polystick

{{Short description|Shapes made from regular line segments}}

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.[http://mathworld.wolfram.com/Polystick.html Weisstein, Eric W. "Polystick." From MathWorld ]

The name "polystick" seems to have been first 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), http://puzzler.sourceforge.net/docs/polytrigs-intro.html and "polytwigs"David Goodger, "An Introduction to Polytwigs (Hexagonal-Grid Polysticks)," (2015), http://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.{{cite web |url=https://mathworld.wolfram.com/Polynema.html |title = Polynema -- from Wolfram MathWorld}}

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.[http://www.solitairelaboratory.com/polyenum.html Counting polyforms, at the Solitaire Laboratory]

class=wikitable align="left" width="350" |align=center colspan=4| Square Polysticks
Sticks	Name	Free {{OEIS2C|A019988}}	One-Sided {{OEIS2C|A151537}}
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

class=wikitable align="right" width="250" |align=center colspan=4| Hexagonal Polysticks
Sticks	Name	Free {{OEIS2C|A197459}}	One-Sided {{OEIS2C|A197460}}
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

class=wikitable style="margin-left: auto; margin-right: auto; border: none;" width="250" |align=center colspan=4| Triangular Polysticks
Sticks	Name	Free {{OEIS2C|A159867}}	One-Sided {{OEIS2C|A151539}}
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

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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).