# 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:

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.

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Col. George Sicherman
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