As a reader pointed out in a comment, the three methods I have shown so far, all traditional Japanese techniques, each are a variation in the theme of constructing a hypotenuse outboard or inboard of the radial sweep of the rafters, dividing that hypotenuse up by the number of required rafters, and then establishing the spacing from there. It seems like no matter how one tries to position or relate that developed hypotenuse to the radial sweep of the rafters, an imperfect result is produced.

Just because the results of a method are imperfect in some way, like a lot of things in carpentry, doesn't mean that there are carpenters who are unwilling to use the methods. After all, considering the problems (see post 1 and 2 in this thread) inherent in fan rafter spacing, it seems like obtaining an approximate result, with some compromises built in, is the best one might hope for, no? All of the previous methods are easily remembered and accomplished with two of the standard tools of the geometer, compass and straightedge, and that is a major plus.

Well, it turns out that there is a tidier solution in fact, and it does not involve constructing a hypotenuse like the other methods. It is a method of modern origin, unlike the others described so far. This is the method I used to obtain the fan rafter arrangement on the bell tower design recently completed:

This method however does not employ a simple geometrical development with compass and straightedge, but makes use of a computer to solve an algorithm, and a unique drawing development to produce the spacing of the rafters. Finding a spacing method that solves the inherently conflicting issues of even rafter tip spacing and even wedge-shaped spaces between rafters is a more complex problem than it may appear on the surface. While I'm not 100% sure that calculus is the method used, the problem, by nature, involves working towards limits that are never actually reached, only approximated. Calculus is after all the study of change. If I learn what the algorithm is at some point, I'll share the information.

The tables of rafter spacing for different arrangements of fan rafters are available in only a few Japanese lay out texts, and I have not at this point taken the time to determine the exact algorithm used to produce the tables. This algorithm is not shown in the texts, just the table of values. So in a sense, like books of rafter tables such as some roofers employ, if you don't know how the numbers are generated, you are entirely dependent upon the tables. So, that is one shortcoming of the method that ought to be acknowledged.

Let's look at the method and how it would apply to our situation of a regular 45˚ hip corner, and with eight fan rafters to be fitted to the space:

Clicking on any of the illustrations in this post, or any other post, will enlarge them for easier viewing.

I have an eave edge measuring '10' in length, and I will call this the Length, 'L'. To determine the radial spacing for the first rafter, I take my length 'L' and divide it by the number of rafter spaces to be fitted, in this case 8 (that is, 7 rafters):

L/8 = 1.25

Now I look up the coefficient in the table for the rafter spaces required, 8, and the value shown is 0.8660156. I multiply my previous result of 1.25 by this coefficient:

1.25 x 0.8660156 = 1.0825195....

That is the rafter space distance.

I then space out a mark for my first rafter along a perpendicular from the starting point (the common rafter line), this spacing distance of 1.0825...:

Then I connect a line from the fan origin point to this mark:

Easy enough! One down, seven to go.

The next step is to draw a perpendicular line from the place where my first fan rafter line meets the eave edge, and measure out along this perpendicular the same space of 1.0825...:

Next I connect a line from the origin to the end point of the line just measured out, and extend it out to the eave edge:

The next rafter is done the same way, again originating the perpendicular at the eave edge:

Done:

The procedure is repeated for the remaining rafters:

If the marking out was accurate, the last perpendicular marked out, from rafter 7, will exactly meet the 45˚ line. If it doesn't, a mistake was made somewhere along the way.

Now let's take a closer look to see what sort of spacing patterns were achieved with this method.

Rafter tip spacing first:

As you can see, the spaces grow in a nice even progression, with no sharp jumps between consecutive rafters. Of the methods looked at so far, the rafter tip progression produced here is the most seamless.

Now a look at the radial spacing, as seen when viewing the rafters from below:

Here we have achieved a nice smooth progression of descending values, 6.17˚ for the first, and down, in fairly even steps, to 4.72˚ for the last space. This is in stark contrast to the other methods we have looked at, in which the angular value would climb and descend as we swept along. The smooth progression that this coefficient method produces is the most seamless and sublime result, I do believe, of the methods available. That's why I chose to use it, but it took some research and analysis to determine that it was in fact the best method available in terms of producing an elegant result.

In the next post in this series, which I had originally planned to be the

Thanks for coming by the Carpentry Way today, and feel free to comment if you like.

-->on to post 7

Labels: fan rafters