Conversation

If you cut out the yellow shape here, you can fold it up along the red lines, and all 11 sharp tips will meet at one point! You'll get a polyhedron with 12 corners, tiled by equilateral triangles. 5 triangles meet at each corner. There are also places where 6 triangles meet: these places are flat. Some triangles get folded along the red lines.

This may seem like a curiosity. But it shows up in a paper by the great mathematician William Thurston, which I'm writing an article about, and there's a lot more to it.

Notice how he got this. He drew a lattice of equilateral triangles on the plane. Then he drew an 11-sided polygon whose corners are lattice points. Along each edge of this polygon, he drew a green equilateral triangle pointing inward. If you cut out these green triangles, you're left with the yellow shape, which you can fold up.

There are lots of ways you can draw such an 11-sided polygon; the picture shows just one. Each way gives a way of triangulating the sphere where the number of triangles meeting at a vertex is always either 5 or 6. In fact - and this is not obvious - you can get *all* such triangulations using this trick.

Is anyone out there willing to draw such an 11-sided polygon and then draw the polyhedron you get when you cut out the green triangles and fold up the left-over yellow shape? A super-genius could write a program where the user could choose 11 points that define a valid polygon, and then the program would show what happens when you fold up the yellow shape. But I'd be thrilled to see just one example.

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