14 CHAPTER 2
The Group of Rotations of a Sphere
A group is a set along with a rule that tells how to combine any two
elements in the set to get another element in the set. We usually use
the word composition to describe the act of combining two elements
of the group to get a third.
We start our consideration of groups by thinking about a beauti-
ful perfect sphere, one foot in radius, made of pure marble. Let it
rest in a spherical container so it just ﬁts exactly. Assume that we
have a perfect map of the earth drawn on the sphere, so we can refer
to points on the sphere by the corresponding latitude and longitude
of points on the earth. We ignore the fact that the real earth is not
a perfect sphere.
To mark the initial position of this sphere in its container, draw a
red dot on the sphere and on the container at the North Pole, and
draw circles on the sphere and on the container where the equator
is. We can also put a green dot on the equator, both on the sphere
and the container, marking the Greenwich meridian. Now, there is
exactly one way to place the sphere in the container so that the North
Pole dots match, the equators match, and the Greenwich dots match.
We have not deﬁned any groups yet. This sphere and container
are just the (idealized) physical set-up we need to deﬁne the group
that is called SO(3). We will ﬁrst deﬁne the set SO(3) by telling
about its elements. An element g of SO(3) is a rotation of the sphere
inside the container. If we rotate the sphere by g, it will come to a
new position in the container, which we can see because the two
dots and the equator will be somewhere else. This is true except in
the case when g is the “neutral element” in the group (see below).
Now, here is a very important point: If we take the sphere out of
the container, toss it around, show it to our friends, and then put
it back carelessly into the container, it will be in a new position. It
is always possible to move the sphere from its original position into
this new position by some rotation.
For instance, g might be rotation about the North Pole by 30
(This is the rotation of the earth in any 2-hour period.) Another
In describing rotations by numbers of degrees, w e shall always mean counterclockwise
as we look down from above, as if we were trying to unscrew a light bulb.