surfaces.Where the ring bends, the surfaces perpendicular to the axis of the bend simply continue around the corresponding corner of the polyomino face. The surfaces parallel to the axis
climbover the inner or outer axis of the bend.
Vélez's best known polycube ring has 22 cells.
It turns so that its cross-section
makes a 90° twist, or quarter turn,
along its full length.
This means that its four
surfaces form one continuous surface.
polycube appeared in Martin Gardner's
column in the August 1978 issue of Scientific American,
and later in Gardner's book
Fractal Music, Hypercards
and More ….
Gardner asked for the smallest such polycube ring
whose surfaces have a quarter turn.
His answer was this ring with just 10 cells:
Vélez and others later made a Wolfram Demonstration
showing both polycubes and their surfaces,
Vélez-Jahn's Möbius Toroidal Polyhedron.
I have not found a polycube ring with a three-quarter twist, a whole twist, or any higher value. So far as I know, this problem is open.
Last revised 2023-10-25.