space. Secondly, with the vector in the body frame that goes through this point fixed, the body can have any amount of rotation around that vector. So in principle, the body's orientation is some point on a toroidal 2-manifold inside the 3-manifold of all orientations. In general, the object will follow a non-periodic path on this torus, but it may follow a periodic path. The time taken for to complete one cycle around its track in the body frame is constant, but after a cycle the body will have rotated by an amount that my not be a rational number of degrees, in which case the orientation will not be periodic, but almost periodic. In general a torus is almost determined by three parameters: the ratios of the second and third moments of inertia to the highest of the three moments of inertia, and the ratio relating the angular momentum to the energy times the highest moment of inertia. But for any such a set of parameters there are two tori, because there are two "tacos" (corresponding to two polhodes). A set of 180° rotations carries any orientation of one torus into an orientation of the other with the opposite point aligned with the angular momentum vector. If the angular momentum is exactly aligned with a principal axes, the torus degenerates into a single loop. If exactly two moments of inertia are equal (a so-called symmetric body), then in addition to tori there will be an infinite number of loops, and if all three moments of inertia are equal, there will be loops but no tori. If the three moments of inertia are all different and but the intermediate axis is not aligned with the angular momentum, then the orientation will be some point on a topological open annulus. Because of all this, when the angular velocity vector (or the angular momentum vector) is not close to the axis of highest or lowest inertia, the body "tumbles". Most moons rotate more or less around their axis of greatest inertia (due to viscous effects), but Hyperion (a moon of Saturn), two moons of Pluto and many other small bodies of the Solar System have tumbling rotations. If the body is set spinning on its intermediate principal axis, then the intersection of the ellipsoid and the sphere is like two loops that cross at two points, lined up with that axis. If