Tilings of Convex Sets by Mutually Incongruent Equilateral Triangles Contain Arbitrarily Small Tiles

  • Christian RichterEmail author
  • Melchior Wirth


We show that every tiling of a convex set in the Euclidean plane \(\mathbb {R}^2\) by equilateral triangles of mutually different sizes contains arbitrarily small tiles. The proof is purely elementary up to the discussion of one family of tilings of the full plane \(\mathbb {R}^2\), which is based on a surprising connection to a random walk on a directed graph.


Perfect tiling Equilateral triangle Convex set Random walk Recurrent Markov chain 

Mathematics Subject Classification

52C20 (Primary) 05C81 51M20 60J10 (Secondary) 



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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of MathematicsFriedrich Schiller UniversityJenaGermany

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