David Anderson
Originally written in May 1998
Once the math of the n-frequency geodesic is worked out, building a structure is fairly straightforward; the question is reduced to construction technique. There are two general directions you can go with a dome - either building a faced structure or an edged structure. A faced structure centers around the construction of geodesic faces as the principal component of the structure. The geometry of the faces is determined as a function of the chord lengths and fastening angles, the faces are constructed individually, and are integrated into a finished dome. The edged structure amounts to building struts which form the edges of the geodesic structure as a set of joined chords. The chord lengths for the geodesic are determined, the struts are fabricated and finally, the structure is constructed by attaching the struts together at geodesic vertices.
Generally, building a edged structure is easier than the corresponding faced structure. After all, the faced structure has surfaces which have to be carefully built to fasten to the adjoining surfaces; if the surfaces are more than simple cardboard triangles connected by duct tape, this construction is not trivial. The edged structure is just a framework. All one needs to do is cut the struts to the right length, determine a means for attaching them, and the dome practically builds itself.
I'd been planning a geodesic greenhouse: I'd figured out the math and built some simple models with straws and pipe cleaners. Now it was time to see if it would work on human scale. I decided on a 2-frequency half-spherical geodesic because I'd only need struts of two sizes; I wanted something fairly straightforward to prove everything before I added complexity. The math worked out to 30 short struts and 35 long ones.
The main considerations for me were materials and technique. I wanted this thing to be cheap since I didn't want to sink money into false starts - but at the same time I wanted it to be solid enough to hold up to the elements. I didn't want to toss this out as a prototype unless I'd really screwed up. I chose PVC pipe because it had a good track record: it was strong, it could be cut easily, and it was remarkably affordable.
I decided I was going to try the flattening technique I had read about in articles on the Internet geodesic mailing list.
By putting some adhesive inside the end of a PVC strut, putting that end in a vice, and screwing it closed for a while, letting the glue set, a good solid strut could be made.
The clamped end would be able to be drilled and bolted to other struts.
To test things out I put an end of a PVC pipe I intended on using into my bench vice and started turning...
Now, although I may not be the strongest guy on my street, I'm certainly not the weakest - this PVC pipe was really hard to crush.
I'm sure Fuller would have something synergetic to say about collapsing a cylinder, but all I know is that there was no way I wanted to vice-close 130 PVC pipe ends.
This was not a reasonable answer for me.
Since at one time I was a woodworker (mostly making cabinetry) I have an old bandsaw.
After a little thinking, I decided the PVC would still be strong enough for my purposes if instead of squeezing the pipe flat, I used the bandsaw to cut it flat.
I tried cutting a few pipes, drilling 1/4" holes 1" from the ends, and connecting them with a bolt.
Voila!
The technique worked.
In fact, it worked better than I'd thought at that moment.
Because the PVC had to bend up to 12 degrees on a 2-frequency dome, cutting the PVC made it bend more easily - without breaking.
The bend put less strain on the smoothly cut strut so the dome was easier to assemble.
The diagram shows the gentle curve of the cut; this is important for two reasons. First, smoothly cut shapes resist cracking or breaking better than sharp cuts. PVC bends when a smooth force is applied along a smooth line, but it's still brittle enough to fracture if too much force is quickly applied or a sharp cut exists in the material. Stress tends to accumulate around sharp cuts. Second, the struts must accommodate each other in the close quarters around a geodesic vertex. If the curve wasn't relatively long and smooth, the struts from below would press up against the struts from above where they came together. Such pressing would locally distort the geodesic near the vertex and increase the stress on the joint. The smooth shape of the curve lets the struts fit nicely against each other and avoids increasing the joint's mechanical stress.
The dome I desired was supposed to be about 14' in diameter - but since the PVC pipe came in 10' lengths, only two struts of a 2-frequency geodesic would be able to be cut per pipe - with a lot of waste. I spent a little time iterating runs of the Java code I wrote to calculate the chord lengths - remembering to add 2" per strut (because of I was giving up 1" on each end) I worked it down to three struts per pipe to get a slightly smaller 11' diameter dome. Smaller but cheaper, and that was better for me on the first try.
Measuring and cutting the struts (adding an inch onto each end) took a little over an hour - I got faster as I went along. Drilling a 1/4" hole 1" from each end took another half hour. The struts were ready.
To fasten the struts together I decided to use a 2" long 1/4" bolt and wingnut at each vertex - with two washers to sandwich the PVC.
I chose wingnuts so I could quickly unjoin and rejoin struts - that way I could make partial geodesic sections and join them together during construction.
The wingnuts also meant I could build the dome quickly by hand and then go tighten things up at the end - very important for the degree of instant gratification I require.
An important thing to realize is that if you're doing a lot of cuts on the bandsaw, and you're basically eyeballing the effort, you don't want to go too shallow. I never cut the end completely flat since that would have given up too much of the pipe's surface. This cut leaves a bit of a tight fit at the bolt, and you're going to wreck your hands trying to tighten the wingnut against the compression resistance of the PVC. What's important is to keep the height of the stacked PVC short enough so you get good purchase with the wingnut. Unless you stack the PVC right, it can be a tight fit.
In the case of the 5-way joints this is still pretty easy; a stack of five struts at a vertex was still manageable on a 2" long 1/4" bolt.
In the case of the 6-way joint, there's trouble unless you realize an underlying geodesic principle: a flattened joint is always rotationally symmetric.
You can get tight joints if you add the struts to the bolt in a particular order.
For a 5-way joint, add the struts in every-other order: if you numbered the struts in clockwise-order, you'd add struts in the sequence 1, 3, 5, 2, and 4.
For a 6-way joint, you make sure opposite pairs of struts are aligned - this way the grooves of the paired pipes are kept together: if you numbered the struts in clockwise-order, you'd add struts in the sequence 1, 4, 2, 5, 3, and 6.
If you don't keep the pairs together, you'll tear your hands to pieces trying to push pipes together while you're pushing a bolt through.
Trust me, this is the voice of sore-handed experience.
Connecting the struts into a geodesic can be accomplished in many ways, but my two suggestions for wingnut-constructed small domes are BTPF (by-the-partial-face) and BLFTTD (by-layers-from-the-top-down.) The BTPF method is done by pre-building the partial geodesic faces with the intention of connecting them together at the construction site. The struts already present in an adjoining face are left out of the one you're building - to connect them you undo the wingnut on the target joint, add the struts, and reconnect the joint with wingnut. If the faces aren't too big, this is very easy. The BLFTTD method is also easy, but is especially nice for not-too-large edged-structure domes. You start with the pentagon at the top of the dome as the first layer. Then you build lattitudes of the successive layers and, one at a time, add them as the layer under the previously completed structure above. If you have a reasonable way to hoist the existing structure up another level each time, everything's simple.
In any case, I was able to put my first dome, 11' in diameter, together myself in about three hours total - less than 5 hours from uncut raw materials to geodesic structure.
Next I've got to build a greenhouse-functional cover for it.