Context: The fractal analysis has been used by landscape ecologists as a shape metric in order to quantify the complexity of landscape patches. However the use of fractal geometry in ecology possesses an unexplored potential. We then developed a broader study of shapes with unitary fractal dimension. Objectives: Our aim is to amplify the use of fractal dimensions as a metric of shape in the analysis and discovery of forestry landscape patches with unitary fractal dimension. Furthermore, we develop a method for monitoring and recovery of forestry patches. Methods: We establish a method of expansion in order to obtain patches with a good perimeter-area ratio, i. e., unitary fractal dimension. In order to do that, to each landscape patch we associate a polygon and, to each side, we define a locus that expands the polygon so that its fractal dimension is equal to one. Results: This study reveals a range of patches’ shapes with unitary fractal dimension. Inspired by the proposed method's recursion we denote them as fractally stable polygons. To each side of the polygon we set a condition of expansion possibility. The locus of expansion was also defined. Additionally, we define a test of global expansion. Conclusions: Through the developed method it is possible to ascertain when the perimeter-area ratio of a landscape patch is compromised. To expandable sides, the method provides the locus of recovery of the perimeter-area ratio. This enables a wider applicability in the analysis of forestry fragmentation through fractal dimension.