Skip to main content
SpringerLink
Log in

Tribone Tilings of Triangular Regions that Cover All but Three Holes

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

This paper extends the literature of tiling \(n\)-row triangular arrays (\(T_n\)) with copies of \(1 \times 3\) trominos discussed by Thurston, Conway, and Lagarias, focusing on tilings which cover all but three holes. We find a set of \(2^{\Omega (n^2)}\) such tilings, disproving the conjecture from 1993 that there are only \(2^{o(n^2)}\) such tilings. Furthermore, we show that if three cells are randomly removed from \(T_n\) when \(n \equiv 0,2 \pmod 3\), then the probability that the remaining region can be tiled by tribones is nonzero.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Conway, J.H., Lagarias, L.C.: Tiling with polyominoes and combinatorial group theory. J. Comb. Theory Ser. A 53(2), 183–208 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  2. Lagarias, J.C., Romano, D.S.: A polyomino tiling problem of Thurston and its configurational entropy. J. Comb. Theory Ser. A 63(2), 338–358 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  3. Thurston, W.P.: Conway’s tiling groups. Am. Math. Mon. 97(8), 757–773 (1990)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

I would like to sincerely thank Rich Schwartz for introducing me to Conway’s tiling groups, for discussing this and related problems with me, and for all his advice throughout the process. I would also like to thank Brown University for funding this research during Summer 2013 with the Karen T. Romer Undergraduate Teaching and Research Award, and an anonymous referee for helpful comments and suggestions.

Author information

Authors and Affiliations

  1. Harvard Business School, Boston, MA, USA

    Michael Thaler

Authors
  1. Michael Thaler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Thaler.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Thaler, M. Tribone Tilings of Triangular Regions that Cover All but Three Holes. Discrete Comput Geom 53, 466–477 (2015). https://doi.org/10.1007/s00454-014-9652-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-014-9652-z

Keywords

Mathematics Subject Classification

Search

Navigation