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The following monstrosity of a shape:

huge icosahedron-wrapping shape

... can be wrapped onto the surface of an icosahedron in a way that completely covers the entire icosahedron with no gaps and no overlaps.

How can it be done?

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Here's a picture of the solution.

enter image description here
Once it was clear that the long hooks had to fold in on themselves rather than enclose something else, the rest followed.

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