Author:  David Goodger <goodger@python.org> 

Date:  20150224 
Revision:  600 
Web site:  http://puzzler.sourceforge.net/ 
Copyright:  © 19982015 by David J. Goodger 
License:  GPL 2 
Contents
Polytwigs (a.k.a. polycules) are polyforms constructed from unit line segments (edges) joined endtoend on a regular hexagonal grid. Unit line segments on a square grid are called "polysticks", so polytwigs could also be called "polysticks on a hexagonal grid" or "hexagonalgrid polysticks" or "hexagonal polysticks". The polytwigs are a subset of the polytrigs (triangulargrid polysticks); polytwigs would fit on a polytrig grid, but there are a lot fewer of them.
Here is a puzzle containing all the polytwigs of order 1 through 5:
And here is a puzzle containing all the hexatwigs (order 6):
See Polytwigs: Puzzles & Solutions for many more puzzles.
Most of the polytwigs of order N can be thought of as the projective duals of the polyiamonds of order N+1 (i.e., joining the centers of the triangles of a hexiamond results in a pentatwig). The exceptions are the polytwigs with loops (e.g. the "O06" hexatwig), which are duals of the same order or lowerorder polyiamonds.
I invented the name "polytwigs" because "hexagonalgrid polysticks" is too unwieldy, especially when combined with orderprefixes (mono, di, tri, tetra, etc.; "hexagonalgrid tristicks"!?). The name comes from "twig" being a small stick, often bent and branching. "Polytwig" is also similar to "polytrig", the name I invented for triangulargrid polytsicks. I hope these names catch on.
When I began working on the hexagonalgrid polysticks, I was unable to find any references to prior research. While (to my knowledge) nothing seems to have been published, I couldn't believe that this was a completely novel idea — an intuition that proved to be true.
In February 2012 I received email from Mr. Colin F. Brown, a puzzle enthusiast from Dudley, England, who shared the results of his unpublished work on hexagonalgrid polysticks from the late 1970s and early 1980s. Mr. Brown wrote,
I did name them ‘Polycules’, from the fact that sea sponges have spicules that resemble a tricule (your 3stick hexagon gridded ‘polytwig’) each segment being set at 120° apart. I also gave each pentacule a name...
See the Wikipedia article on "sponge spicules" for an image demonstrating the resemblance. In fact, Mr. Brown used the term "polycules" for sqaregrid polysticks and triangulargrid polytrigs as well (prefixed with "rectangular", "triangular", and "hexagonal"), but:
... for constructional problems, I chose the hexagonal stick forms, partly because the rectangular and triangular types admitted tilings and constructions with ‘crossings’ – impossible with polytwigs.
I'll use "polycules" as a synonym for "polytwigs" here. Mr. Brown's pentacule names are listed in the pentatwigs section below. See Polytwigs: Puzzles & Solutions for examples of several of Mr. Brown's puzzle constructions.
Mr. Brown also explored the quasitritwigs with gaps between segments limited to length1; see QuasiPolyforms below.
At the Gathering for Gardner 10 (G4G10, 2012), Les Shader gave me a copy of a page from a Russian journal "Charade" ("ШАРАДА"), N6(6), December 1993. It featured a solution to the "Hexonet" puzzle ("ГЕКСОСЕТИ"), exactly the pentatwigs triangle (side length 5). Thanks to Tanya Khovanova of MIT for translating.
If you know of any other work on this type of puzzle, please let me know!
The number and names of the various orders of polytwigs are as follows:
Order  Polyform
Name

Free
Polytwigs

OneSided
Polytwigs


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 
The numbers of polytwigs can also be found in the following sequences from The OnLine Encyclopedia of Integer Sequences: A197459 (free) and A197460 (onesided).
Examples of the polytwigs from order 1 (monotwig) to order 6 (hexatwigs) are given in the tables below.
The polytwigs are named with a letternumber scheme, like "I1", "C3", and "R16". The letter part is the letter of the alphabet that the polytwig most closely resembles. In some cases, that resemblance is weak, and the letters are arbitrary. The final digit of the number represents the polyform order (how many line segments are in the polytwig). There are more hexatwigs than letters in the alphabet, so their names have an extra middle digit (numbered from 0) to differentiate the variations.
In the tables below, "Aspects" refers to the number of unique orientations that a polyform may take (different rotations, flipped or not). This varies with the symmetry of the polyform.
The "OneSided" column identifies polyforms that are asymmetrical in reflection. Treating the flipped and unflipped versions of asymmetrical polytwigs as distinct polyforms (and disallowing further reflection or "flipping"), results in "onesided" polytwigs and puzzles.
There are 3 free tritwigs (order3 polytwigs) and 4 onesided tritwigs:
Name  Image  Aspects  OneSided 

