Polyhex ( English polyhex ) [1] [2] , or a hexagonal monster ( English hexagonal animal ) [3] [4] - a geometric figure in the form of a polygon made up of several regular hexagons connected by sides. Polyhexes can be considered as finite subsets of a hexagonal parquet with a connected interior.
Along with other polyforms - polymino and polyamondas , polyhexes are widely used in entertaining mathematics, mainly in problems of drawing figures. The name was proposed by D. Clarner by analogy with the names of other polyforms [2] .
The polyhexes resemble the structural formulas of polycyclic aromatic hydrocarbons in shape (each hexagon corresponds to a benzene ring ).
Content
- 1 Number of polyhexes
- 2 See also
- 3 notes
- 4 References
Number of polygons
As in the case of polymino , they distinguish between “free” polygons (when the turns and reflections are considered the same figure), “one-sided” when the figures are considered different when mirror images are used, and “fixed”, which are also distinguished during turns.
The number of “free” n- hexes for n = 1, 2, 3, 4 ... is given by the sequence
- 1, 1 , 3 , 7 , 22 , 82 , 333 , 1448, ... (
A000228 ).
Other polyhex OEIS sequences:
- the number of polyhexes with holes -
A038144 ;
- the number of polyhexes without holes -
A018190 ;
- the number of fixed polyhexes -
A001207 ;
- the number of one-sided polyhexes -
A006535 .
| Monohex | |
| Digex | |
| Three trihexes | |
| Seven tetrahexes | |
| 22 pentagex | |
| 82 hexagex |
See also
- Polimino
- Polyamond
Notes
- ↑ Weisstein, Eric W. Polyhex on the Wolfram MathWorld website.
- ↑ 1 2 Gardner M. Mathematical short stories / Per. from English Yu.A. Danilova. Ed. Ya.A. Smorodinsky .. - M .: Mir, 1974. - S. 267 - 281.
- ↑ Golomb S.V. . Polyminino = Polyominoes / Per. from English V. Firsova. Foreword and ed. I. Yagloma . - M .: Mir, 1975 .-- S. 143 - 147. - 207 p.
- ↑ George E. Martin. Polyominoes: a guide to puzzles and problems in tiling. - MAA , 1996. - ISBN 0-88385-501-1 . The animals.
Links
- Maths library Triangular and hexagonal "monsters"