Warped Passages - Lisa Randall [21]
Figure 17. In a two-dimensional universe, when a dimension is curled up there is a circle at every point along the infinite dimension of space.
Figure 18 presents a different example: here there are two infinite dimensions rather than one, plus a single additional dimension curled up into a circle. In this case, there is a circle at each and every point in the two-dimensional space. And if there were three infinite dimensions, the rolled-up dimensions would exist at every point in three-dimensional space. You might liken the points in extra-dimensional space to the cells in your body, each of which carries your entire DNA sequence. Similarly, each point in our three-dimensional space could host an entire compactified circle.
Figure 18. In a three-dimensional universe, if one of the three dimensions is curled up you have a circle at every point in the plane.
So far, we’ve only considered a single additional dimension, which is rolled up into a circle. But everything we’ve said would hold true even if that curled-up dimension took some other shape—any shape at all. And it would also be true if there were two or more tiny, rolled-up dimensions of any shape at all. Any and all dimensions that are sufficiently small would be completely invisible to us.
Let us consider an example with two rolled-up dimensions. There are many possible shapes that these rolled-up dimensions could take. We’ll choose a torus, a donut-like shape in which the two additional dimensions are both simultaneously rolled into a circle. This is illustrated in Figure 19. If both circles—the one that winds through the donut hole and the one that winds around the donut itself—are sufficiently small, the additional two rolled-up dimensions would never be seen.
Figure 19. When two out of four dimensions are curled into a donut, you have a donut at every point in space.
But that’s just one example. With more dimensions there are a huge number of conceivable compact spaces—spaces with rolled-up dimensions, distinguished by the precise manner in which the dimensions are rolled up. One category of compact spaces important to string theory are the Calabi-Yau manifolds, named after the Italian mathematician Eugenio Calabi, who first proposed these particular shapes, and the Chinese-born Harvard mathematician Shing-Tung Yau, who showed that they are mathematically possible. These geometric shapes roll up and wind together extra dimensions in a very special way. The dimensions are curled up into a small size, as with all compactifications, but they are tangled in a way that is more complicated and difficult to draw. 4
Whatever shape the rolled-up extra dimensions take, and however many there are, at each point along the infinite dimensions there would be a small compact space containing all the curled-up dimensions. So, for example, if string theorists are right, everywhere in visible space—at the tip of your nose, at the North Pole of Venus, at the spot above the tennis court where your racket hit the ball the last time you served—there would be a six-dimensional Calabi-Yau manifold of invisibly tiny size. The higher-dimensional geometry would be present at every point in space.
String theorists often suggest—as Klein did—that curled-up dimensions are as small as the Planck length, 10-33 cm. Planck-length-size compact dimensions would be extraordinarily well hidden; there is almost certainly no way for us to detect something so small. Therefore, Planck-length extra dimensions would very likely leave no visible trace of their existence. So even if we live in a universe with Planck-length extra dimensions, we would still register only the three familiar dimensions. The universe could have many such tiny dimensions, but we might never have the resolving power to find out.
Newton’s Gravitational