### Abstract

The Wang tiling is a classical problem in combinatorics. A major theoretical question is to find a (small) set of tiles which tiles the plane only aperiodically. In this case, resulting tilings are rather restrictive. On the other hand, Wang tiles are used as a tool to generate textures and patterns in computer graphics. In these applications, a set of tiles is normally chosen so that it tiles the plane or its sub-regions easily in many different ways. With computer graphics applications in mind, we introduce a class of such tileset, which we call sequentially permissive tilesets, and consider tiling problems with constrained boundary. We apply our methodology to a special set of Wang tiles, called Brick Wang tiles, introduced by Derouet-Jourdan et al. in 2016 to model wall patterns. We generalise their result by providing a linear algorithm to decide and solve the tiling problem for arbitrary planar regions with holes.

Original language | English |
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Journal | Japan Journal of Industrial and Applied Mathematics |

DOIs | |

Publication status | Published - Jan 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Engineering(all)
- Applied Mathematics

### Cite this

**A linear algorithm for Brick Wang tiling.** / Derouet-Jourdan, Alexandre; Kaji, Shizuo; Mizoguchi, Yoshihiro.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - A linear algorithm for Brick Wang tiling

AU - Derouet-Jourdan, Alexandre

AU - Kaji, Shizuo

AU - Mizoguchi, Yoshihiro

PY - 2019/1/1

Y1 - 2019/1/1

N2 - The Wang tiling is a classical problem in combinatorics. A major theoretical question is to find a (small) set of tiles which tiles the plane only aperiodically. In this case, resulting tilings are rather restrictive. On the other hand, Wang tiles are used as a tool to generate textures and patterns in computer graphics. In these applications, a set of tiles is normally chosen so that it tiles the plane or its sub-regions easily in many different ways. With computer graphics applications in mind, we introduce a class of such tileset, which we call sequentially permissive tilesets, and consider tiling problems with constrained boundary. We apply our methodology to a special set of Wang tiles, called Brick Wang tiles, introduced by Derouet-Jourdan et al. in 2016 to model wall patterns. We generalise their result by providing a linear algorithm to decide and solve the tiling problem for arbitrary planar regions with holes.

AB - The Wang tiling is a classical problem in combinatorics. A major theoretical question is to find a (small) set of tiles which tiles the plane only aperiodically. In this case, resulting tilings are rather restrictive. On the other hand, Wang tiles are used as a tool to generate textures and patterns in computer graphics. In these applications, a set of tiles is normally chosen so that it tiles the plane or its sub-regions easily in many different ways. With computer graphics applications in mind, we introduce a class of such tileset, which we call sequentially permissive tilesets, and consider tiling problems with constrained boundary. We apply our methodology to a special set of Wang tiles, called Brick Wang tiles, introduced by Derouet-Jourdan et al. in 2016 to model wall patterns. We generalise their result by providing a linear algorithm to decide and solve the tiling problem for arbitrary planar regions with holes.

UR - http://www.scopus.com/inward/record.url?scp=85066636245&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85066636245&partnerID=8YFLogxK

U2 - 10.1007/s13160-019-00369-z

DO - 10.1007/s13160-019-00369-z

M3 - Article

JO - Japan Journal of Industrial and Applied Mathematics

JF - Japan Journal of Industrial and Applied Mathematics

SN - 0916-7005

ER -