Wednesday, August 25, 2010

Fan of the Fan (V)

So far in this series I have taken an in-depth look at the issues inherent in laying out fan rafters, and have presented a couple (post III, post IV) of different traditional Japanese methods for spacing the rafters. Each of those methods has proven to be decent in one aspect of spacing (at the rafter tips), but not so good at the other (that being the view of the inter-rafter wedge-shaped spaces from directly underneath).

Today I will examine another traditional Japanese approach to the problem. This is one of the more commonly-described techniques in the layout texts: the infamous '2.7 slope' method.

Again we start with a 45˚ plan view, that is, a regular hip rafter corner:


This is a right triangle of course, labeled 'ABC'.

Now I drop a perpendicular down from the eave edge, line AB, measuring 2.7 in length. The end of this line is marked 'D', and a line is connected from it back over to 'B':


Next, the hypotenuse of this 2.7/10 triangle, line BD, is subdivided into the desired number of rafter spaces - again we choose 8 divisions, just as we had with the preceding examples:


The next step should be predictable enough by now - we connect lines from the origin of the fan, point 'A', through to the division marks on line BD:


It's time to analyze the results. First we look at the eave edge, line AB:


Clicking on the above picture will render it larger and more readable. While 10/8 gives an average 'ideal' spacing of 1.25, when using the 2.7/10 method it can be seen that the spacing of our rafter tips along the eave edge follows a pattern of gradually increasing value:

1.0111
1.0678
1.1296
1.1967
1.2701
1.3504
1.4386
1.5357

Whats nice about this pattern is its smooth progression. In fact, each rafter tip space is roughly 1.06 times larger than the one preceding it. Our average 'ideal' value of 1.25 for the spacing falls in the middle of the series of values (more accurately: it would be a median value if placed into the series). That gives a pleasing look for the rafter tip spaces it seems to me.

Next, let's see what this method does for the pattern of wedge-shaped spaces - the view from under the eave:


An even division of 45˚ into 8 equal angles would be 5.625˚. As you can see, with the 2.7/10 method, the angular values start out in excess of that with 5.7736˚, and then proceed as follows:

5.7736˚... 5.9708˚ ... 6.0446˚ ...

The spaces are climbing steadily for the first three intervals, however we started out somewhat over the 5.625˚ value, so something will have to give soon - and it does with space #4:

5.7736˚... 5.9708˚ ... 6.0446˚ ... 5.9857˚ ... 5.8016˚ ... 5.5144˚ ... 5.1547˚... and 4.7543˚

So, this fan pattern exhibits a similar behavior to the previous method in post IV: the spaces between rafters climb up from the start, then drop back down as we sweep over to the hip. In fact, this 2/7/10 method gives a greater extreme in the range of values for these wedge-shaped spaces than the post IV method. The only advantage this method has over the method in post IV is that the eave rafter tip spacing is on a more regular progression and does not exhibit the same extremes. This method is the simplest really in terms of the steps of layout, so that is a point in its favor. Still, it's not ideal. Choosing between this method and that in the previous post is a bit of a toss-up if you ask me. Therefore, the search continues.

Is there any solution to this conundrum? The three methods we have looked at so far all have their shortcomings. Well, the answer to the rhetorical question is, 'yes', and we'll be looking at what is undoubtedly the penultimate method for spacing fan rafters in this thread's next post. I hope you'll return for that installment.

Thanks for your visit to the Carpentry Way today. --> on to post 6

3 comments:

  1. All three methods so far boil down to the same fundamental principle, the only difference is how the hypotenuse is established. For interest's sake, I took the two preceding methods and figured out their equivalent definition using this method.

    If you call this the "2.7 slope method", then the method in Post #3 could also be called the "1.72 slope method", while that of Post #4 is the "4.15 slope method". Unless your last method is something else entirely, it'll be interesting to see how it compares to the previous methods.

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  2. Chris

    If the next post on fan rafter layout is the penultimate, then the next after that must be the ultimate. Can't wait! You mentioned something earlier about Chinese fan rafter layout, too. Will that be the postultimate?

    Tom

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  3. Daniel, good things come to those who wait, and yes, your observation that the three methods illustrated so far vary only in regards to how the hypotenuse is established is quite correct.

    Tom,

    yes, the next post is the second to last in this thread, however we'll see if the following post on the Chinese method is actually 'ultimate' or not.

    ~Chris

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