Playing with an icosahedron, and thinking of it as a kind of sphere broken down into simple pieces, got me wondering what would be the smallest piece you could break a sphere down into, so that all the pieces were exactly the same size and shape. That way, if you wanted to assemble a big dome or sphere, you could just carry portable little pieces around with you.
By having all the pieces be identical, you wouldn’t need to worry about numbering or ordering them in any way. Of course you could make a sphere or dome by inflating something, which would also be cool, but I’m thinking of how to make a rigid sphere.
The best solution I’ve come up with is 60 little identical pieces. If you inflate an icosahedron out to a sphere, then you can divide each of its twenty now-curved triangles into three equal pieces, like so:
As you can see from the figure above, we can put little matching protrusions and notches into the edges of each piece, so that they will snap firmly together. Now we have a very compact way to “carry around” a rigid sphere as sixty identical little almost-flat pieces, and then assemble it together as needed. Of course if you only need a hemispherical dome (eg: as a portable planetarium) then you’d only need thirty pieces.
Unless somebody can think of a better solution.