Bad Astronomy: Misconceptions and Misuses Revealed, from ...

148 SKIES AT NIGHT ARE BIG AND BRIGHT
of space is closed, because it curves back onto itself. There is a
boundary to it; it’s finite.
Open space would be one that curves the other way, away from
itself, so it takes on a saddle shape. If you lived in open space, you
could walk forever and never get back to where you started.
These three spaces—open, closed, and flat—have different properties.
For example, if you remember your high school geometry,
you’ll recall that if you measure the three inside angles of a triangle
and added them together, you get 180 degrees. But that’s only
if space is flat, like a page in this book. If you draw a triangle on
the surface of a sphere and do the same thing, you’ll see that the
angles always add up to more than 180 degrees!
Imagine: take a globe. Start at the north pole and draw a line
straight down to the equator through Greenwich, England. Then
go due west to, say, San Francisco. Now draw another line back
up to the north pole. You’ve drawn a triangle, but each inside
angle is 90 degrees, which adds up to 270 degrees, despite what
your geometry teacher taught you. Actually, your teacher was just
sticking with flat space; closed and open space can be quite different.
In open space, the angles add up to less than 180 degrees.
So that ant, if it were smart enough, could actually try to figure
out if its space is open, closed, or flat just by drawing triangles
and carefully measuring their angles.
This is all well and good if you’re an ant, but what about us,
in our three-dimensional space? Actually, the same principles
apply. Since space itself is warped, it can take on one of these three
shapes, also called geometries. And, just like the ant, you could try
taking a walk to see if you come back to where you started. The
problem is that space is awfully big, and even the fastest rocket we
can imagine would take billions or even trillions of years to come
back. Who has that kind of time?
There’s an easier way. Karl Friedrich Gauss was a nineteenthcentury
mathematician who worked out a lot of the math of the
geometry of the universe. He actually tried to measure big triangles
from three hilltops, but was unable to tell if the angles added up to
more or less than 180 degrees.