Thursday, July 22, 2010

Fan of the fan

The Japanese term for fanning, or radially-arranged rafters is ōgi-daruki (扇垂木). The latter two characters, '垂木' read taruki/~daruki literally means drooping tree/wood, a reference to rafters. The first kanji of that compound, '', is a straightforward one to describe, so why not?

An earlier form of it looks like this:

Forming a frame around the left and top is '', which is a pictograph of a single [swinging] door. That element, or radical as it is termed, is in fact the left half of the character with the meaning of door/gate, namely '' (read: 'mon'). This character is therefore about something which is like a door in some way.

The two side-by-side elements enclosed within the character, '', is an independent character of its own, and stems from a pictograph of a bird's wings. Here's an even earlier version of that character so you can see, perhaps, how the modern form derived:

So, '' is feather/wing/plumage, and combining it with door, '', you get '', which means a feathered door. Well, not quite - I jest. It means: (folding) fan, the panels of which fold into each other when closed. I guess the early Chinese saw the folding fan as analogous structures to bird wings, which are door like panels unfurled from the body. Fair enough, that makes sense.

The rafters which fan around the corners do look much like a hand-held fan:

Now, fanning rafters arranged radially around a circular wall plan are fairly simple to do, as each rafter is the same as the next, and spacing them is a straightforward matter of dividing the circumference of the wall plate into even increments to locate the rafters upon, or placing the rafters in a constant angular relation to one another, say, one rafter every 5˚ or something along those lines. The biggest problem in such situations relates to forming a curved wall plate and purlins (if any), bridging between rafters, and tying the rafters together at the roof's apex in some decently clean manner. That's not a topic of consideration in this post.

When you want to place fanning rafters at the the corner of a polygonal (3, 4, 5, 6, 7, 8 sides or more) building however, you enter a world of confusion, pain and hurt, for the circle and the square, as it were, do not always cooperate. This may not be apparent at first inspection, so in today's blog I want to explore this matter in some detail and consider also why, despite the difficulties, fan raftering might be a good idea. This is an involved topic, so I expect this to spill over into another post or two.

The circle and the square...actually, let's simplify that a bit. With most buildings we are dealing with squares or rectangles and the walls meet at 90˚ angles and the hip rafters are regular, that is, at a 45˚ relative to the wall plan. What happens on one side of a hip is the same as what happens on the other side, so we only need to look at one 45˚ section, not an entire square, and instead of a full circle, only that section of circle which sweeps through 45˚. Here's what I mean:

If you click on the picture it will enlarge and you should be able to see the scribbles more clearly.

Next, I'll cut out the extraneous detail, including the wall outline and focus in on that 45˚ slice of the roof plan we are considering:

The roof terminates at the eave edge, so that forms the boundary for the triangle at which we are going to look. A 1:1 ratio produces the 45˚ angle for the hip rafter, and the length of the hip rafter in plan, by Pythagorus's well-known method, is √2 times the side.

Now, let's deal with the simplest case of placing a fan rafter, where we wish to place one rafter in the middle of our 45˚ section. One apparently obvious way to do that would be to bisect the 45˚ so as to produce two 22.5˚ angles, and that line is then the centerline of our fan rafter:

While that divided our 45˚ pie wedge into two perfectly equal slices of 22.5˚ each, take a look at what happened along the eave edge. By measurement, we find that the run of 1 unit there has been divided into two subsections, one measuring 0.4142136... and the other measuring 0.585 or so. Those sections are not symmetrical at all. This means that if we were looking at the building eave from below, while the division of the rafters would look just fine, but if we instead viewed the edge of the roof from out in the yard, that fan rafter tip would not be even remotely centered between the adjacent common rafter and the hip. That's no good - looks like a hack job.

Let's do it the other way then - we'll divide the side length of 1 into two equal portions of 0.5 each and connect a line from the origin to that point:

While that now gives us a rafter tip perfectly centered between the common rafter and the hip, if we stood under the eave and looked up at the rafters, we would see that the two pie wedges to which our 45˚ has been divided are hardly equal at all - one is 26.565˚ or so, and the other 18.434˚ or so. That looks ungainly.

So that's the nub of the problem right there folks. The circle and the square are not getting along so well. What divides a given arc evenly does not do so in a right triangle, and vice versa.

Consider the above problem again: if one has, say, a 10/10 triangle forming a 45˚, one might jump to the conclusion that if we want a triangle with half as much rise, ie., a 5/10, then we would simply divide the 45˚ in half to make 22.5˚ and that would be the same thing. But it isn't so.

Let's check that with the calculator:

5 divided by 10 gives us 0.5; use the arc-tan ('2nd' + TAN) to find the angle value: 26.565...˚.

The trick to fan rafters lies in determining a method of placing them so that both the view from below (of the 'pie wedges') and the end-wise view of the eave edge out in the yard both produce even patterns. This is actually impossible to do perfectly, so we have to find a method that gets us closest. This entire issue, by the way, is analogous to that of constructing a world map (or any other map) - when trying to represent a circle (the globe) on a flat rectangle with square corners (like most maps) there is going to be some distortion no matter how you, er, slice it. Some maps distort the size of the continents, some distort the poles, etc. I imagine there are similar problems in the world of optics too.

So there we have the first few tentative steps into the swamp of the fan rafters. I hope you'll return to see what other alligators we can wrestle with in part II of this thread.

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