In the previous post, I had just traversed the section of the La Menuiserie Volume III text covering the intersection of solids.
The next section tackles hoppers of various configurations and and hip rafters/posts fitted to their arrises, or Les arêtiers droits. This section begins with a number of pages dealing with the basics of stereotomy, which I'll leave off getting into in this post. Funny enough, if you look at the foregoing link, and see how short the entry for that subject is on Wikipedia, for what a vast field of study stereotomy comprises, and then take a look for the Wiki entry for stereotomy (The Alan Parson's Project album of the same name), you'll note how much more detailed the latter is. Amusing, kind of....
Anyway, the first exercise in this chapter deals with problems of fitting sloped boards (or panneaux) into corners of spaces:
In the 90˚ corner of the above space is a parallelepiped, we could call it a 2x4.
The sloped board, at any plan angle and slope, is to be fitted thusly:
Now these kinds of problems come up at times in both rough framing and finish carpentry work, and I estimate that 95% of the time, in this country at least, the worker's solution is cut-and-fit, trial-and-error, and then, if the result is not too tidy, liberal amounts of filler or caulking are applied, followed by paint, and the hopeful estimation that "from x-feet away nobody will notice". Nothing to see here folks, move along.
Some minority of carpenters might try to find a way to scribe-fit the angled board, however descriptive drawing is, I would say, the more elegant and less time-consuming method, once you know how to do it. It just takes a bit of study.
Here's a look at the rudiments of the drawing method used to produce a plan drawing of the sloped board:
The sloped board then superimposed upon the plan to demonstrate the connection between drawing and piece:
Note that many of the trace lines have been removed from the sketch for clarity.
The next problem in the book is much the same, however instead of a parallelepiped in the corner, we have a stick which is 5-sided, and the plan is no longer that of a square corner, but an obtuse one in plan:
Removing the sloped board to show the guts of the drawing:
After completing that one, I moved out into making up a couple of related sketches of my own, to apply the method to slightly different circumstances.
In the first, I have made both the shape of the stick and the plan angle of the sloped boards alignment more extreme:
Then I did one with yet another random shape of corner post:
I liked that once the method was understood, it could be applied to a wide variety of situations.
The next section in the chapter deals with casing around a pipe, say, with a flared base:
Here's the sketch for that, in a stripped-down view:
Notice that the model under study covers all the bases for the intersections between the boards, as we have miter joints on the left half of the model and butt joints on the right. Efficient.
The next model ups the ante some, still a problem involving wrapping boards around some object, however this time the surface with which you are dealing and fitting the boards against slopes inward, as would be seen, for instance if it were the inside face of a roof plane:
Note also that the plan on the floor is no longer square.
Another view shows the slope issue with the upper section, and how the boards on the left side of the model, upper and lower, meet with a miter while the ones on the right side meet with butt joints:
As mentioned, the left half is mitered, while the right uses butt joints. Here is an overhead view of the lower flared portion only:
The developed drawing solution is much the same as with the preceding example:
However, there is a problem with this model. One of those nit-picky sort of things which hung me up like a nail sticking up from the floor.
If you consider the left (all-mitered) side of the construction only, you can see that that considering the meeting between only the lower sloped board and the upper pipe casing, there would be a certain bisection angle which would produce the right miter between the pieces, if one kept the thicknesses of the boards the same. And one would imagine that in most cases it would be most convenient to keep the board thicknesses the same. You show up with a pile of boards, all planed to the same dimension, and proceed to cut out some casing. What could be more straightforward?
Here's what an elevation view of the left hand section looks like, given a board thickness of 40.0mm:
Then consider the pairing the front lower board and its associated upper casing board, which are also mitered. There, the angle at which the parts meet would be completely different than that of the left side board and associated upper casing board:
Therefore the miter bisection is different as well - another way of putting that, is that if you measured from the floor to the hight of the miter on the front face, it would be the same for both left and front pairings, however the height from the floor to the rear wall miter line of each pairing would not be the same.
There's where we run into the problem. If one keeps the lower left and center boards the same thickness as one another, as would be natural, then the upper left and center pipe casing boards cannot be the same thickness and meet with a perfect miter to the lower boards. If you think about the fact that the lower pair of boards slope the same and meet one another 90˚ in plan, then there would be a specific miter angle between them, BUT, in the case of the upper pair of boards, one has plumb side while the other slopes forward. THE upper pair has a different miter angle where they meet than the lower pair.
That might be hard to follow just from reading, so allow me to illustrate.
Here's a front view of the meeting between the lower and upper boards on the left side, where everything miters:
The miter line at the front is at the same elevation all the way around, seamless and clean. Happy times indeed.
Now let's look at the backside of the same assembly:
It looks okay from a distance, but if we zoom right in to the point where all four boards meet, there is a hole:
This hole is the result of the miter angles between upper and lower board pairs being different from one another. As mentioned previously, as the angular relations between top and bottom boards with each pair are different, and the front miter lines at the same height, the rear miter lines cannot be the same height. The result is clearly shown above.
A look at the pairing of the left side boards only shows one of the offset on the upper boards face miter:
Similarly, the front board pairing, viewed from below, has an offset at the miter as well:
Something has to give somewhere. Either one makes at least one of the lower casing boards of different thickness (in which case the miter between those boards must be unequal), or one makes at least one of the upper casing boards a different thickness.
I played around with many different solutions just trying to see if there was another way, even though I had already surmised that there wasn't. The only way to obtain a clean solution is to make one of the boards a different thickness than the others, namely the upper front board:
A near 3mm difference - which is a lot in my view - in board thickness allows for an unequal miter between the front pair of boards, and also between the top two boards there is a second unequal miter, and clean connections therefore everywhere else. It's a clean solution aesthetically, but a complex one, cut-out-wise.
And maybe, as some might aver, all is all is a trifling issue and who cares? What's little hole on the back of the junction going to hurt anybody?
Yah, you could stuff some putty in there I'm sure. To be particular about it, it is a problem of concern only at a detail level and it would matter if one was constructing such a piece as a finished work in which all faces and intersections were visible, (as it would be, for example, if constructing something like a speaker's podium of similar shape). The trouble is, as the text makes no mention of the issue and the drawing also elides the issue, I am left unsure of how such a situation is intended to be resolved. I tried to figure out the best solution I could once I discovered the problem, which is not supposed to be my job when studying examples from a text like this.
This is like the Mazerolle trépied établi, where intersections between posts and top have exposed ugly shoulder offsets that cannot be avoided if the drawing is followed, and no good solution seems to present itself. Or the unconstructable cornice molding on Mazerolle's dormer on a bias. Those study examples can be found elsewhere on this blog.
I like beating my head against French carpentry drawing from time to time, what can I say?
At that juncture of mild frustration/irritation with the material, I took a break from the text. I will resume my study soon enough, as some interesting stuff lies ahead. I'm just going to ignore the last study, and simply think of it as, well, flawed.