The Grand Design

by Stephen Hawking

According to string theory, they are curved up into a space of very small size. To picture this, imagine a two-dimensional plane. We call the plane two-dimensional because you need two numbers (for instance, horizontal and vertical coordinates) to locate any point on it. Another two-dimensional space is the surface of a straw. To locate a point on that space, you need to know where along the straw’s length the point is, and also where along its circular dimension. But if the straw is very thin, you would get a very good approximate position employing only the coordinate that runs along the straw’s length, so you might ignore the circular dimension. And if the straw were a million-million-million-million-millionth of an inch in diameter, you wouldn’t notice the circular dimension at all. That is the picture string theorists have of the extra dimensions—they are highly curved, or curled, on a scale so small that we don’t see them. In string theory the extra dimensions are curled up into what is called the internal space, as opposed to the three-dimensional space that we experience in everyday life. As we’ll see, these internal states are not just hidden dimensions swept under the rug—they have important physical significance.