The Duocylinder
Construction
We have seen that the prismic cylinders are formed by taking the limit of the m,n-duoprisms as m approaches infinity. We need not stop with the prismic cylinders; we can now take the limit of n-gonal prismic cylinders as n approaches infinity.
As n increases, so do the number of constituent cylinders; but their height also decreases to zero. The number of sides in the ZW torus also increase without bound, and it becomes increasingly close to being circular. At the limiting case, the ring of cylinders forms into a circular torus, and the ZW polygonal torus also forms into a circular torus. These two torii are in fact identical (although they appear different in the projections we see here).
This resulting object is called a duocylinder. It is the 4D volume described by the equations:
x2 + y2 ≤ r2
z2 + w2 ≤ r2
It is bounded by two equivalent, mutually perpendicular, circular torii, described respectively by the equations:
x2 + y2 ≤ r2,
z2 + w2 = r2;
z2 + w2 ≤ r2,
x2 + y2 = r2
The following animation shows the duocylinder rotating in the YW plane. The pink part represents half of the duocylinder, and the black part the other half.
Properties
The duocylinder is a peculiar object. Its two torus-shaped boundaries are surfaces that it can roll on, like a wheel. They are mutually perpendicular, so when rolling on one side, the duocylinder can only cover the space of a line. But if you tip it sideways on the other side, it will roll along a perpendicular line. It can always roll no matter which side you stand it on, but in different perpendicular directions depending on which side it is.
