Dissections of Polygons into Convex Polygons

In the paper we present purely combinatorial conditions that allow us to recognize the topological equivalence (or non-equivalence) of two given dissections. Using a computer program based on this result, we are able to generate a set which contains all topologically non-equivalent dissections of a p0-gon into convex pi-gons, i = 1, ..., n, where n, p0, ..., pn are integers such that n ≥ 2, pi ≥ 3. By analyzing generated structures, we are able to find all (up to similarity) dissections of a given type. Since the number of topologically non-equivalent dissections is huge even if the number of parts is small, it is necessary to find additional combinatorial conditions depending on the type of sought dissections, which will allow us to exclude the majority of generated structures. We present such conditions for some special dissections of a triangle into triangles. Finally we prove two new results concerning perfect dissections of a triangle into similar triangles.