I'm thinking of using this: http://www.northerngreenhouse.com/products/polys/clear.htm for the outer layer of plastic. It has good light transmission characteristics (and strong wind, collision, and UV resistance, which I need), but as a woven material will strongly scatter the light. Even having a second layer of plastic on the inside of the struts will provide some degree of light scattering, though, preventing strong point shadowing from the struts. There will be reflective radiant insulation on the north wall as well, which should brighten things up a bit further by limiting pass-through and providing light from the other side of the structure. The dome shape helps reflect horizontal winter sun down onto the plants from all facets, anyway. The dome is about 13' high, too, and the upper structure captures and bends that light back down.
As far as general constraints on permissivity due to strut coverage, a worst-case back of the envelope calculation would give
1/4'x4' per strut
~2/5 dome surface exposure to side radiance
60 struts blocking
=60 out of 718 sq ft, or 8% coverage
1/4'x4' per strut
~2/3 dome surface exposure to top radiance
100 struts blocking
=100 out of 1257 sq ft, or 8% coverage
so, 8% is my worst-case blockage due to struts. The actual blockage will be less because this inflates the size of the struts a bit and assumes that none of the radiance on the sides of the struts will make it back into the dome. This will need to be added to shading from the glazing to find final permissivity numbers. Hopefully top dome light gathering and north wall reflection will make up for a bit of this as well.
I have some cutting to do. I've spent a lot of time considering how to cover a sphere with flat plastic without coming to a clear, obvious answer. Covering with a single sheet is not going to happen, because the size of the sheet would necessarily be enormous (better than 80' square) and the waste would be tremendous around the sides. If this were a shelter or something other than a greenhouse I might consider covering with five pieces around the dome (it is 5-way radially symmetric) and refolding wherever needed as I worked my way down the dome. Folding involves greater shading though, and I need to keep the plastic tight for a number of reasons, including standing up to the wind and shedding snow and rain.
At this point I plan to cut the plastic into the shape of the individual icosahedral patches. If this were a sphere, that would require 20 patches to cover, but it's considerably less for my dome, and should guarantee that everything fits nicely on the structure.
I'm using the woven plastic material in 10' width, which is just wide enough to accommodate the height of the full triangular patches with sufficient overage to line everything up on the dome and still have overlap. However, I didn't really need the top triangle of the pattern given above on most of the pieces. The top half of the dome requires five of these patches, but by leaving out the top triangle on each one I can use a single pentagon as a cap and not have to worry about any of those relatively flat seams leaking. The bottom of the dome is a bit wider so I need twice as many patches, but the upside down ones don't require the top triangle because that's where the dome ends, and the patches in between, while right side up, don't actually need the whole bottom row, which makes them simple smaller triangles. (I hope this explanation is clear. The following pictures and steps may help.)
I'm going to use a tarp to trace a template directly on the dome itself, and then lay it under the real plastic to mark each section. I will leave a 6-8 inches of overlap plastic around the pieces to accommodate the seams.
The actual attachment to the dome will be done with plastic lathing on the overlapped seam, secured by long staples and wide top nails at strategic points. The trick, of course, will be accessing the top of the dome to put these in...
Here are the pieces being cut out. The plastic is 10' wide X very long, so I unrolled it and restacked it accordion-style to be able to pull it off linearly. After the first couple of pieces I found it easier to use a previous piece as the template for each subsequent piece. The trapezoidal patches are rotated to fit with each other as they come off the roll. Unfortunately, I ran out of plastic, due mostly to my failing to account for the large amounts of overlap that I ended up using on each patch, and I needed to reorder to get a bit more plastic to finish off the external covering.
In the above picture, I had just finished putting down a new floor in my living room, so I had enough room to lay the patterns out and cut all of the pieces. When the second order of plastic came, I had to cut it outside, and that proved more difficult but possible. I would recommend finding a large indoor space if you can, though.
This time my helpers were a bit younger. The rocks are to keep the wind from blowing the edges around.
The sheets of plastic were initially attached with handclamps. Each patch is up to 12' x10', but the center point and interior cut have to be positioned to within 1/2" vertically, horizontally, and rotationally to place the end of the cut right in the middle of the hub and running straight up the strut. I didn't actually cut out the small wedge in the patch diagram; I slit the wedge down the middle so that the two edges would overlap when the piece was wrapped around the (non-flat) hexagon at the center.
When the positioning was right I stapled the plastic lathing onto the center seam by overlapping the cut from above, to allow it to shed water. There is 6-12" of margin around the outer edge of each piece which greatly simplifies positioning but also allows each of the corners to be secured under all of the other seams of the vertex, giving me greater confidence in the integrity of these points.
