, its 4D nature emerging with time, from an event which lies a finite time in our past but does not need any sort of infinite Creator? Can the universe be a closed system, containing the reason for its very existence within itself?

Perhaps -- to put it mildly. Theory stands mute. Yet this latest outcome of our wrestling with dimensions a.s.sumes that there are laws to this universe, mathematically expressed in a stew of coordinates and algebra and natural beauties.

But whence come the laws themselves? Is that where a Creator resides, making not merely s.p.a.cetime but the laws? Of this mathematics can say nothing -- so far.

Edwin Abbott would no doubt be astonished at the twists and turns his Lewis Carroll-like narrative has taken us to, only a bit more than a century beyond his initial penning of Flatland. The questions still loom large.

So such matters progress, sharpening the questions without answering them in final fashion. We can only be sure that the future holds ideas which he, and we, would find stranger still.

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