We consider a twisting solid torus with sides, approximated by a ring of blocks. The cross section, a regular n-sided polygon, has n-fold symmetry. The solid torus must twist a multiple of times over its length for its sides to remain aligned, that is, full twists where k is an integer.

Twisting blocks

Let’s focus on one of the solid torus’s sides by selecting one face from the color models dropdown for some . One side is painted the current color whereas the remaining sides are light gray. For the painted face to return to itself, forming a closed band, the number of full twists must be a whole number, in other words, k must be a multiple of n. Appearances to the contrary, the painted side, regarded as a surface separate from the solid torus, is two-sided and orientable.

For the case , the program constructs a band rather than a solid torus. As we’ve seen, when this band undergoes twists, it is a Mobius band for odd values of k and a cylinder for even values of k.