After I had two of these patches in position I secured the seam between them, reaching the top half from above with a ladder inside the dome, and the bottom half from a ladder outside the dome. On each seam I left the top and bottom edges loose to be able to slide in the adjoining top and bottom plastic.
Next, I traced out a pentagon onto my tarp in the same way I had done the previous pieces, including the center cut. Just like the trapezoids, no sliver is cut out of the pentagon--it is just a single cut where the plastic increasingly overlaps outward from the center, in the same way you might make a cone out of a circle with a radial cut in it.
This piece was poked up through the center and positioned around all of the edges and along the cut. Note that on all of these pieces the position of the cut on one of the struts is vital, since the overlap is very small in the middle. Unlike the side pieces, the direction of overlap is unimportant since it does not need to shed water downwards.
Once positioned, I started pinning down the seam of the top pentagon from the open side of the dome where I had left off the fifth side patch. Yes, I'm pretty high in the air here. Note that I'm leaning on the dome itself, and have someone below steadying my 10' ladder.
We decided to cover the center of the pentagon, where the seam comes together in the middle of a hub on a horizontal plane (i.e. it isn't sloped to shed water) with a circular "hat". Thus, a ~14" circle was attached with small strips of lathing on each hub at the top, and the seam lathing came down over the top of that. The pentagon itself is sloped down on each face, so should shed water (though not snow!). Eventually one or more vents may be cut out from the inside around this pentagon, but that will wait for summer when we can better judge how much ventilation will be needed.
After the hat and seam were attached I went to work around the sides. Each edge was attached from the next triangle over, all around the pentagon. You can see the circular "hat" here as it appears from the inside.
Next triangle. The beginning of the seam was started from within the triangle being covered itself, and then the ladder was moved and I came up from within the next triangle so that it could be pulled flat.
Topside. I was using extra long staples to be sure that the seams would hold. The electric staple gun allowed me to easily place all of the staples at full arm extension, but because of the long 9/16 staples they didn't fully penetrate the wood unless I really held the gun tightly to the surface. For this reason, I had a hammer up there to pound in anything that wasn't flush. I can't imagine only doing this with the hammer, though, since I generally had to keep two plastic surfaces in tension (opposite directions) under a tensioned plastic lathing strip, and I could not have done this if I needed both hands in order to place a fastener with one and use the hammer with the other. The hand clamps were in constant use and I occasionally tacked a bit of plastic in place with a single staple to keep it there while I held an overlapping piece.
You may be able to see on this picture that the plastic lathing at the ends of each strut is attached on either edge. This is to go around the metal straps that hold the struts to the hubs. Since the edge is thin and weak here, I used more staples. In fact, I used a lot of staples throughout. We've had quite a few wind storms (this area is famous for them) and nothing has loosened in the 65mph gusts. I will also mention here that this woven plastic is quite nice to work with--much sturdier and more resistant to creasing and kinking than any other flexible poly that I've used.
When we were sure of the position, the top and middle seam were clamped and the whole side was rolled up and inserted through the top middle triangle so that I could work on the top seam of the patch. Here I am temporarily tacking the top into place.
The pentagon was folded back down over the side patch, and the lathing on each side of the open pentagon triangle was finished out to the edge over the top of the plastic of the side patch. Then, the last edge of the pentagon was finally attached.
The side patch plastic was unrolled back over the side of the dome, and I moved over to the next triangle to secure the center seam. At this point, the patch was not attached on either side, just at the top. After starting the center seam from inside the triangle, it was finished from a ladder outside the dome.
With the top half of the dome completely covered, the bottom is sealed with four trapezoids (basically the tessellated patch from the diagram above, minus the top triangle, turned upside down) and five triangle patches (just the top four triangles from the same patch). Then there is the door and the portions around it which will need to be closed up as well.
Here is the first of the lower trapezoids clamped into place on the frame, with the center seam hanging open. Each lower piece is tucked under the top pieces, whose bottom edges and corners were left open for this purpose.
As in the top patches, the center seams have to be positioned, secured and lathed before the outer seams.
At the edges, the bottom patches have to be leaved under the top patches, and the triangles have to be under the trapezoids. There are only five points around the greenhouse where the corners of the patches meet, and for each of these joins there are five corners which have to be layered in the right order. All of the edge seams around one of these vertices are under the lathing before I work on the corner. I tack down the lower layers with a single staple while I pull the upper layers into place so that they can all be pinned under the same lathing. As I mentioned before, this results in the corners all being tacked under the their two opposite seams across the hub, making it even less likely that anything will come loose.
In between the trapezoids are triangle shapes, as I mentioned in the beginning of this thread, which hardly bear mention since they are trivial to attach from the ground around the dome. The only other oddities were the two pieces on either side of the door which were small and custom shaped to fit. At this point an overlapping flap door or similar arrangement of plastic on the open hexagon would make this a functional closed structure.