In the applet above the white triangle corresponds to the set of all possible triangles, the blue X the current base triangle (the triangle that this corresponds to is shown on the right). The corresponding colored dots are the daughters. In the upper left you can choose a subdivision method (now allow for subdividing into 3 parts as well). You can also decide the number of generations down to consider.
By clicking in the main triangle and dragging you can see what happens to the behavior as the base triangles are changed.