I spend a lot of my free time doing woodworking. My dad and I have a workshop that we try to use as much as possible. We recently acquired a used v-nailer for doing picture frames, and when I discovered that the backstops could be moved from a 4 sided frame to a 5, 6, or 8 sided frame, I knew I had to try to make some kind of geometric shape.
Step 1: V-Nailer/Underpinner
This project would have been difficult (but not impossible) without a v-nailer. Ours is an older used model, but it still works quite well. We got it off ebay (with some other framing equipment) for a few hundred dollars. The backstop can be adjusted for different frame shapes, and there is a quick adjustment to change the location of the fastener.
First you cut your stock to the appropriate angle (45 degrees for rectangle, 36 for pentagon, and 30 for hexagon). You hold your stock in the machine and press a foot peddle. A pneumatic cylinder brings down an arm to hold your joint in place while a v shaped piece of metal is driven into the joint. It is especially strong if you first glue the faces of the joint.
Step 2: Shapes and Angles
A truncated icosahedron is made up of 20 hexagons and 12 pentagons. I used 5" long pieces of pine ripped to 3/4" x 3/4". I cut the ends of the hexagons at 30 degrees, and the pentagons at 36 degrees. The hardest part of the whole project was calculating the bevel of the edges of the shapes. If you cut this right, they will join together perfectly and end up with a uniform shape. Do it wrong and your edges won't line up at all. Small mistakes are exaggerated as the project progresses.
The angle between hexagon and hexagon should be 20.9 degrees (I used 21, since my table saw isn't particularly accurate) and the angle between hexagon and pentagon is 18.7 degrees (I used roughly 18.5). I first cut the stock to 5" pieces, then cut the angles for the joint, then ran the pieces through the table saw set for the right angle. I labeled every piece either H-H (hexagon to hexagon, and 21 degrees), H-P (hexagon to pentagon, and 18.5 degrees), or P-H (pentagon to hexagon, and 18.5 degrees). This made assembly much easier.
After I had a few shapes made, I started assembly by applying glue to the edges and using a brad nailer to keep everything in place. Clamps are pretty much worthless, so be prepared to hold things at weird angles while you get a nail in a joint.
Step 3: Taking Shape
This part is really satisfying. It was a huge relief to get past the half-way point and find that my joints were still lining up. Some weren't perfect, and I ultimately found that my life was easier if I ran the corners on the belt sander before putting a shape in place. Lots of glue, lots of brad nails.
Step 4: Assembled
I apologize for the lousy picture. At this point it was all assembled, and I used wood putty to fill in cracks, knot holes, nail holes, and anywhere that didn't have a perfect joint. It took a lot of sanding with a handheld electric sander (a Black and Decker Mouse) to get everything smooth and take some of the sharp angles off the joints. I knew I would paint it eventually, so I was generous with the wood putty.
Step 5: Finishing
After it was all sanded, I painted everything (inside and out) with two coats of primer. I got a coat of better quality white paint and did two coats on everything after it was primed. The finished product has a really smooth finish.
I decided to turn it into a light fixture. I got a simple opaque globe light from the hardware store. I made a top place (a hexagon) big enough to cover one cell. I mounted the light on the underside of the plate. I made a column out of an 18" long piece of PVC with another hexagon plate at the other end of the PVC. I held it all together with a long piece of threaded rod running through the middle. I also ran the wiring for the light through the column. I sanded the plates and column and painted them flat black. I also painted the light fixture hardware flat black. I haven't hung it in my house yet, but I have tall ceilings and plan on replacing a recessed can light on one end of the room with this. For the moment it is clamped to an I-beam in my workshop.