Friday, October 1, 2010

Fan of the Fan (VI)

In the previous post in the series from a few weeks back, I described a third method of laying out the radial spacing/angling of fan rafters. Like the methods described before it (method 1, method 2), the third one described, the 2.7/10 slope method, only provided a decent progression of rafter tip spaces while the interior angular spaces between rafters, as would be visible from below on an exposed eave, were, well, higgledy-piggledy, climbing and falling abruptly.

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 last (but I've decided to add one more beyond it :^)), I will take a look at a Chinese temple carpentry method for determining and making fan rafters. As you will see, they take quite a different approach than the Japanese do to the same problem.

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

-->on to post 7

12 comments:

  1. I'm always impressed with what I read here - very excited to see tools hit the wood on your bell tower.

    ReplyDelete
  2. I appreciate the comment anonymous, however I do not generally allow comments lacking their author's name (as stated below the heading, "Post a Comment").

    I certainly don't want to discourage anyone from commenting, however anonymity on the internet sometimes leads certain people to make quite angry or cruel comments about various things, including, I have found, carpentry.

    So, to everyone reading this, in the future, please append your name even if posting (according to the various ways the google comment bot categorizes comments for posting) as 'Anonymous'.

    Comments without the author's name, even if they are the sweetest words ever recorded, will be disallowed. And, in the opposite vein, even if your words are condemnatory and critical, if you put your name to it, I will publish.

    Thanks again,

    Chris

    ReplyDelete
  3. http://www.purplefrog.com/~thoth/philosophy/radial-rafters.html
    I decided to throw together a mathematical analysis of your rafter fans. I don't know if you've already gotten an explanation in the 3 years since this post was written.

    ReplyDelete
  4. Robert,

    excellent work! I bookmarked your page - glad this post inspired you to investigate.

    ~C

    ReplyDelete
  5. Chris,

    I was compelled deeper into the fan rafter topic the more I read your posts. Like Robert have done earlier in the replies, I also was curious into the Mathematical basis behind this method of fan rafter layout and have written a computer code that outputs coefficients. As I do not own any Japanese layout texts, I want to ask for verifications as to the correctness of the coefficient outputs of my program.

    3 rafters : 0.8579320496202825
    4 rafters : 0.8603009839320475
    5 rafters : 0.8625206355575021
    6 rafters : 0.8644195622291062
    And of course 7 rafters as outlined above is 0.8660155926188962 noting that more decimals is output for precision.

    Secondly, I noticed that the length mentioned above is 10 and is a straight line. However, if a roof starts curving at the eave edge corners, it also sweeps forward (outwards near the corners as viewed from the top). How does this scenario apply to the problem discussed above?

    Thanks,
    Frank

    ReplyDelete
    Replies
    1. Franks,

      thanks for the comment and your work on the math.

      Your numbers are correct, yet they appear to be one rafter count out. For example, the value you give for 3 rafters is the value given in the book for 4 rafter spaces. The value you give for 4 rafters is what the book gives for 5 rafter spaces, etc. might be just the difference as to whether one counts rafters or rafter spaces.

      I would appreciate it if you would share your math with me, by email. I haven't devoted any attention to this topic for a while but it remains of great interest.

      ~C

      Delete
  6. Hi Chris,

    Yes, I counted rafters instead of rafter spaces above. Thanks for the verifications. As for the Math, its quite simple. From the above diagram of yours, each time a rafter is laid, it creates a right angle triangle, so it can be solved using trigonometry functions. However, the tricky part is that the solution is not closed form as Robert mentioned on his web page earlier. As such, there are many ways to work around this. Newton's method was suggested by Robert. For me, I opted using Numerical Analysis techniques that I recently learned in my university courses.

    As for my second question from earlier, if the roof curves up in the corner producing an outward sweep when viewed from the top, how does one do the measurement for the eave edge? The example in this post describes a case with a flat eave edge measuring 10.

    Frank

    ReplyDelete
    Replies
    1. That eave lift my friend is all about the chūkō

      Delete
  7. Hi Chris,

    Since I started reading your site, I've learned alot. Many interesting information. I have a question through my observations regarding the layouts I'm seeing. I realized the example you shared above assumes the first rafter starts out perpendicular and starts fanning out to the 45degree hip rafter. However, your Bell Tower design, and possibly the vast majority of Japanese Architecture does not start their fanning rafter in that manner. More precisely, the first rafter already starts fanning a little bit. How do you go about with that type of layout with respect to this method?

    Thanks for your sharing,
    Michael

    ReplyDelete
    Replies
    1. Michael,

      thanks for your comment.

      The layout of the fanning rafter proceeds from one which is 90˚ to the eave line, whether or not that 90˚ rafter is included in the rafters. However it is not true that "possibly the vast majority of Japanese architecture does not start their fanning rafter in that manner". I'm not sure what you base that assumption on.

      The vast majority of decorative eaves, I would observe, employing fanning rafters to begin with (a small number of examples - parallel rafters are by far the more common), employ them on an eave in which there are confined to the hip corner, and thus they are preceded by parallel rafters. On a smaller building, or roof over a genkan, say, then it would be possible to have the hips sufficiently close to one another that the fan rafters from one hip radiate over to the next without a parallel rafter occupying a place. In such a situation, it would be a choice to make on the part of the carpenter, whether to include a parallel rafter or not.

      In the case of the belltower design, I wanted the sweep of the fan rafters to be continuous from one side to another and wanted the eave curve up to be continuous as well, and thus no parallel rafter was used. The layout of the rafters however still proceeded from a parallel rafter over to the 45˚ hip- the layout method is the same regardless.

      If i had chosen instead to make the eave flat in the middle portion, and only curve up the last couple of meters each side, then I would have employed a few parallel rafters and made the zone in which the rafters are swept a narrower one. It's more a question of aesthetics really. The most common Japanese roof with curvature is one in which the eave line is flat for the most part and only sweeps up modestly at the end.

      ~C

      Delete
  8. Hi Chris,

    The fan rafters from your bell tower was seamless without the use of parrellel rafter and that was the question I was trying to find (not exactly easy to formulate after a day at work ;) ). For your example Bell Tower, to be precise, where did you place that "invisible" parrelel rafter to begin the fanning?

    And does the fan absolutely need to begin when the roof starts curving?

    Thank you so much and Merry Christmas,
    Michael

    ReplyDelete
    Replies
    1. Michael,

      the 'missing' parallel rafters would simply be one position over from the starting fan rafter. The rhythm of the rafter tips tells you where the rafter would be located.

      And no, the fan does not need to start where the roof starts curving. Again, that is a question of aesthetics.

      ~C

      Delete

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