From Checkerboard to Cloverfield: Using Wang Tiles in Seamless Non-Periodic Patterns
Inhaltsverzeichnis
Reference
Tuomas Nurmi: From Checkerboard to Cloverfield: Using Wang Tiles in Seamless Non-Periodic Patterns. In: Bridges 2016, Pages 159–166.
DOI
Abstract
This article discusses Wang tiles from an artistic point of view. Mathematically Wang tiles are unit squares with colored edges and a local rule saying that two edges may only be adjacent, if they have the same color. We will present some known methods to use and abuse this concept, ultimately to force aperiodic tilings of arbitrary size, optionally containing irregular tiles and continuity at the tile corners. This paper also introduces a previously unpublished aperiodic set of 30 Wang tiles with colored corners. This is the smallest such set known today.
Extended Abstract
Bibtex
@inproceedings{bridges2016:159,
author = {Tuomas Nurmi},
title = {From Checkerboard to Cloverfield: Using Wang Tiles in Seamless Non-Periodic Patterns},
pages = {159--166},
booktitle = {Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture},
year = {2016},
editor = {Eve Torrence, Bruce Torrence, Carlo S\'equin, Douglas McKenna, Krist\'of Fenyvesi and Reza Sarhangi},
isbn = {978-1-938664-19-9},
issn = {1099-6702},
publisher = {Tessellations Publishing},
address = {Phoenix, Arizona},
url = {http://de.evo-art.org/index.php?title=From_Checkerboard_to_Cloverfield:_Using_Wang_Tiles_in_Seamless_Non-Periodic_Patterns },
note = {Available online at \url{http://archive.bridgesmathart.org/2016/bridges2016-159.html}}
}
Used References
[1] R Berger. The undecidability of the domino problem. 1966. (document)
[2] Michael F. Cohen, Jonathan Shade, Stefan Hiller, and Oliver Deussen. Wang Tiles for Image and Texture Generation, 2003. (document)
[3] Emmanuel Jeandel and Micha¨el Rao. An aperiodic set of 11 Wang tiles. CoRR, abs/1506.0, 2015. (document)
[4] Jarkko Kari. A small aperiodic set of Wang tiles. Discrete Mathematics, 160(1-3):259–264, 1996. (document)
[5] A Lagae, J Kari, and P Dutr´e. Aperiodic sets of square tiles with colored corners. Report CW, 2006. (document)
[6] Ares Lagae and Philip Dutr´e. An alternative for Wang tiles: colored edges versus colored corners. ACM Transactions on Graphics (TOG), 25(4):1442–1459, 2006. (document)
[7] Hao Wang. Proving theorems by Pattern Recognition II. Bell Systems technical journal, (40):1–41, 1961. (document)
Links
Full Text
http://archive.bridgesmathart.org/2016/bridges2016-159.pdf
