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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Beauquier, D., Nivat, M.: On Translating one polyomino to tile the plane. Discrete Comput. Geom. 6, 575–592 (1991)
Brleck, S., Frosini, A., Rinaldi, S., Vuillon, L.: Tilings by translation: enumeration by a rational language approach. The Electronic Journal of Combinatorics 13(1) (2006)
Brlek, S., Provençal, X.: An optimal algorithm for detecting pseudo-squares. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 403–412. Springer, Heidelberg (2006)
Nivat, M.: On a tomographic equivalence between (0,1)-matrices. In: Karhumäki, J., Maurer, H., Păun, G., Rozenberg, G. (eds.) Theory Is Forever. LNCS, vol. 3113, pp. 216–234. Springer, Heidelberg (2004)
Frosini, A., Nivat, M.: Binary matrices under the microscope: a tomographical problem. TCS 370, 201–217 (2007)
Herman, G.T., Kuba, A.: Discrete tomography: Foundations algorithms and applications. Birkhauser, Boston (1999)
Nivat, M.: Sous-ensembles homogenes de \(\mathbb{z}^2\) et pavages du plan. C. R. Acad. Sci. Paris. I 335, 83–86 (2002)
Provençal, X.: Combinatoire des mots, geometrie discrete et pavages. Ph.D. thesis, D1715, Université du Québec a Montréal (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Battaglino, D., Frosini, A., Rinaldi, S. (2011). Planar Configurations Induced by Exact Polyominoes. In: Aggarwal, J.K., Barneva, R.P., Brimkov, V.E., Koroutchev, K.N., Korutcheva, E.R. (eds) Combinatorial Image Analysis. IWCIA 2011. Lecture Notes in Computer Science, vol 6636. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21073-0_25
Download citation
DOI: https://doi.org/10.1007/978-3-642-21073-0_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21072-3
Online ISBN: 978-3-642-21073-0
eBook Packages: Computer ScienceComputer Science (R0)