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Suppose I have a tiling of the plane with parallelograms where the sides of the parallelograms come from a specified finite set of vectors. If I only have access to the vertices of this tiling I may not be able to recover the tiling. For example the triangular lattice could come from different tilings when the vectors are the sixth roots of unity.

Suppose that we choose a specific vertex in our point set. Can we answer the question of how many parallelograms contain it as a vertex? Are such questions decidable?

The original question was motivated by a physics project where the tiling was 3 dimensional and there were 6 vectors available to construct the parallelohedra. I already don't know the answer for 2 dimensional tilings.

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  • $\begingroup$ (1) When you say, "how many parallelograms contain it as a vertex," must the parallelograms be empty of points of your set? (2) Are you imagining the point set given as an infinite list of points, or by the finite set of vectors that generates that list? $\endgroup$
    – Joseph O'Rourke
    Commented Sep 10, 2013 at 23:43
  • $\begingroup$ I am imagining the point set as an infinite list of points. You know that these are the vertices of some tiling but you don't know which tiling. $\endgroup$
    – Gjergji Zaimi
    Commented Sep 10, 2013 at 23:49
  • $\begingroup$ does your question also cover aperiodic tilings, like Penrose tilings? see en.wikipedia.org/wiki/Penrose_tiling $\endgroup$
    – Manfred Weis
    Commented Sep 11, 2013 at 9:36

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For example, in the case of the triangular lattice there are tilings by rhombi where a given vertex can be a vertex of $3$, $4$, $5$ or $6$ rhombi. I'll draw a picture.

enter image description here

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