Uploaded April 2015 | Updated September 2026, 2 weeks ago
This video illustrates how a rotation of a sphere by two full turns can be deformed continuously into a non-rotation.
This is a sequential version showing the same deformation as in this other version: youtube.com/watch?v=HK7KksT1nYw
This video consists of 25 segments, each 8 seconds long: in the first segment, the sphere performs a rotation by two full turns (=4π radians) around a constant axis. Each of the following segments modifies the movement a little bit until, in the end, the sphere is not moving at all.
It would not be possible to similarly deform to nothing a rotation of a circle (whatever the number of turns), because the "number of turns" is a well-defined invariant. Perhaps more surprisingly, it would also be impossible to deform to nothing a rotation of the sphere by a single full turn: this is because a single-turn rotation defines a non-trivial element of the spin group, whereas a rotation by two full turns gives the identity (=trivial) element of the spin group. The spin group is simply connected, so any loop therein can be contracted to nothing, and this is what is being done here.
This video illustrates how a rotation of a sphere by two full turns can be deformed continuously into a non-rotation.
This is a sequential version showing the same deformation as in this other version: youtube.com/watch?v=HK7KksT1nYw
This video consists of 25 segments, each 8 seconds long: in the first segment, the sphere performs a rotation by two full turns (=4π radians) around a constant axis. Each of the following segments modifies the movement a little bit until, in the end, the sphere is not moving at all.
It would not be possible to similarly deform to nothing a rotation of a circle (whatever the number of turns), because the "number of turns" is a well-defined invariant. Perhaps more surprisingly, it would also be impossible to deform to nothing a rotation of the sphere by a single full turn: this is because a single-turn rotation defines a non-trivial element of the spin group, whereas a rotation by two full turns gives the identity (=trivial) element of the spin group. The spin group is simply connected, so any loop therein can be contracted to nothing, and this is what is being done here.
