Denote by gdist(p) the least number of cells that have to be changed to get a latin square from the table of addition modulo prime p. A conjecture of Drápal, Cavenagh and Wanless states that there exists c > 0 such that gdist(p) ≤ c log(p). In this work we prove the conjecture for c ≈ 7.21, and the proof is done by constructing a dissection of an equilateral triangle of side n into O(log(n)) equilateral triangles. We also show a proof of the lower bound c log(p) ≤ gdist(p) with improved constant c ≈ 2.73. At the end of the work we present computational data which suggest that gdist(p)/ log(p) ≈ 3.56 for large values of p.

[pt] O presente trabalho desenvolve um sistema computacional que, baseado em conceitos algébricos bem simples, discretiza domínios planos de contornos irregulares em elementos triangulares equilaterais ou próximas a estes. A técnica utilizada consiste em gerar faixas consecutivas de elementos triangulares a partir do contorno externo, de tal forma a diminuir consecutivamente a área restante a ser discretizada. O método apresentado envolve várias modificações feitas sobre o método de Sadek como a atualização instantânea do contorno após o processo de geração, cortes sequenciais, teste de interferência, sequência do processo de geração e o cálculo da posição do nó gerado. Vários exemplos gráficos são mostrados. / [en] The development a computional algorithm for the generation of triangular meshes for plane surfaces with irregular boundarys is described. The method used is based on the generation of successive layers of elements starting from the external boundary and the consecutive reduction of the remaining area. It contains modifications of the method of Sadek such as continous update of the boundary, sequential cuts and test of interference. Several graphical examples are included.

The target of the this diploma thesis called ''The Napoleon's theorem'' is a detailed concentration on this theorem, where the process of so called ''regularization'' is described. Under the investigation of the Napoleon's theorem this diploma thesis is concerned with a lot of proofs, properties and then their generalization in a plane and in space. Pictures, which can help the reader to understand this problem are supplemented in this diploma thesis.