222 Chapter 5 Our Ever-Expanding Universe
Consider a triangle where the apex is the North Pole of Earth and the base is on the
equator (like the triangle in model 1 in Figure 5-5). In this case, the angles created by the
base (equator) and the sides connecting the apex and base (that is, the meridians) are right
angles (90°). Therefore, just the sum of the two interior angles created by the base is 180°,
and when the interior angle of the vertex is added, the sum has to exceed 180°. You can
intuitively see that the opposite is true of a triangle drawn on a plane with negative curva-
ture, as shown in model 3.
Friedmann’s Dynamic Universe
The three-dimensional universe that we live in could also take any of three types of shapes
when viewed from the fourth dimension, with positive, zero, or negative curvature. The
famous Friedmann models of the universe were created from this kind of analysis.
The Russian astrophysicist Alexander Friedmann (1888–1925) hypothesized a dynamic
universe; that is, a universe that is continuously subjected to forces that cause it to expand
or contract. He considered what would happen if the curvature of this dynamic space was
positive, zero, or negative. The results of these three different curvatures are modeled in
three dimensions in Figure 5-6. Each letter S affixed to the surfaces of the models repre-
sents a galaxy.
Figure 5-7 shows Friedmann’s predictions about what happens in these three models
over time. The y-axis represents the average distance between galaxies in the universe,
while the x-axis represents time elapsed. A scale factor of 1 on the y-axis indicates that the
distance between galaxies exists as it is now, while 2 indicates that the distance between
galaxies has doubled.
Astronomers don’t typically refer to the specific curvature of space but rather to the
overall geometry of space. A universe with positive curvature, like a sphere, is called a closed
universe. If you traveled in a straight line in a closed universe, your journey would be a
closed loop; you’d eventually come back to your original location. As shown in Figure 5-7,
1. Curvature is positive.
Sum of the interior angles is
greater than 180˚.
2. Curvature is zero.
Sum of the interior angles is
equal to 180˚.
3. Curvature is negative.
Sum of the interior angles is
less than 180˚.
Figure 5-5: Each representation of the universe’s curvature has different implications.
The Edge, Birth, and End of the Universe... 223
a closed universe will eventually collapse in on itself. A universe with a negative curvature
is called an open universe, and a universe with zero curvature is called a flat universe.
Figure 5-7 also demonstrates Friedmann’s prediction that while a universe with zero or
negative curvature would slow down its rate of expansion over time, it would continue to
expand forever.
In summary, there are three ways in which space can curve: positively, not at all (zero
curvature), or negatively. Those three types of curvatures give us three types of universes to
consider: closed, flat, or open, respectively. For now, that’s all you need to keep in mind.
1. Curvature is positive. 2. Curvature is zero. 3. Curvature is negative.
Figure 5-6: Friedmann’s models of the universe—the S shapes in each model represent galaxies.
Scale factor
1. Curvature is negative
(an open universe).
2. Curvature is zero
(a flat universe).
3. Curvature is positive
(a closed universe).
Figure 5-7: Friedmann predicted a change over time for the three models of the universe.

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