 Our second main result characterizes pairs of triangulations , containing sets of edges E and E ' respectively, such that the minimum diagonal flip path between them contains the minimum number of diagonal flips over all minimum diagonal flip paths between pairs of triangulations , containing sets of edges E and E' respectively. Remarkably, all of our results are derived from a simple relationship between edges in triangulations and Farey sequences .
<meaning-changed>  Our second main result characterizes pairs of triangulations , containing sets of edges E and E ' respectively, such that the minimum diagonal flip path between them contains the minimum number of diagonal flips over all minimum diagonal flip paths between pairs of triangulations , containing sets of edges E and E' respectively. Remarkably, all of our results are derived from a simple relationship between edges in triangulations and Farey sequences .
 Our second main result characterizes pairs of triangulations , containing sets of edges E and E ' respectively, such that the minimum diagonal flip path between them contains the minimum number of diagonal flips over all minimum diagonal flip paths between pairs of triangulations . Finally, we demonstrate how our results can model crack propagation in crystalline materials caused by Stone-Wales defects. Notably, the above results follow from simple number-theoretic and geometric concepts .