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Here's an article I wrote about the math of this business. The great geometer Bill Thurston proved a bunch of amazing theorems about it, which I explain here. The way I'm describing it here it sounds like a fun little game, but he builds it up into something quite profound.

But he doesn't seem to answer my question here: whether the small equilateral triangles ever need to be bent in the process of folding up the star!

My article starts with a different example picture. @GerardWestendorp tells me that in *this* example, when you fold up the star, the triangles do not get bent.... and you get a very nice polyhedron, called the hexagonal antiprism. I'll show you his picture of it in my next post.

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https://johncarlosbaez.wordpress.com/2024/10/30/triangulations-of-the-sphere-part-2/