On Minimal Moment of Inertia Polyominoes

  • Srečko Brlek
  • Gilbert Labelle
  • Annie Lacasse
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4992)


We analyze the moment of inertia Open image in new window , relative to the center of gravity, of finite plane lattice sets S. We classify these sets according to their roundness: a set S is rounder than a set T if Open image in new window . We show that roundest sets of a given size are strongly convex in the discrete sense. Moreover, we introduce the notion of quasi-discs and show that roundest sets are quasi-discs. We use weakly unimodal partitions and an inequality for the radius to make a table of roundest discrete sets up to size 40. Surprisingly, it turns out that the radius of the smallest disc containing a roundest discrete set S is not necessarily the radius of S as a quasi-disc.


Discrete sets moment of inertia polyominoes lattice paths 


  1. 1.
    Minsky, M., Papert, S.: Perceptrons: An Introduction to Computational Geometry. MIT Press, Cambridge (1969)zbMATHGoogle Scholar
  2. 2.
    Brlek, S., Labelle, G., Lacasse, A.: The discrete green theorem and some applications in discrete geometry. Theoret. Comput. Sci. 346(2), 200–225 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Altshuler, Y., Yanovsky, V., Vainsencher, D., Wagner, I.A., Bruckstein, A.M.: On minimal perimeter polyminoes. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 17–28. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Heck, A.: Introduction to Maple, 3rd edn. Springer, New York (2003)zbMATHGoogle Scholar
  5. 5.
    Lacasse, A.: Contributions à l’ analyse de figures discrètes en dimension quelconque. PhD thesis, Université du Québec à Montréal, Montréal (2008)Google Scholar
  6. 6.
    Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman lectures on physics. Mainly mechanics, radiation, and heat, vol. 1. Addison-Wesley Publishing, Reading (1963)Google Scholar
  7. 7.
    Stanley, R.P.: Enumerative combinatorics. Cambridge Studies in Advanced Mathematics, vol. 1, vol. 49. Cambridge University Press, Cambridge (1997), with a foreword by Gian-Carlo Rota, Corrected reprint of the 1986 originalGoogle Scholar
  8. 8.
    Klette, R., Rosenfeld, A.: Digital straightness—a review. Discrete Appl. Math. 139(1-3), 197–230 (2004)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Srečko Brlek
    • 1
  • Gilbert Labelle
    • 1
  • Annie Lacasse
    • 1
  1. 1.Laboratoire de Combinatoire et d’Informatique MathématiqueUniversité du Québec à MontréalMontréalCanada

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