Null-Homotopy of the Double-Twist in SO(3)
A Closed Path in SO(3)
The "crow's foot" at right travels at uniform speed down the length of the box,
its orientation varying smoothly with time. This trajectory can be seen as a (closed) path
f(t) in SO(3), where the coordinate axes at time t are rotated
by the corresponding matrix f(t). Here translation of the axes only
helps to mark the passage of time.
Controls
Mouseover to animate,
click and drag to rotate,
double-click to pause,
SHIFT + drag to zoom,
CONTROL + drag to change perspective
Deforming the Path
Now we consider the whole path at once, and let it be deformed continuously in SO(3),
with its endpoints kept fixed at the identity.
The initial path traces out two full rotations about the "red" axis, the so-called
"double-twist". This is deformed to the constant path in SO(3)
Continuity in SO(3) is evident in that the bars of a given color vary continuously
in 3-space, with t and s. We claim that
- Every path can be deformed into some number of twists around the red axis, and
- The single-twist can not be deformed to the identity.
Thus the fundamental group of SO(3) is Z2.
The red, green or blue axes may be viewed
individually, for better insight into this deformation.
These figures were created in Mathematica embedded in this
page using
LiveGraphics3D , a "non-commercial Java applet" by Martin Kraus.