Binary matrices under the microscope: A tomographical problem
Theoretical Computer Science
On the tiling by translation problem
Discrete Applied Mathematics
A linear time and space algorithm for detecting path intersection in Zd
Theoretical Computer Science
An optimal algorithm for detecting pseudo-squares
DGCI'06 Proceedings of the 13th international conference on Discrete Geometry for Computer Imagery
A parallelogram tile fills the plane by translation in at most two distinct ways
Discrete Applied Mathematics
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An unknown planar discrete set of points A can be inspected by means of a probe P of generic shape that moves around it, and reveals, for each position, the number of its elements as a magnifying glass. All the data collected during this process can be naturally arranged in an integer matrix that we call the scan of the starting set A w.r.t. the probe P. In [10], Nivat conjectured that a discrete set whose scan w.r.t. an exact probe is k-homogeneous, shows a strong periodical behavior, and it can be decomposed into smaller 1-homogeneous subsets. In this paper, we prove this conjecture to be true when the probe is a diamond, and then we extend this result to exact polyominoes that can regarded as balls in a generalized L"1 norm of Z^2. Then we provide experimental evidence that the conjecture holds for each exact polyomino of small dimension, using the mathematical software Sage [13]. Finally, we give some hints to solve the related reconstruction problem.