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DEGREES OF FREEDOM
-11
Although philosophers have suggested the implausibility of a fourth dimension,
you will see in the following sections that higher dimensions probably
provide the basis for the existence of everything in our universe.
Hyperspace and Intrinsic Geometry
The fact that our universe, like the surface of an apple, is curved in an
unseen dimension beyond our spatial comprehension has been experimentally
verified. These experiments, performed on the path of light
beams, shows that starlight is bent as it moves across the universe.
—Michio Kaku, Hyperspace
Imagine alien creatures, shaped like hairy pancakes, wandering along the surface
of a large beach ball. The inhabitants are embedded in the surface, like microbes
floating in the thin surface of a soap bubble. The aliens call their universe "Zarf."
To them, Zarf appears to be flat and two-dimensional partly because Zarf is large
compared to their bodies. However, Leonardo, one of their brilliant scientists,
comes to believe that Zarf is really finite and curved in something he calls the
third dimension. He even invents two new words, "up" and "down," to describe
motion in the invisible third dimension. Despite skepticism from his friends,
Leonardo travels in what seems like a straight line around his universe and returns
to his starting point—thereby proving that his universe is curved in a higher
dimension. During Leonardo's long trip, he doesn't feel as if he's curving, although
he is curving in a third dimension perpendicular to his two spatial dimension.
Leonardo even discovers that there is a shorter route from one place to another.
He tunnels through Zarf from point A to point B, thus creating what physicists
call a "wormhole." (Traveling from A to B along Zarf's surface requires more time
than a journey that penetrates Zarf like a pin through a ball.) Later Leonardo discovers
that Zarf is one of many curved worlds floating in three-space. He conjectures
that it may one day be possible to travel to these other worlds.
Now suppose that the surface of Zarf were crumpled like a sheet of paper.
What would Leonardo and his fellow pancake-shaped aliens think about their
world? Despite the crumpling, the Zarfians would conclude that their world
was perfectly flat because they lived their lives confined to the crumpled space.
Their bodies would be crumpled without their knowing it.
This scenario with curved space is not as zany as it may sound. Georg Bernhard
Riemann (1826—1866), the great nineteenth-century geometer, thought
constantly on these issues and profoundly affected the development of modern