These methods are very useful when you for whatever reason want to split a surface. I use to split things quite often to add details that are based on the original properties of the object and it is a principle of mine to always create new things in relationship to something already there.
In the images red lines are the ones drawn in that step, blue lines are part of the final result and gray lines are previously drawn lines. The yellow spots mark what intersection you need to use to place the lines drawn on that step.
Each step is made so that all the new lines on it can be drawn in any order; meaning that all the references they need are already in place. This is also what decides how many steps a method needs.
First are guides to split a four corner surfaces into a number of parts of equal size, the range being from two to nine. Click the small images to show the full size version.
If you look over the different steps you can see that some patterns repeat over several different methods. For example you can find the pattern for thirds in 5, 6, 7 and 9! As you probably understand you can get a wide variety of splits by combining methods.
Another thing you might have noticed is the familiar pattern in the methods for 4 and 8 parts. That is actually my pattern for infinite subdivisions of a grid. To the left is a short walk through of that to display how it works. After step 8 you could go on forever!
This guide is split into two. The first row shows how to create the pattern and the rest shows how to use it. I tweak my line color convention a bit here and for the large boxes the pattern is still blue while the drawn grid is gray, even if those lines are what you are really after. Intersections are still yellow dots!
The method can be used on any rectangle, not just squares, but then it will also be divided into rectangles.
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