Planar configurations induced by exact polyominoes

Abstract

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. When the probe is a rectangle, a set A whose scan is homogeneous shows a strong periodical behavior, and can be decomposed into smaller homogeneous subsets. Here we extend this result, which has been conjectured true for all the exact polyominoes, to the class of diamonds, and we furnish experimental evidence of the decomposition theorem for exact polyominoes of small dimension, using the mathematical software Sage. © 2011 Springer-Verlag Berlin Heidelberg.