How scientists visualize the Fourth Dimension. Computer graphics make a baffling concept tangible
Thomas Banchoff struts over to a bookshelf and retrieves the seminal work that helped launch his career as a mathematics teacher and prominent theoretician: ``America's Greatest Comics.'' Inside is a 1940s ``Captain Marvel'' strip in which the Einstein of the day, Dr. Kiddin, takes boy reporter Bill Batson to see the world of tomorrow, where scientists are working on the seventh, eighth, and ninth dimensions. ``I wonder what happened to the fourth, fifth, and sixth dimensions,'' the young scribe thinks to himself.
Banchoff, about 10 when he first read it, wondered the same thing -- and has much of the time since.
Now chairman of Brown University's mathematics department, he is in the forefront of scientists today making strides in visually depicting the fourth dimension -- and thus making this mind-bending concept more tangible.
The idea of a direction beyond length, width, and depth has long intrigued everyone from artists to philosophers to science-fiction buffs. But even though researchers have routinely dealt with it as a serious mathematical concept, the public (and some scientists) have been slow to accept something they couldn't see or feel, or even pinpoint what and where it was.
But now that they are able to illustrate it with computer graphics, scientists are retrieving the fourth dimension from the abstract world of mathematics and the imagination -- as well as finding practical uses for it in everything from geology to oceanography.
``There is a rebirth of interest in it because we can actually achieve with graphics what people in the past only aspired to,'' says the bearded Banchoff.
What he and his colleagues do at Brown is use computers to simulate four-dimensional objects moving through three-dimensional space. The results are captivating. In one film, ``The Hypercube,'' a standard 3-D cube with its eight corners and six square faces suddenly changes into an object with 16 vertices and eight sides -- a basic 4-D cube. In another, a blue hemisphere is shown rotating against a black backdrop. The figure distorts, twists into new shapes, collapses into a dot, and finally reappears in original form.
These kaleidoscopes are not actual objects in the fourth dimension, or ``hyperspace.'' They are, in essence, three-dimensional shadows cast by theoretical 4-D shapes. The objects are rotated on the screen so scientists can get different views and begin to understand them.
What's happening is analogous to casting a shadow on a wall. To flat (2-D) creatures on the wall surface, the shadow of your hand (a 3-D object) would appear mysterious indeed. It would change shape, growing fatter or thinner, as you turned your hand. If you moved it out of the light, it would disappear altogether. In reality, of course, your hand isn't changing. It only appears to do so to the 2-D creatures.
Similarly, a four-dimensional creature invading our world would presumably appear just as odd: contorting, turning inside out, appearing and disappearing. No one knows, of course, exactly what would happen.
The fourth dimension is sound mathematical theory. But whether it will ever translate into something earthlings can experience directly is where fiction takes over for science.
``No one pooh-poohs it as fantasy stuff,'' says Banchoff of the theory. ``It's accepted. But whether there are beings that will visit us from the fourth dimension -- that's another question.''
The affable mathematician dodges such speculation (``I'm just a theoretician''). But he and colleagues have come as close to anyone else to ``seeing'' what this baffling world might be like.
Early on in his science, Banchoff himself was inspired by fiction. It came in the form of a satirical little novel written by Edwin Abbott in 1884 (right year?) I thought it was much later called ``Flatland.'' It depicts a world of 2-D creatures who won't accept the idea of another dimension, even though they're visited by a sphere from a 3-D world.
At one point the protagonist, a square, visits the third dimension and sees what he's been missing. He suggests to his spherical friend that maybe there is even a fourth dimension. The sphere scoffs. The square returns to his 2-D domain and tries to tell friends of his journey, eventually ending up jailed for his heresy.
In our world, doubters and devotees of a higher dimension have existed throughout history. In the second century, the astronomer Ptolemy flatly rejected the idea of a fourth dimension. Earlier in this century, Einstein decreed that, yes, there was a fourth dimension, but designated it as time. This satisfied many scientists. In recent years, though, the concept of a fourth spatial dimension has again gained credence, aided in part by some contemporary theories on the structure of the universe, as well as the computer graphics work.
Along with the serious speculation has come a rebirth of interest from other quarters as well. This is as it has always been, too: Everyone from Biblical figures (St. Paul) and philosophers (Plato) to artists (Dali) and poets (T. S. Eliot) has been captivated by the notion of an extra dimension to life. At Brown, both the serious and curious -- several hundred scientists, artists, writers -- recently turned up for the first world conference on the subject.
``The fourth dimension is almost a prototype of what we mean by a mind-stretching experience,'' says Dr. Banchoff.
Still, to most laymen, something intangible like hyperspace seems fanciful. ``It is still mysterious to the general public,'' Banchoff says. But like the ``square'' in Flatland, he continues to challenge those perceptions and try to broaden people's thinking.
Among today's youth, in particular, he may be winning enthusiasts. After all, they have grown up in a world of 3-D graphics and ``Star Wars'' heroes like Han Solo, who routinely blasts his starship into hyperspace. ``Students nowadays are more sophisticated visually,'' he says. ``They were brought up on television. We have a whole generation of students willing to think about the idea.''
Even if they don't, plenty of others today are finding down-to-earth applications for 4-D. Indeed, the serious research done in computer visualizations here and elsewhere is being used increasingly by scientists to interpret vast amounts of data with four or more variables. Oceanographers, for example, use them to plot such things as wind speed, pressure, humidity, and temperature in studying atmospheric and oceanographic phenomena. Sociologists use them in drawing conclusions about human behavior. Geologists at Brown are using them to probe ancient climates.
``We are coming up with totally new ways of examining data,'' Banchoff says.
In the end, Dr. Kiddin may have been right, too: Physicists are now studying 11-dimensional structures that might give a unified account of the basic forces of nature. An occasional column Diagam: Hatching a hypercube 1. To imagine this basic 4-D object, first extend lines from each corner of a square, as though making a cube. 2. Then draw lines from each of these eight corners in another direction -- one that suggests right angles to the height, width, and depth of the cube. 3. The result is a hypercube with 16 corners and eight faces. (In reality, the `shadow' of a 4-D object on a two-dimensional surface.)