SO(3) is a schematic of the group of rotations in three dimensions. Any rotation can be specified by a vector pointing along the axis of rotation, with a length equal to the amount of rotation; using this correspondence, each cube here has been rotated by its own position vector, relative to the central cube. The group is drawn as a sphere — with a wedge removed to reveal the interior — but the true topology identifies opposite points on the surface, which represent rotations of 180° around opposite axes.
Since the applet applies these rotations to a spinning central cube, the overall effect is equivalent to moving the centre of the sphere around the group in a constant direction; the sphere always encompasses the entire group, but the particular element lying at the centre changes.
Click on the applet to pause or redraw.