A polyhex is a polyform consisting of a finite nonempty edge-connected union of regular hexagons from
the regular hexagonal tessellation . Its cells are
joined along complete sides, and holes are allowed. Polyhexes are analogs of polyominoes
and polyiamonds and are also called hexes, hexas,
or polyfrobs (Beeler 1972). For the 4-hexes (tetrahexes), the possible arrangements
are known as the bee , bar ,
pistol , propeller ,
worm , arch , and wave .
The numbers of free translations , rotations ,
and reflections identified, for A000228 ; Klarner 1967, Balaban
and Harary 1968, Harary and Read 1970, Lunnon 1972, Gardner 1978, Knop et al.
1984, Gardner 1988).
The numbers of A038144 ;
Myers), the first few of which are illustrated above.
One-sided polyhexes identify translations and rotations , while reflections
are not allowed. Mirror image pieces are therefore
distinct unless a rotation identifies them. The numbers
of A006535 ).
The skeleton of a polyhex without holes is a benzenoid
and hence a fusene (Brinkmann et al. 2002).
See also Benzenoid ,
Fullerene ,
Fusene ,
Polyform ,
Polyhex
Tiling ,
Polyiamond ,
Polyomino ,
Polyplet
Explore with Wolfram|Alpha
References Balaban, A. T. "Enumeration of Cyclic Graphs." In Chemical
Applications of Graph Theory Balaban, A. T. and Harary,
F. "Chemical Graphs V: Enumeration and Proposed Nomenclature of Benzenoid Cata-Condensed
Polycyclic Aromatic Hydrocarbons." Tetrahedron 24 , 2505-2506,
1968. Balasubramanian, K.; Kauffman, J. J.; Koski, W. S.; and
Balaban, A. T. "Graph Theoretical Characterization and Computer Generation
of Certain Carcinogenic Benzenoid Hydrocarbons and Identification." J. Comput.
Chem. 1 , 149-157, 1980. Beeler, M. Item 112 in Beeler, M.;
Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial
Intelligence Laboratory, Memo AIM-239, pp. 48-50, Feb. 1972. https://www.inwap.com/pdp10/hbaker/hakmem/polyominos.html#item112 . Beineke,
L. W. and Pippert, R. E. "On the Enumeration of Planar Trees of Hexagons."
Glasgow Math. J. 15 , 131-147, 1974. Brinkmann, G.; Caporossi,
G.; and Hansen, P. "A Constructive Enumeration of Fusenes and Benzenoids."
J. Algorithms 45 , 155-166, 2002. https://doi.org/10.1016/S0196-6774(02)00215-8 . Brinkmann,
G.; Caporossi, G.; and Hansen, P. "A Survey and New Results on Computer Enumeration
of Polyhex and Fusene Hydrocarbons." J. Chem. Inf. Comput. Sci. 43 ,
842-851, 2003. Clarke, A. L. "Polyhexes." http://www.recmath.com/PolyPages/PolyPages/Polyhexes.html . Cyvin,
S. J.; Brunvoll, J.; Xiaofeng, G.; and Fuji, Z. "Number of Perifusenes
with One Internal Vertex." Rev. Roumaine Chem. 38 , 65-77, 1993. Dias,
J. R. "A Periodic Table for Polycyclic Aromatic Hydrocarbons. 1. Isomer
Enumeration of Fused Polycyclic Aromatic Hydrocarbon." J. Chem. Inf. Comput.
Sci. 22 , 15-22, 1982. Dias, J. R. "A Periodic Table
for Polycyclic Aromatic Hydrocarbons. 2. Polycyclic Aromatic Hydrocarbons Containing
Tetragonal, Pentagonal, Heptagonal, and Octagonal Rings." J. Chem. Inf. Comput.
Sci. 22 , 139-152, 1982. Dias, J. R. "A Periodic
Table for Polycyclic Aromatic Hydrocarbons. 3. Enumeration of All the Polycyclic
Conjugated Isomers of Pyrene Having Ring Sizes Ranging from 3 to 9." Math.
Chem (Mülheim/Ruhr) 14 , 83-138, 1983. Gardner, M. "Polyhexes
and Polyaboloes." Ch. 11 in Mathematical
Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind
from Scientific American. Gardner,
M. "Tiling with Polyominoes, Polyiamonds, and Polyhexes." Ch. 14 in
Time
Travel and Other Mathematical Bewilderments. Golomb, S. W. Polyominoes:
Puzzles, Patterns, Problems, and Packings, 2nd ed. Harary, F. "Graphical Enumeration
Problems." In Graph
Theory and Theoretical Physics Harary, F. Graph
Theory. Harary,
F. and Palmer, E. M. Graphical
Enumeration. Harary, F. and Read,
R. C. "The Enumeration of Tree-Like Polyhexes." Proc. Edinburgh
Math. Soc. 17 , 1-13, 1970. House of Graphs. Polyhex Graphs.
C6 , The
theta graph theta(1,5,5) , Elementary
2-Wall , Hex Grid 2,2,2 ,
Hex Grid 3,3,3 , and broken
wheel 13, spoke positions 1,5,9 . Keller, M. "Counting Polyforms."
https://web.archive.org/web/20030803065351/http://members.aol.com/wgreview/polyenum.html . Klarner,
D. A. "Cell Growth Problems." In Canad. J. Math 19 ,
851-863, 1967. Knop, J. V.; Szymanski, K.; Jeričević,
Ž.; and Trinajstić, N. "On the Total Number of Polyhexes." Match:
Commun. Math. Chem. , No. 16, 119-134, Aug. 1984. Lunnon, W. F.
"Counting Hexagonal and Triangular Polyominoes." In Graph
Theory and Computing Myers, J. "Polyomino Tiling." https://www.polyomino.org.uk/mathematics/polyform-tiling/ . Palmer,
E. M. "Variations of the Cell Growth Problem." In Graph
Theory and Applications: Proceedings of the Conference at Western Michigan University,
Kalamazoo, Mich., May 10-13, 1972 Mathematica J. 9 , 609-640, 2005. https://content.wolfram.com/sites/19/issue/v9i3.pdf .Sloane,
N. J. A. Sequences A000228 /M2682,
A006535 /M2846, and A038144
in "The On-Line Encyclopedia of Integer Sequences." Vicher,
M. "Polyforms." https://web.archive.org/web/20250831090700/https://www.vicher.cz/puzzle/polyforms.htm . von
Seggern, D. CRC
Standard Curves and Surfaces. Weisstein, E. W. "Books about Polyominoes." http://www.ericweisstein.com/encyclopedias/books/Polyominoes.html . Referenced
on Wolfram|Alpha Polyhex
Cite this as:
Weisstein, Eric W. "Polyhex." From MathWorld https://mathworld.wolfram.com/Polyhex.html
Subject classifications More... Less...
-polyhexes,
with translations, rotations,
and reflections identified, for 
, 2, ... are 1, 1, 3, 7, 22, 82, 333, 1448, 6572, 30490,
143552, ... (OEIS A000228; Klarner 1967, Balaban
and Harary 1968, Harary and Read 1970, Lunnon 1972, Gardner 1978, Knop et al.
1984, Gardner 1988).
-polyhexes
with holes for 
,
7, 8, ... are 1, 2, 13, 67, 404, ... (OEIS A038144;
Myers), the first few of which are illustrated above.
-hexagon one-sided polyhexes for 
, 2, ... are 1, 1, 3, 10, 33, 147,
620, 2821, 12942, 60639, 286190, 1364621, 6545430, ... (OEIS A006535).
Rangel-Mondragón, J. "Polyominoes and Related Families."
Mathematica J. 9, 609-640, 2005.