Polystick

In recreational mathematics, a polystick (or polyedge) is a polyform with a line segment (a 'stick') as the basic shape. A polystick is a connected set of segments in a regular grid. A square polystick is a connected subset of a regular square grid. A triangular polystick is a connected subset of a regular triangular grid. Polysticks are classified according to how many line segments they contain.[1]

The name "polystick" seems to have been first coined by Brian R. Barwell.[2]

The names "polytrig"[3] and "polytwigs"[4] has been proposed by David Goodger to simplify the phrases "triangular-grid polysticks" and "hexagonal-grid polysticks," respectively. Colin F. Brown has used an earlier term "polycules" for the hexagonal-grid polysticks due to their appearance resembling the spicules of sea sponges.[5]

There is no standard term for line segments built on other regular tilings, an unstructured grid, or a simple connected graph, but both "polynema" and "polyedge" have been proposed.[6]

When reflections are considered distinct we have the one-sided polysticks. When rotations and reflections are not considered to be distinct shapes, we have the free polysticks. Thus, for example, there are 7 one-sided square tristicks because two of the five shapes have left and right versions.[7][8]

The set of n-sticks that contain no closed loops is equivalent, with some duplications, to the set of (n+1)-ominos, as each vertex at the end of every line segment can be replaced with a single square of a polyomino. For example, the set of tristicks is equivalent to the set of Tetrominos. In general, an n-stick with m loops is equivalent to a (n−m+1)-omino (as each loop means that one line segment does not add a vertex to the figure).