Until Wednesday I will only post soothing articles not related to the election. And right now I could use some help. If you cut out the green star here, you can fold it so all its tips meet at one point, and make a convex polyhedron. Here's my question: are all the necessary fold lines along the gray edges - the edges of the small equilateral triangles?
I would print it out and fold it up myself, but my printer is broken!
[Edit: my question above is now answered in the thread below, thanks to Gerard Westendorp.]
In fact I have a more general question. Suppose you draw any 11-sided polygon whose corners lie on an equilateral triangular grid. Draw yellow equilateral triangles pointing in from the edges as shown here. If those yellow triangles crash into each other or share an edge, start over! Otherwise, if you cut them out you'll get an 11-pointed star. You can fold it up so its points touch, and you'll get a convex polyhedron tiled by the small equilateral triangles coming from the grid. But here's my question: do those equilateral triangles ever need to get *bent*?
That is, does every such triangle lie wholly on one face of your polyhedron, or does the folding process ever make it lie on two or more faces?
I don't know how to resolve this question mathematically, so I want to make a bunch of examples, print them out, fold them up, and see what happens. What a bad time for my printer to stop working!
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