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