An ''odd'' formula for the volume of three-dimensional lattice polyhedra
type: Item
instance of: scholarly article
title: An ''odd'' formula for the volume of three-dimensional lattice polyhedra
zbMATH Open document ID: 0858.52005
DOI: 10.1007/BF00150027
author: Krzysztof Kołodziejczyk
published in: Geometriae Dedicata
publication date: 1996-09-18T00:00:00Z
review text: Let \(L_1\) be the fundamental lattice with integer coordinates in \(\mathbb{R}^3\), and let \(L_n\) be the lattice consisting of all points \(x\) in \(\mathbb{R}^3\) such that \(nx\) belongs to \(L_1\). For proper lattice polyhedra, i.e. polyhedra which have their vertices in \(L_1\) and which are equal to the closure of their interior, \textit{J. E. Reeve} [Proc. Lond. Math. Soc., II. Ser. 7, 378-395 (1957; Zbl 0080.26701)] established a formula which expresses the volume \(V(P)\) in terms of \(B_n= |L_n\cap \partial P|\), \(I_n= |L_n \cap \text{int} P|\) and the Euler characteristics \(\chi(P)\) and \(\chi (\partial P)\). In the simplest case of the lattices \(L_1\) and \(L_2\) this formula involves 27 points in each unit cube. The author establishes an analogous formula which involves only 9 points in each unit cube. This formula uses only the number of points lying in the interior and in the relative interior of the faces of proper lattice polyhedra. Moreover it does not make use of the Euler characteristic. Its proof is a modification of the proof of Reeve's formula. The proof is slightly shorter as it is sufficient to count interior points and points in the relative interiors of the faces of \(P\).
reviewed by: Charles Leytem
Mathematics Subject Classification ID: 52B20
Mathematics Subject Classification ID: 52B11
Mathematics Subject Classification ID: 11H06
zbMATH DE Number: 912996
zbMATH Keywords: lattice polyhedra
zbMATH Keywords: lattice point
zbMATH Keywords: Euler characteristic
zbMATH Keywords: proper lattice polyhedron
MaRDI profile type: Publication
cites work: Q4274019
cites work: Areas of lattice figures in the planar tilings with congruent regular polygons
cites work: A new formula for the volume of lattice polyhedra
cites work: Q5333037
cites work: On the Volume of Lattice Polyhedra
cites work: A Further Note on the Volume of Lattice Polyhedra
cites work: The Fascination of the Elementary
full work available at URL: https://doi.org/10.1007/bf00150027
OpenAlex ID: W2021848002
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description: scientific article; zbMATH DE number 912996