Credits out of the way, geodesic domes are beautiful structures. Spheres give the greatest interior volume with the minimum of material. Their basic geometry can be broken up into hexagons and pentagons, or further broken up into triangles. See the rotating geogedisic spheres on the Buckminster Fuller page.
This model is made up of triangle units, which I then combined into hexagons and pentagons. Those units were then combined to make the sphere. I used bamboo shish-kabob skewers from the supermarket (called pincho sticks where I live), and hot melt glue.
Step 1: GEOMETRY
In a flat pentagon (5 sides) the two sides of the triangles that radiate from the center will be shorter than the third side. To more closely approximate a sphere, you have to lengthen them also. As with the hexagon, the amount that those sides are lengthened will determine the height of the peak in the center.
I don't know the math for making triangles that result in a near-perfect sphere, but I find the idea of lumpy domes and peaky geodesic spheres more interesting anyway. Just playing it by feel, this is the design I came up with.
I used the same triangles for making the hexagons and the pentagons. The dimensions I used for the triangles are: 2 inches for the short side, 2 1/4 inches for the long sides. Whatever variations you do with sides that radiate from the center, the outer sides have to be the same so that the hexagons and pentagons will join correctly.
(I have plans for making a next-generation model with the hexagon peaks going inward and the pentagon peaks going outward, to make a pollen grain-like structure.)
Step 2: TOOLS AND MATERIALS
Hot melt glue gun
Materials: Flat working surface
Non-stick work surface cover (Teflon cloth)
Vinyl (or cardboard) to make a holding jig for the sticks while gluing them
Bamboo shis-kebab ("pincho") sticks from the supermarket.
Hot melt glue
Step 3: CUT STICKS TO LENGTH
Unless you are really enamored by the way mine turned out using 2 inch and 2 1/4 inch pieces, have fun and create your own variations.
When I decided on the dimensions I was going to use, I cut a sample stick of the two sizes and used those sticks as guides for cutting all the other sticks of the same length. That way, all I had to do was match up stick ends, butt the cutters up against the other end of the guide stick and snip.
If you use different lengths for the radiating triangle sides of the hexagons and pentagons, you will have to cut three different lengths, not two.
Step 4: MAKE a HOLDING JIG
Over the Teflon cloth, I stapled down some vinyl material as guides to hold the bamboo stick sections. You could probably use cardboard, or maybe ice cream sticks. Leave the corner areas open, since that is where you will be using the glue and you don't want to gum up your holding jig.
Once I got the sticks positioned in the jig, I set a weight on them to hold them in place. (I used a big round washer with some magnets stuck to it as the weight.) The sticks are so light that any pressure from the glue gun can dislodge them if they are not held in place.
Step 5: ASSEMBLE THE HEXAGON AND PENTAGON UNITS
Putting in the last triangle of each unit is the trickiest part, since you have to fold the assembled polygons some as you raise the center peak. Fortunately, the hot melt glue is flexible, so joints hinge nicely. The glue also cools and solidifies quickly, so you can hold it by hand until the last glue job hardens up. As you hold it, keep the base of the hexagon or pentagon unit pressed against the flat base of the work surface so that the base of the hexagon or pentagon will be in a flat plane.
Step 6: ASSEMBLING THE SPHERE
As the sphere grows, it passes through stages that can be used as architectural models for dome structures.