Abstract
There are many interesting combinatorial results and problems dealing with lattice polygons, that is, polygons in ℝ2 with vertices in the integral lattice ℤ2. Geometrically, ℤ2 is the set of corners of a tiling of ℝ2 by unit squares. Denote by H the set of corners of a tiling of the plane by regular hexagons of unit area and call a polygon P a Hex-polygon or an H-polygon if all vertices of P are in H. Our purpose is to study several combinatorial properties of H-polygons that are analogous to properties of lattice polygons. In particular we aim to find some relationships between the numbers b and i of points from H on the boundary and in the interior of an H-polygon P with the numbers v and c of vertices and the so-called boundary characteristic of P. We also pose three open problems dealing with convex Hex-polygons.
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References
Arkinstall, J. R.: Minimal requirements for Minkowski's theorem in the plane I. Bull. Austral. Math. Soc. 22, 259–274 (1980)
Coleman, D. B.: Stretch: a geoboard game. Math. Mag. 51, 49–54 (1978)
Ding, R., Kołodziejczyk, K., Murphy, G., Reay, J.: A fast Pick-type approximation for areas of H-polygons. Amer. Math. Monthly 100, 669–673 (1993)
Ding, R., Kołodziejczyk, K., Reay, J.: A new Pick-type theorem on the hexagonal lattice. Discrete Math. 68, 171–177 (1988)
Ding, R., Reay, J.: The boundary characteristic and Pick's theorem in the Archimedean planar tilings. J. Combin. Theory Ser. A 44, 110–119 (1987)
Ding, R., Reay, J., Zhang, J.: Areas of generalized H-polygons. J. Combin. Theory Ser. A 77, 304–317 (1997)
Ehrhart, E.: Propriétés arithmo-geometriques des polygones. C. R. Acad. Sci. Paris 241, 686–689 (1955)
Kołodziejczyk, K.: Areas of lattice figures in the planar tilings with congruent regular polygons. J. Combin. Theory Ser. A 58, 115–126 (1991)
Kołodziejczyk, K.: Hex-triangles with one interior H-point. Ars Combin. 70, 33–45 (2004)
Kołodziejczyk, K., Olszewska, D.: A proof of Coleman's conjecture. Discrete Math. (2006), DOI:10.1016/j.disc.2006.09.033
Kołodziejczyk, K., Reay, J.: Primitive and mensurable Hex-triangles. Geom. Dedicata 43, 233–241 (1992)
Rabinowitz, S.: A census of convex lattice polygons with at most one interior lattice point. Ars Combin. 28, 83–96 (1989)
Rabinowitz, S.: On the number of lattice points inside a convex lattice n-gon. Congr. Numer. 73, 99–124 (1990)
Scott, P. R.: On convex lattice polygons. Bull. Austral. Math. Soc. 15, 395–399 (1976)
Scott, P. R.: The fascination of the elementary. Amer. Math. Monthly 94, 759–768 (1987)
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Kołodziejczyk, K. Realizable Quadruples for Hex-polygons. Graphs and Combinatorics 23, 61–72 (2007). https://doi.org/10.1007/s00373-006-0688-6
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DOI: https://doi.org/10.1007/s00373-006-0688-6