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]

The name "polystick" seems to have been first coined by Brian R. Barwell.[2]

The names "polytrig"[3] and "polytwigs"[4] 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.[5]

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.[6]

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.[7][8]


Square Polysticks

SticksNameFree OEIS: A019988One-Sided OEIS: A151537
1monostick11
2distick22
3tristick57
4tetrastick1625
5pentastick5599
6hexastick222416
7heptastick9501854

Hexagonal Polysticks

SticksNameFree OEIS: A197459One-Sided OEIS: A197460
1monotwig11
2ditwig11
3tritwigs34
4tetratwigs46
5pentatwigs1219
6hexatwigs2749
7heptatwigs78143

Triangular Polysticks

SticksNameFree OEIS: A159867One-Sided OEIS: A151539
1monostick11
2distick33
3tristick1219
4tetrastick60104
5pentastick375719
6hexastick26135123
7heptastick1907437936


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

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. ↑ Weisstein, Eric W. "Polystick." From MathWorld
  2. ↑ Brian R. Barwell, "Polysticks," Journal of Recreational Mathematics volume 22 issue 3 (1990), p.165-175
  3. ↑ David Goodger, "An Introduction to Polytrigs (Triangular-Grid Polysticks)," (2015), http://puzzler.sourceforge.net/docs/polytrigs-intro.html
  4. ↑ David Goodger, "An Introduction to Polytwigs (Hexagonal-Grid Polysticks)," (2015), http://puzzler.sourceforge.net/docs/polytwigs-intro.html
  5. ↑ David Goodger, "An Introduction to Polytwigs (Hexagonal-Grid Polysticks)," (2015), http://puzzler.sourceforge.net/docs/polytwigs-intro.html
  6. ↑ "Polynema -- from Wolfram MathWorld".
  7. ↑ Weisstein, Eric W. "Polystick." From MathWorld
  8. ↑ Counting polyforms, at the Solitaire Laboratory
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