The figure-8 Klein bottle of the previous page is formed by rotating a figure-8 around an axis while applying a single half-twist. To produce a family of closed surfaces, we rotate a figure-8 while applying n half-twists for integer n. For example, setting n = 1 produces the figure-8 Klein bottle whose constituent Mobius bands are right-handed, and setting n = –1 produces the figure-8 Klein bottle whose two Mobius bands are left-handed. In general, applying n half-twists produces a topological torus when n is even, and a Klein bottle when n is odd.
You might compare these surfaces to the bands produced on the more bands page. The current program glues the boundary edge of each n-band to the edge of a second n-band, thereby forming a closed surface. The bands can be glued together due to the figure-8 cross section’s curvature. Larger values of setting a flatten the figure-8. Setting a = 4 produces a very flat cross section in which the closed surfaces resemble the bands of the previous program.