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This article has presented an approach for extracting ordered and structured patterns from a 3-D triangular mesh surface. The construction process of these patterns has been demonstrated, which starting from a ring of adjacent and ordered triangular facets, builds a more complex patterns that include concentric rings and sequence of facets aιτanged in a spiral-wise fashion. Thanks to their structured and organized aspects, these patterns have great potential to be exploited in a variety of applications that include the assessment of the quality of the triangular mesh tessellation, the computation of geodesic features, and the extraction of particular features of triangular facets, which in turn can be used for advanced modeling of the surface shape, as well as for matching surfaces. This framework presents some limitations. At the current stage, it does not handle closed surfaces. It is sensitive to border effects; as a matter of fact, facets on the border cannot be treated as roots for facet spirals. In terms of applications, this framework is not fit for optimized triangular mesh surfaces. Although it can handle those surfaces, it is hard to see any significant information that can be derived from the out come.