For years I have wrestled to explain in classes the conceptual issues involved in the question: "If the Universe expands, into what does it expand?" I have even recently been asked that question by a local helpful-household-hints columnist on behalf of her teenage son. I have given the standard answer guided by the mathematics and physics of Einstein's theory.

The three-dimensional space of the universe is all the three-dimensional space there is. It could be finite, closed and doomed to re-collapse, or it could be infinite and destined to expand forever, but the universe is everything. Three-dimensional physicists and astronomers and teenagers can ask about the nature and behavior of this three-dimensional universe in the context of their own dimensionality. In that context, the question of what the universe is expanding into, what is "outside" the universe is a meaningless question.

I also carefully explained to students and to the columnist (who was not satisfied), that if the universe were expanding into anything, it had to be into a higher dimension, not into "ordinary" three-dimensional space. The analogy is the surface of a balloon as you blow it up. The two-dimensional rubber surface of the balloon is all the two-dimensional surface (read "universe") there is. If you were a two-dimensional astronomer or teenager living on that surface, you could determine everything there was to know about it -- its shape, its size, its rate of expansion -- by doing two-dimensional experiments and observations, careful two-dimensional geometry. That is basically the analogy of what we do when we train our telescopes outward in our three-dimensional space. In the balloon analogy, however, we can clearly perceive the space around the balloon into which the balloon expands; hence the intuitively completely justified question: "Into what does the universe expand?" The answer has been as I've said: You don't need to ask that question, or to know whether it expands into anything, to study the dynamics, origin and fate of the universe.

I have recently come to recognize that physicists have had a much more concrete reason for arguing that there was no outer, higher dimensional space into which our universe expands. This insight comes in the context of a revolution that is raging on right now, one that has overthrown those arguments.

The argument goes back to Newton. Newton deduced that for a spherical object like Earth, the Moon or the Sun, the force of gravity falls off with distance at the rate of 1 over the distance squared. Go twice as far away, and the force is one-quarter as strong.

This has a profound implication. It basically says that gravity permeates three-dimensional space, but no higher dimension, perceived or otherwise. The reason is similar to why the brightness of a light bulb diminishes as the inverse square of the distance from it. The light spreads out over an ever larger surface. In a three-dimensional world, surfaces have two dimensions, and the light is diluted by the size of the surface it must fill, hence by distance squared.

Gravity works is a similar way. Thus, since gravity is the fundamental creature of space-time, the fact that Newton's, and Einstein's gravity displays this 1-over-distance-squared behavior was long interpreted as meaning that there could be no higher dimension into which gravity could leak. If, for instance, there were a four-dimensional "space" surrounding our familiar three-dimensional universe, and into which our universe were expanding, gravity would weaken at a rate of 1 over the distance cubed, since "surfaces" in that four-dimensional world scale as length cubed. Gravity does not work that way, hence no higher dimension.

Now, this argument is under severe and excited assault. The new perspectives have emerged from work on String theory and the mathematically constrained result that there must be 10 or 11 spatial dimensions to make a self-consistent theory. At first, the argument was made that these extra dimensions, if they exist, must be microscopically wrapped up and imperceptible to us, much as we usually do not perceive the thickness of a two-dimensional sheet of paper, though we know it is really a three-dimensional object.

A little over a year ago, however, Lisa Randall of Princeton and the Massachusetts Institute of Technology, and Raman Sundrum of Boston University found the hole in the "Newton" argument. At its mathematical base, this elegant argument contained an assumption that gravity's relation to space -- how one describes the curvature of space -- is independent of that space. Randall and Sundrum realized that, mathematically at least, that need not be so. More complicated couplings could exist that are completely consistent with the mathematics of Einstein's theory and of String theory. There could be, they argued, a true, large (that is to say effectively infinite four -- or higher) dimensional space surrounding our three-dimensional universe. With proper choices, gravity could, nevertheless, leak only a tiny way into that extra space. Gravity would then fall off like distance squared plus a little and that little could be so small that it has evaded detection up to now. Maybe more precise experiments could find it.

In the higher-dimensional spaces of String theory, surfaces are known as "branes," a word derived from "membranes," and carry the notion that they can be stretched and twisted, as befits an elastic space-time. In this evolving view, our three-dimensional universe would be a three-dimensional brane immersed in the surrounding four-dimensional space. All this has a long way to play out from mathematical possibility to experimental reality, but at long last we may have a glimmer of an answer to the intuitively forceful question: "Into what does our universe expand?" The answer may be a "multi-verse" of higher dimension. What else, one then wonders, could lie "out there?"