Polystick

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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]

Square Polysticks

Sticks Name Free One-Sided
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

Triangular Polysticks

Sticks Name Free
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[edit]

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

References[edit]

  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

External links[edit]