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.
The name "polystick" seems to have been coined by Brian R. Barwell.Brian R. Barwell, "Polysticks," Journal of Recreational Mathematics volume 22 issue 3 (1990), p.165-175
The names "polytrig"David Goodger, "An Introduction to Polytrigs (Triangular-Grid Polysticks)," (2015), https://puzzler.sourceforge.net/docs/polytrigs-intro.html and "polytwigs"David Goodger, "An Introduction to Polytwigs (Hexagonal-Grid Polysticks)," (2015), https://puzzler.sourceforge.net/docs/polytwigs-intro.html 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.
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.
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.
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
Hexagonal Polysticks Sticks Name Free One-Sided 1 monotwig 1 1 2 ditwig 1 1 3 tritwigs 3 4 4 tetratwigs 4 6 5 pentatwigs 12 19 6 hexatwigs 27 49 7 heptatwigs 78 143
Triangular Polysticks Sticks Name Free One-Sided 1 monostick 1 1 2 distick 3 3 3 tristick 12 19 4 tetrastick 60 104 5 pentastick 375 719 6 hexastick 2613 5123 7 heptastick 19074 37936
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).
Diagram
References
External links Polysticks Puzzles & Solutions, at Polyforms Puzzler Covering the Aztec Diamond with One-sided Tetrasticks, Alfred Wassermann, University of Bayreuth, Germany Polypolylines, at Math Magic