Rotating Cubes about Axes of Symmetry; 3D Rotation Is Non-Abelian

maximum τ

AB,BA

reset

rotation A

rotation B

Rotating about 2a Reference Cube Rotating about 3b

A cube has rotational axes of symmetry through the centers of a pair of opposite edges (two-fold symmetry, six axes), two opposite corners (three-fold symmetry, four axes), or the centers of two opposite faces (four-fold symmetry, three axes). The middle cube shows two examples of each axis; it is slightly transparent. Markers 0, 1, 2, and 3 are always visible, marking the vertices

(0,0,0)

(0,0,1)

(0,1,0)

, and

(1,0,0)

of each cube. Move the

slider. The left cube rotates about the

axis and the right rotates about the

axis; the axes appear as stubs on the cubes. When

is an integer, the edges are once more parallel to the central cube, demonstrating the symmetry, but the original coloring is not recovered until

equals the axis number. Try different pairs of axes and change

. You can click the picture and drag to alter the viewpoint.

Now click "AB, BA" in "maximum

" and activate the

slider. (Slow the animation down!) The rotation pauses after one complete turn (snapshot 3), giving orientations

and

. Then the axes of rotation are swapped. The new axes appear and the second rotations are performed. The image pauses again (snapshot 4), with the left cube in orientation

and the right in orientation

. These are different; in 3D rotations

is not necessarily

. In real and complex numbers

AB=BA

(they are Abelian or commutative); 3D geometry (like many aspects of physics) is non-Abelian.