Realizable quadruples for hex-polygons. Combinatorics of honeycombs
type: Item
instance of: scholarly article
title: Realizable quadruples for hex-polygons. Combinatorics of honeycombs
zbMATH Open document ID: 1121.52005
author: Krzysztof Kołodziejczyk
published in: Graphs and Combinatorics
publication date: 2007-04-26T00:00:00Z
review text: Let \( H \) be the set of the vertices of the hexagons in a tiling of \( \mathbb R^{2} \) by regular hexagons with unit area. An \textit{\( H \)-polygon} is a simple polygon whose vertices are all in \( H \). \( H \)-polygons arise in the study of honeycombs, which were introduced by \textit{A. Knutson} and \textit{T. Tao} [J. Am. Math. Soc. 12, No. 4, 1055--1090 (1999; Zbl 0944.05097)] to prove the saturation theorem for representations of the Lie group \( GL_{n}(\mathbb C) \). The article under review proves several results for \( H \)-polygons analogous to theorems of \textit{G. Pick} [Sitzungsber. Lotos Prag (2) 19, 311--319 (1900; JFM 31.0215.03)], \textit{E. Ehrhart} [C. R. Acad. Sci., Paris 241, 686--689 (1955; Zbl 0065.03106)], and \textit{P. R. Scott} [Bull. Aust. Math. Soc. 15, 395--399 (1976; Zbl 0333.52002)] for polygons with vertices in \( \mathbb Z^{2} \). The article also includes three open problems. Given an \( H \)-polygon, let \( v(P) \) be the number of its vertices, let \( b(P) \) be the number of \( H \)-points on its boundary, and let \( i(P) \) be the number of \( H \)-points in its interior. Define the \textit{boundary characteristic} \( c(P) \) of \( P \) as follows. For each \( x \in H \cap \partial P \), let \( c(x, P) \) be the number of edges of the hexagonal tiling that extend locally from \( x \) into the exterior of \( P \), minus the number of such edges that extend locally from \( x \) into the interior of \( P \). Let \( c(P) = \sum_{x \in H \cap \partial P} c(x, P) \). An \( H \)-polygon \( P \) is called \textit{mensurable} if its area \( A(P) \) satisfies the following Pick-type formula: \[ A(P) = \frac{b(P)}{4} + \frac{i(P)}{2} + \frac{c(P)}{12} - 1. \] Finally, let \( q(P) = (v(P), b(P), i(P), c(P)) \in \mathbb Z^{4} \). The paper is devoted to proving several inequalities and parity conditions satisfied by the subsets \( \mathcal Q \), \({\mathcal{Q}}_{\text{co}}\), and \({\mathcal{Q}}_{\text{m}}\) of \( \mathbb Z^{4} \), wh…
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52C20
zbMATH DE Number: 5146109
zbMATH Keywords: Lattice point
zbMATH Keywords: Lattice polygon
zbMATH Keywords: Hexagonal tiling
zbMATH Keywords: H-point
zbMATH Keywords: Hex-polygon
zbMATH Keywords: Pick-type theorem
zbMATH Keywords: Boundary characteristic
reviewed by: Tyrrell B. McAllister
MaRDI profile type: Publication
full work available at URL: s00373-006-0688-6
OpenAlex ID: W2067605785
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DOI: 10.1007/S00373-006-0688-6
description: scientific article; zbMATH DE number 5146109