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]

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.[2][3]

Sticks Name Free One-Sided Square Polysticks 1 monostick 1 1 2 distick 2 2 3 tristick 5 7 4 tetrastick 16 25 5 pentastick 55 99 6 hexastick 222 416 7 heptastick 950 1854
Sticks Name Free Triangular Polysticks 1 monostick 1 2 distick 3 3 tristick 12 4 tetrastick 60 5 pentastick 375 6 hexastick 2613 7 heptastick 19074

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. In general, an n-stick with m loops is equivalent to a (nm+1)-omino (as each loop means that one line segment does not add a vertex to the figure).

Diagram

The free square polysticks of sizes 1 through 4, including 1 monostick (red), 2 disticks (green), 5 tristicks (blue), and 16 tetrasticks (black).

References

1. ^ Sloane, N.J.A. (ed.). "Sequence A019988 (Number of ways of embedding a connected graph with n edges in the square lattice)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
2. ^ Weisstein, Eric W. "Polystick." From MathWorld
3. ^ Counting polyforms, at the Solitaire Laboratory