C3  6  
S3  6  yes  
Y3  2 
There are 4 free tetratwigs (order4 polytwigs) and 6 onesided tetratwigs:
Name  Image  Aspects  OneSided 

C4  6  
P4  12  yes  
W4  6  
Y4  12  yes 
There are 12 free pentatwigs (order5 polytwigs, a.k.a. pentacules) and 19 onesided pentatwigs. Colin F. Brown's original pieces and their evocative names are listed in the "Pentacule" and last "Name" columns.
Name  Image  Aspects  OneSided  Pentacule  Name 

C5  6  Bowl  
H5  12  yes  Chair  
I5  6  yes  Stick  
L5  12  yes  Signpost  
P5  12  yes  Hook  
R5  12  yes  Pipe  
S5  6  yes  Snake  
T5  6  Bird  
U5  6  Bridge  
W5  12  yes  Club  
X5  6  Trestle  
Y5  6  Fork 
There are 27 free hexatwigs (order6 polytwigs) and 49 onesided hexatwigs:
Name  Image  Aspects  OneSided 

C06  12  yes  
F06  12  yes  
H06  12  yes  
H16  12  yes  
I06  6  
J06  12  yes  
L06  12  yes  
L16  12  yes  
L26  12  yes  
M06  6  
O06  1  
Q06  12  yes  
Q16  12  yes  
Q26  12  yes  
R06  12  yes  
R16  12  yes  
S06  12  yes  
S16  12  yes  
S26  12  yes  
U06  6  
V06  6  
W06  12  yes  
X06  12  yes  
X16  12  yes  
Y06  4  yes  
Y16  12  yes  
Y26  12  yes 
Quasipolyforms are polyforms where the requirement that all unit shapes be connected has been removed. In other words, quasipolyforms are polyforms where some or all unit shapes may be separate from the others.
Without limits on the distance between unit shapes there would be an infinite number of quasipolyforms (for orders 2 and above). We will limit the quasipolyforms we consider to those with length1 gaps between segments.
The number and names of the various orders of quasipolytwigs are as follows:
Order  Polyform
Name

Free
Polytwigs

OneSided
Polytwigs


1  monotwig [*]  1  1 
2  quasiditwig  3  4 
3  quasitritwigs  17  28 
4  quasitetratwigs  114  214 
5  quasipentatwigs  966  1885 
[*]  With only 1 line segment, there can be no disconnected quasimonotwig. The set of the regular (fullyconnected) monotwigs is identical to the set of quasimonotwigs, and consists of one piece: the "I1" monotwig. 
Examples of the quasipolytwigs of order 2 (quasiditwigs) and order 3 (quasitritwigs) are given in the tables below. See the table legend above for column descriptions.
There are 3 free quasiditwigs (order2 quasipolytwigs) and 4 onesided quasiditwigs:
Name  Image  Aspects  OneSided 

C2  6  
L2  6  
S2  6  yes 
In his prior research, Colin F. Brown showed that in addition to the 3 free connected tritwigs, there are 5 semiconnected (one gap) quasitritwigs and 9 disconnected (two gaps) quasitritrigs, for a total of 17 free quasitritwigs (order3 quasipolytwigs). There are 28 onesided quasitritwigs.
Name  Image  Aspects  OneSided 

C03  6  
C13  12  yes  
C23  2  
H13  12  yes  
I13  6  yes  
P13  12  yes  
P23  12  yes  
P33  12  yes  
S03  6  yes  
S13  6  yes  
T13  6  
U13  6  
W13  12  yes  
W23  12  yes  
Y03  2  
Y13  12  yes  
Y23  6 
Polytwig puzzles use a pseudo3D skewed coordinate system, where the X and Y axes are 60° apart instead of the usual 90°. The typical representation (as seen in the Polyform Puzzler solution data files) positions the Y axis vertically with the X axis 30° counterclockwise from horizontal. Solution graphics are often rotated so the X axis is horizontal, and the Y axis 30° clockwise from vertical.
The (X,Y) 2dimensional coordinate identifies the lowerleft corner of the (X,Y) hexagon in a polyhex grid (a honeycomb with hexagons with horizontal tops & bottoms). The three line segments emanating from this point share the (X,Y) coordinate, and the Z dimension is used for the direction of the line segment:
Since only three line segments may emanate from each (X,Y) intersection, there is no possibility for polytwigs to cross each other and no need to keep track of intersections. This separates polytwigs from polysticks and polytrigs, where such considerations are crucial.