It’s possible that you can cut the shape out of an appropriate flexible and stretchable material, which will “bulge out” appropriately when stressed by being fit together with the other 59 pieces of the sphere. I think one would need to experiment with various materials and thicknesses to see if there is one that works well.

]]>Actually a criterion that is closer to the original intent of my post (note the word “portable” in the post’s title) is “a collection of identical pieces covering the sphere such that each piece has the smallest possible linear extent.” In other words, the set of pieces that is the most portable if you disassemble the sphere and carry around the pieces. Note that this immediately disqualifies the degenerate case.

]]>triangles: take spherical triangle with one vertex at the pole and two

other vertices at the equator. It is probably not what you are looking for.

The largest number of non-pathological identical pieces are 60 pieces,

you describe.

The boundaries of the pieces can be arbitrary curve starting and

ending at the symmetry points of icosahedron.

There are also 24 and 12 pieces configurations having rotational

symmetry of cube and tetrahedron. ]]>

I can build a sphere out of squares, if the squares are small enough, and out of two hemispheres at the other extreme. What is the requirement? Fewer pieces, smaller pieces, or (I suspect) both?

Given the nature of the beast, I’m minded to suggest you apply a quality formula like q(n,s) = ns, where n is the number of pieces and s is their size.

Or perhaps the quest is to minimise the size of some bounding structure (a cuboid?) by volume?

Intrigued as usual.

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