220 Chapter 5 Our Ever-Expanding Universe

Will You Return to the Same Location in a Plane,

aCylinder, and a Sphere?

When the curvature of space is zero (meaning it’s flat, like a sheet of paper), it can be drawn

with straight lines. But the greater the curvature of space, the more it is bent, meaning it

must be drawn using more sharply curving lines.

When a perfectly flat world like that represented on the graph paper in Figures 5-1

and 5-2 is viewed from our perspective in three dimensions, we can see a two-dimensional

space with zero curvature. But even graph paper is seldom perfectly flat—it would be dif-

ficult to maintain a curvature of exactly zero.

So let’s assume that the curvature is no longer zero in the x direction. In Figure 5-3,

the graph paper is bent horizontally. What will happen in this case?

If two-dimensional space extends infinitely but is curved, and if we assume that the

curvature is constant, then it will curl around and eventually arrive back where it started,

taking on a cylindrical shape as in Figure 5-3. This cylindrical shape will be created when

the x-coordinate’s positive and negative directions meet.

The inhabitants of this two-dimensional world will have no idea that it is a cylinder. But

if they walk in a straight line, looking for the address x = ∞, they will eventually experience

the oddity of x becoming negative.

Moreover, a two-dimensional space that bends only in the direction of the x-axis is a

rather special condition. If the curvature in the direction of the y-axis is also positive, then

the two-dimensional shape that is derived is a sphere, as in Figure 5-4. Even if the curva-

ture in the direction of the x- or y-axis is not necessarily constant, if two-dimensional space

continues curving in a fixed direction, it will ultimately intersect in both the x and y direc-

tions to form a closed shape like a sphere.

5

4

3

2

1

x54321

y

Figure 5-1: A simple plane, expanding in all directions—imagine

that this model does not have an edge

x5

4

3

2

1

5

4

3

2

1

y

Figure 5-2: A 2-D plane viewed from a 3-D perspective

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