Week 140 (Annex Soffit Framing) - Manifold Recording

In the Annex Control Room the carpenters have cut and installed the subfascia for the rear soffit. Here you can see one of them putting up the decking on top:

And here are a variety of perspectives to try to show what this curvature really looks like:

As you can see in the second and third of these three images, the curved sub-fascia has the same reverse-plumb that the other sub-fascias express: -22.5°. And you may also notice that not only is the top surface of the sub-fascia level (it conforms perfectly to the decking above), but so is the bottom. How was this accomplished? With a very sharp pencil and a very keen mind. As Dr. Mamikon Mnatsakanian explains in the article Unwrapping Curves from Cylinders and Cones:

For any ﬁnite portion of the cone with a circular base, as shown in Figure 16a, the unwrapped
image of the base is a circular arc with center at the cone’s vertex and radius equal to
the slant height s of that ﬁnite portion. In this simple case, the basic problem is easily
solved because the radial distance R(θ) is constant, R(θ) = R(0) = s .

Figure 16 also reveals a basic fact that plays a key role in solving the general prob-
lem. Let ρ denote the radius of the base in Figure 16a. When the base rolls through an
angle of ϕ radians, the corresponding portion of the base of arclength ρ ϕ unwraps onto
a circular arc of radius s and central angle that we denote by θ (Figure 16b). Because
the cone is a developable surface, distances are preserved when the cone is unrolled
onto a plane, so we have

s θ = ρ ϕ

In our case, the sub-fascia terminates standard 2×4 framing members (which are 3-1/2″ high). Since the reverse plumb is 22.5°, angle α is 22.5° and thus the sub-fascia is 3-25/32″ along the s dimension. The arc radius ρ defined by our framing studs is 12′ 3-1/2″. The sub-fascia is specified to be 1-1/2″ thick, but because it is difficult to bend 1-1/2″ thick wood to such a radius, the sub-fascia is composed of three layers of 1/2″ plywood, glued and screwed to the soffit beams. The first first layer is cut according to a 12′ 3-1/2″ radius, the second according to a 12′ 3″ radius, and the third according to a 12′ 2-1/2″ radius. What Figure 16b teaches is that the first arc we need to cut from flat 1/2″ plywood has a outer radius of 12′ 3-1/2″ and an inner radius of 11′ 11 23/32″. The second and third respectively deduct 1/2″ and 1″ equally from both the inner and outer radii. Here is some board waste showing their work:

When these circular arcs are cut with a bevel of 22.5° and installed in the proper order, the happy result is achieved:

But the work is not yet done as the posts holding up the soffit are only temporary. When it is finished, it will hang from the ceiling, which, as far as the soffit is concerned, appears to be two 5:12 planes canted 30° away from the centerline. The suspension of this circular arc can be seen as a cylinder which is bisected by the angled ceiling facets, resulting in an ellipse whose semi-minor axis is 12′ 3-1/2″ and whose semi-major axis is 12′ 3-1/2″ × 13/12. (Why 13/12^ths? Because of the 5:12 roof pitch.) The elliptical plates are put into position:

And here is a detail of the resulting projection onto the ceiling:

Now here is the whole picture, albeit distorted through a fisheye lens:

Phew! But that’s not all…progress on the Music Room cloud was also very consistent this week. Making good use of the scissors lift, the electricians install the lighting fixtures and conduits to power them: