Note: it seems that this post is the first thing you find on google about the Villard diagram. And after it was linked on AisleOne in an excellent text about grids, a lot of people landed on my humble home. I must say that the ‘diagram’ you can find here was my interpretation on what I’ve found about Villard diagram, it may be not-so-accurate. What is interesting is that it is a very practical way to divide segments, and it is practical precisely because it use in a clever way a recursive procedure.
I mentioned, in the recusion post, Villard de Honnecourt. it turns out that the so-called Villard diagram is, in itself, an example of nice recursion. This technique is useful to divide a segment: you only need two “triangle rulers”.
You can start drawing a rectangle, then trace the first diagonal (black, in the image). Now, draw another diagonal, at the intersection draw a line parallel to the segment you need to divide. The intesection point divide this segment in two halves; moreover the segment divide the height of the rectangles in two halves. And now? Use the new line to draw another diagonal (red), you find another intersection with the first, black, diagonal; draw ar line passing through the new intersection. The yellow triangle is similar to the blue one, scaled by a factor of 2, and the intesection you found divides the diagonal in two parts with the same proportions, i.e. the horizontal segment, as well as the height of the original rectangle, is divided in two parts: 1/3 and 2/3. You repeat the process starting from the new division and find 1/4, 1/5, and so on, recursively.
In the image below the segment is divided in seven parts.