More ways to tile with only one shape polygon
- 102 Downloads
Monotiles with arbitrarily large isohedral number do exist;
The additional topological possibilities afforded in 3D allow construction of a simply connected monotile with a rule enforced by shape only, which is impossible for the hexagonal parquet in 2D;
The precise statement of the tiling problem matters— whether color matching rules are allowed; whether multiply connected shapes are allowed; whether spacefilling is required as opposed to just maximum density. So what about the quest for thek = ∞ monotile? Schmitt.
KeywordsMatching Rule Density Tiling Black Edge Mathematical Intelligencer Figure Tile Edge
Unable to display preview. Download preview PDF.
- R. Berger, “The undecidability of the Domino problem,”Mem. Amer. Math. Soc. 66 (1966), 1–72.Google Scholar
- J. S. Myers, “Polyomino tiling,” http://www.srcf.ucam.org/jsm28/ tiling/ (2005).Google Scholar
- L. Danzer, “A family of 3D-spacefillers not permitting any periodic or quasiperiodic tiling,” inAperiodic ’94, edited by G. Chapuis World Scientific, Singapore (1995).Google Scholar
- 13.S. Torquato and F. H. Stillinger, “New conjectural lower bounds on the optimal density of sphere packings,” to appear,Experimental Mathematics 15, Issue 3 (2006).Google Scholar