A Note on Planar Monohedral Tilings

Oswin Aichholzer; Michael Kerber; István Talata; Birgit Vogtenhuber

A Note on Planar Monohedral Tilings

Oswin Aichholzer, Michael Kerber, István Talata, Birgit Vogtenhuber

Research output: Chapter in Book/Report/Conference proceeding › Conference paper

Abstract

A planar monohedral tiling is a decomposition of $R^2$ into congruent tiles. We say that such a tiling has the flag property if for each triple of tiles that intersect pairwise, the three tiles intersect in a common point. We show that for convex tiles, there exist only three classes of tilings that are not flag, and they all consist of triangular tiles; in particular, each convex tiling using polygons with $ngeq 4$ vertices is flag. We also show that an analogous statement for the case of non-convex tiles is not true by presenting a family of counterexamples.

Original language	English
Title of host publication	Proc. 34th European Workshop on Computational Geometry EuroCG '18
Place of Publication	Berlin, Germany
Pages	31:1-31:6
Publication status	Published - 2018
Event	34th European Workshop on Computational Geometry: EuroCG 2018 - FU Berlin, Berlin, Germany Duration: 21 Mar 2018 → 23 Mar 2018 https://conference.imp.fu-berlin.de/eurocg18/home

Conference

Conference	34th European Workshop on Computational Geometry
Abbreviated title	EuroCG 2018
Country/Territory	Germany
City	Berlin
Period	21/03/18 → 23/03/18
Internet address	https://conference.imp.fu-berlin.de/eurocg18/home

Access to Document

https://conference.imp.fu-berlin.de/eurocg18/download/paper_31.pdf

Cite this

@inproceedings{1930f941fcee4122bdf4064662acff37,

title = "A Note on Planar Monohedral Tilings",

abstract = "A planar monohedral tiling is a decomposition of $R^2$ into congruent tiles. We say that such a tiling has the flag property if for each triple of tiles that intersect pairwise, the three tiles intersect in a common point. We show that for convex tiles, there exist only three classes of tilings that are not flag, and they all consist of triangular tiles; in particular, each convex tiling using polygons with $ngeq 4$ vertices is flag. We also show that an analogous statement for the case of non-convex tiles is not true by presenting a family of counterexamples.",

author = "Oswin Aichholzer and Michael Kerber and Istv{\'a}n Talata and Birgit Vogtenhuber",

year = "2018",

language = "English",

pages = "31:1--31:6",

booktitle = "Proc. 34th European Workshop on Computational Geometry EuroCG '18",

note = "34th European Workshop on Computational Geometry : EuroCG 2018, EuroCG 2018 ; Conference date: 21-03-2018 Through 23-03-2018",

url = "https://conference.imp.fu-berlin.de/eurocg18/home",

}

TY - GEN

T1 - A Note on Planar Monohedral Tilings

AU - Aichholzer, Oswin

AU - Kerber, Michael

AU - Talata, István

AU - Vogtenhuber, Birgit

PY - 2018

Y1 - 2018

N2 - A planar monohedral tiling is a decomposition of $R^2$ into congruent tiles. We say that such a tiling has the flag property if for each triple of tiles that intersect pairwise, the three tiles intersect in a common point. We show that for convex tiles, there exist only three classes of tilings that are not flag, and they all consist of triangular tiles; in particular, each convex tiling using polygons with $ngeq 4$ vertices is flag. We also show that an analogous statement for the case of non-convex tiles is not true by presenting a family of counterexamples.

AB - A planar monohedral tiling is a decomposition of $R^2$ into congruent tiles. We say that such a tiling has the flag property if for each triple of tiles that intersect pairwise, the three tiles intersect in a common point. We show that for convex tiles, there exist only three classes of tilings that are not flag, and they all consist of triangular tiles; in particular, each convex tiling using polygons with $ngeq 4$ vertices is flag. We also show that an analogous statement for the case of non-convex tiles is not true by presenting a family of counterexamples.

M3 - Conference paper

SP - 31:1-31:6

BT - Proc. 34th European Workshop on Computational Geometry EuroCG '18

CY - Berlin, Germany

T2 - 34th European Workshop on Computational Geometry

Y2 - 21 March 2018 through 23 March 2018

ER -