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But not flexible in 3 dimensions...

(Ie. You can curve it around a cylinder, but not around a sphere)

If you want to dive deeper into the issues look into Developable Surfaces https://en.wikipedia.org/wiki/Developable_surface

Wouldn't different thicknesses allow this?

Take a sheet of paper and wrap it around a ball. Notice how it folds?

If you could take a flat surface and have it represent the surface of a sphere without warping (or vice-versa), we wouldn't have constant debates about what kind of map projections are best.

Yes, I agree. What I mean is that by varying the thickness, you can get it very close to or exactly a sphere.

Maybe you could make it like an orange peel and bend it mostly sideways? Depends on how rigid it is. Or cut ridges like a fresnel lens? Or like lasik? All depends on the material properties.

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