Investigation of Computer Generated Clusters on Square Lattice Using Box Dimension Characterization

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Author(s) Salau T.A.O | Hashim M.K | Abayomi K
Pages 563-569
Volume 5
Issue 11
Date October, 2015
Keywords Percolation, Threshold Probability, Computer Generated Cluster, Box Counting Dimension, Fractal Dimension, Occupation Probability.


The geometry of fractals and the mathematics of fractal dimension have provided valuable tools for variety of scientific applications. This study modelled a square lattice on 2-dimensional Euclidean plane, populated it with Boolean matrix, labelled it with Hoshen-Kopelman (HK) algorithm and determined geometric variation of five largest clusters if any by estimating their Average Estimated Fractal Dimensions (AEFD) at different scales of resolution and occupation probabilities. The randomly generated matrices according to specified occupation probabilities within the square lattice were labelled, counted and the number of cluster existed within the lattice were identified. The average box counting dimension was obtained by implementing the least square regression procedures on the number of boxes counted per different cluster at different scales of observation across the clusters. The (AEFD) of first five largest clusters increases when the occupation probabilities increases. However, the fractal dimensions of the first largest cluster dominated and maintained a steady value beyond the critical probability (0.593) while the fractal dimension of the remaining four largest clusters started decreasing rapidly for all occupation probabilities above the critical probability. All the five clusters enjoy the same complex degree of geometrical characteristics before the reach of critical probability

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