Twisted Polycube Rings

Twisted Prisms

In 1948, in the Mathematical Notes section of the American Mathematical Monthly, H. T. McAdams raised the possibility of somehow joining the regular polygonal bases of a prism face to face. If the prism is twisted lengthwise, the resulting theoretical solid may have only one face, like a Möbius Strip.

Twisted Polycube Rings

In 1968 Gonzalo Vélez Jahn, of the University of Caracas, began to study polycubes whose cells form a closed loop or ring. A right cross-section of any straight segment of such a polycube is a square. A ring polycube may be regarded as having four 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 climb over 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.

Vélez's 22-cell polycube appeared in Martin Gardner's Mathematical Games 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.

Half Twist

In October 2017 I read Gardner's book and became interested in polycube rings. I wondered whether a polycube ring could have a half twist instead of a quarter twist. Such a ring would have two linked surfaces. The article did not provide one, and a web search turned up nothing. Eventually I found a unique minimal solution with 12 cells:

Here is a wooden model of it:

Other Degrees of Twist

By stretching this half-twist ring, breaching one of the stretched lengths, and joining the ring to a copy of itself, we can get a polycube ring with a full twist:

It is likely that a twist of three-quarters can be obtained in a similar way. I do not know the minimal polycube rings for these or other degrees of twisting. So far as I know, this problem is open.

See also Robert Dickau's One-Sided Toroidal Polyhedra.

Last revised 2025-04-07.

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Col. George Sicherman [ HOME | MAIL ]