An analog of the polyominoes and polyiamonds in which collections of regular hexagons are arranged with adjacent sides. They 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 geometrically planar n-polyhexes for n=1, 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),


The numbers of n-polyhexes with holes for n=6, 7, 8, ... are 1, 2, 13, 67, 404, ... (OEIS A038144; Myers), the first few of which are illustrated above.


"One-sided" polyhexes are considered to be fixed in the plane, and so mirror images are counted separately. The numbers of n-hexagon one-sided polyhexes are 1, 1, 3, 10, 33, 147, 620, 2821, 12942, 60639, 286190, 1364621, 6545430, ... (OEIS A006535).

A simple connected polyhex is called a fusene.

See also

Benzenoid, Fullerene, Fusene, Polyform, Polyhex Tiling, Polyiamond, Polyomino, Polyplet

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Balaban, A. T. "Enumeration of Cyclic Graphs." In Chemical Applications of Graph Theory (Ed. A. T. Balaban). London: Academic Press, pp. 63-105, 1976.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., 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.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.", 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. New York: Vintage, pp. 146-159, 1978.Gardner, M. "Tiling with Polyominoes, Polyiamonds, and Polyhexes." Ch. 14 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 175-187, 1988.Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton University Press, pp. 92-93, 1994.Harary, F. "Graphical Enumeration Problems." In Graph Theory and Theoretical Physics (Ed. F. Harary). London: Academic Press, pp. 1-41, 1967.Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 178-197, 1994.Harary, F. and Palmer, E. M. Graphical Enumeration. New York: Academic Press, 1973.Harary, F. and Read, R. C. "The Enumeration of Tree-Like Polyhexes." Proc. Edinburgh Math. Soc. 17, 1-13, 1970.Keller, M. "Counting Polyforms.", 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 (Ed. R. C. Read). New York: Academic Press, pp. 87-100, 1972.Myers, J. "Polyomino Tiling.", 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 (Ed. Y. Alavi, D. R. Lick, and A. T. White). New York: Springer-Verlag, pp. 214-223, 1972.Sloane, N. J. A. Sequences A000228/M2682, A038144, and A006535/M2846 in "The On-Line Encyclopedia of Integer Sequences."Vichera, M. "Polyforms." Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 342-343, 1993.Weisstein, E. W. "Books about Polyominoes."

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Weisstein, Eric W. "Polyhex." From MathWorld--A Wolfram Web Resource.

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