Editing 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.<ref name="auto">[http://mathworld.wolfram.com/Polystick.html Weisstein, Eric W. "Polystick." From MathWorld ]</ref>

The name '''"polystick"''' seems to have been first coined by Brian R. Barwell.<ref>Brian R. Barwell, "Polysticks," Journal of Recreational Mathematics volume 22 issue 3 (1990), p.165-175</ref>

The names '''"polytrig"'''<ref>David Goodger, "An Introduction to Polytrigs (Triangular-Grid Polysticks)," (2015), http://puzzler.sourceforge.net/docs/polytrigs-intro.html</ref> and '''"polytwigs"'''<ref name="auto1">David Goodger, "An Introduction to Polytwigs (Hexagonal-Grid Polysticks)," (2015), http://puzzler.sourceforge.net/docs/polytwigs-intro.html</ref> 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 [[Sponge spicule|spicules]] of [[sea sponges]].<ref name="auto1" />

There is no standard term for line segments built on other [[List of uniform tilings|regular tilings]], an [[unstructured grid]], or a simple [[connected graph]], but both '''"polynema"''' and '''"polyedge"''' have been proposed.<ref>{{cite web |url=https://mathworld.wolfram.com/Polynema.html |title = Polynema -- from Wolfram MathWorld}}</ref>

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.<ref name="auto" /><ref>[http://www.solitairelaboratory.com/polyenum.html ''Counting polyforms'', at the Solitaire Laboratory]</ref>



{| 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 
|}


{{clear}}
The set of ''n''-sticks that contain no closed loops is equivalent, with some duplications, to the set of [[Polyomino|(''n''+1)-ominos]], as each [[Vertex (graph theory)|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==

[[Image:Polysticks.svg|thumb|600px|center|The free square polysticks of sizes 1 through 4, including 1 monostick (red), 2 disticks (green), 5 tristicks (blue), and 16 tetrasticks (black).]]

==References==
{{reflist}}

==External links==
* [http://puzzler.sourceforge.net/docs/polysticks.html ''Polysticks Puzzles & Solutions'', at Polyforms Puzzler]
* [http://did.mat.uni-bayreuth.de/wassermann/tetrastick.pdf ''Covering the Aztec Diamond with One-sided Tetrasticks'', Alfred Wassermann, University of Bayreuth, Germany]
* [http://www2.stetson.edu/~efriedma/mathmagic/0806.html ''Polypolylines''], at [http://www2.stetson.edu/~efriedma/mathmagic/ Math Magic]

{{Polyforms}}

[[Category:Polyforms]]