The graphic may take 10-30 seconds to load, but once it does, you can rotate it and zoom in or out (for the former, on a PC, just use the left mouse button, holding it down while dragging the mouse across the graphic; for the latter, hold down the shift key before you drag, and while you drag, the mouse vertically across the graphic; and finally, holding the shift key while dragging the mouse horizontally rotates the figure about an axis perpendicular to the screen).
Some background: Here, SO(3), the group of 3x3 orthogonal matrices with real entries and determinant one, representing the group of all (proper) rotations of R3, is represented by the ball centered at the origin, of radius pi, with antipodal points on the boundary identified. A given rotation is rendered as the point ru, and this is possible because every rotation of R3 is determined by an axis of rotation, i.e. a unit vector u, and an amount of (counterclockwise) rotation r about the axis. Using other terminology, the figure displays SO(3), topologically speaking, as RP3. In this interpretation, the curves depicted are left cosets of the isotropy group, in SO(3), of (0,0,1)T.
The "Live" rotation of the Mathematica-generated graphic uses the LiveGraphics3D software. For more info on how to rotate Mathematica-generated graphics on the web, visit LiveGraphics3D.
Besides depicting left cosets of SO(2) in SO(3), this figure depicts many other important objects in mathematics and physics: for example it depicts some geodesics of SO(3) with respect to its bi-invariant metric. Equivalently, some flow lines of the geodesic flow in the unit tangent bundle of S2 (since SO(3) can be identified with the unit tangent bundle of S2). Equivalently, some fibers in the Hopf fibration (since the fibration S1->S3->S2 double-covers the fibration SO(2)->SO(3)->S2). Equivalently, some projected trajectories of the Robinson null congruence of Minkowski spacetime. Equivalently ...
Finally, related graphics are available on this page. The graphic on this page and pages that this page links to were created by Rick Kreminski; some appeared in volume 6, number 1, of Mathematica in Education and Research, Winter 1997, pages 9--14. This page's graphic also appeared on the cover of the Notices of the American Mathematical Society, May 1997. In those publications, the dots were placed in the graphic to aid in visualization; the dots lie in the plane y=0.