The Edge, Birth, and End of the Universe... 221
We can extend this idea to three-dimensional space and predict similar behavior. If the
three x-, y-, and z-coordinate axes that we can set are perfectly straight when viewed from
the fourth dimension, we can keep traveling forever and ever in the universe. But if these
axes are bent just a little, as in our cylinder or our sphere, we will eventually return to the
place where we started.
But, as we’ve said before, there can be three types of curvature: zero, positive, and nega-
tive. What does it mean for the value representing curvature (that is, the degree to which a
curved line or curved surface bends) to be negative? First, recall the three diagrams of 2D
universe models that appeared in the lecture given by Professor Sanuki on page 210. They
were a spherical surface (positive curvature), a plane (zero curvature), and a shape that
looked like a saddle (negative curvature).
Just as our flat (zero-curvature) two-dimensional plane wasn’t a rectangle with defined
edges, we say a universe with negative curvature is “sort of” like a horse’s saddle because it
doesn’t have a definite edge but instead continues to spread out infinitely both vertically and
Let’s draw triangles on these three models to show the effects that differently curved
types of space have on geometry. On the “plane” in model 2, the sum of the triangle’s inte-
rior angles is 180°, as is normal in basic geometry.
But what happens on the sphere in model 1? Here, the sum of the interior angles is
greater than 180°. And on the saddle-shaped surface in model 3, the sum of the interior
angles is less than 180°.
Figure 5-3: A cylinder results
from a positive curvature in the
Figure 5-4: A sphere results
from positive curvature in the x
and y directions.