# Mathematical Progress in Expressive Image Synthesis IIIWang Tile Modeling of Wall Patterns

Mathematical Progress in Expressive Image Synthesis III: Wang Tile Modeling of Wall Patterns [Wall patterns are essential in the creation of textures for visually rich buildings. Particularly, irregular wall patterns give an organic and lively feeling to the building. In this chapter, we introduce a modeling method for wall patterns using Wang tiles which are known for creating aperiodic tiling of the plane under certain conditions. We introduce a class of Wang tiles and prove that any rectangle with border constraints and bigger than a 2×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\times 2$$\end{document} rectangle can be tiled. We use this proof to derive a tiling algorithm that is in linear time. Finally, we give some results of our algorithm and compare the computation time with previous Wang tiling algorithms introduced in computer graphics.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# Mathematical Progress in Expressive Image Synthesis IIIWang Tile Modeling of Wall Patterns

Part of the Mathematics for Industry Book Series (volume 24)
Editors: Dobashi, Yoshinori; Ochiai, Hiroyuki
10 pages      /lp/springer-journals/mathematical-progress-in-expressive-image-synthesis-iii-wang-tile-z0ZmxUO12U
Publisher
Springer Singapore
ISBN
978-981-10-1075-0
Pages
71 –81
DOI
10.1007/978-981-10-1076-7_9
Publisher site
See Chapter on Publisher Site

### Abstract

[Wall patterns are essential in the creation of textures for visually rich buildings. Particularly, irregular wall patterns give an organic and lively feeling to the building. In this chapter, we introduce a modeling method for wall patterns using Wang tiles which are known for creating aperiodic tiling of the plane under certain conditions. We introduce a class of Wang tiles and prove that any rectangle with border constraints and bigger than a 2×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\times 2$$\end{document} rectangle can be tiled. We use this proof to derive a tiling algorithm that is in linear time. Finally, we give some results of our algorithm and compare the computation time with previous Wang tiling algorithms introduced in computer graphics.]

Published: May 22, 2016

Keywords: Texture synthesis; Wall patterns; Wang tiles; Tiling algorithms