Israel Journal of Mathematics

, Volume 52, Issue 1–2, pp 140–146 | Cite as

Tilings whose members have finitely many neighbors

  • Marilyn Breen


Let ℭ be a tiling of the plane such that each tile of ℭ meets at most finitely many other tiles. Then exactly one of the following must occur:
  1. (1)

    Uncountably many boundary points of ℭ belong to no nondegenerate edge of ℭ, hence ℭ has uncountably many singular points; or

  2. (2)

    Every boundary point of ℭ belongs to a nondegenerate edge of ℭ, moreover, ℭ has no singular points.


Furthermore, ifS is the set of singular points of ℭ andW={t:t∈bdry ℭ andt belongs to no nondegenerate edge of ℭ}, thenS=clW.


Singular Point Boundary Point Jordan Curve Reverse Inclusion Obvious Induction 
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  1. 1.
    Marilyn Breen,A characterization theorem for tilings having countably many singular points, J. Geometry21 (1983), 131–137.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    B. Grünbaum and G. C. Shephard,Tilings and Patterns, W. H. Freeman, San Francisco (to appear).Google Scholar
  3. 3.
    Alain Valette,Tilings of the plane by topological disks, Geometriae Dedicata11 (1981), 447–454.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Weizmann Science Press of Israel 1985

Authors and Affiliations

  • Marilyn Breen
    • 1
  1. 1.Department of MathematicsUniversity of OklahomaNormanUSA

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