I got a challenge from two children (3rd and 4th graders). They made two congruent solids using some snap triangles (polydrons, see figures). They thought it should be half a regular icosahedron but could not put them together to make one. Is it really half a regular icosahedron?
No. They are close but not half an icosahedron in terms of the 3D solid. But the surface is indeed true. The dihedral angle of a regular tetrahedron is about 70.53 degrees; that of a regular icosahedron is 138.19 degrees. The former is close to 1/2 of the latter but is not exactly one half. That is why the children's models are so close.
How could I explain it to them? I made a real half icosahedron. It is easy to show where they are correct and where they are wrong! It turns out to be a nice problem to me, too.
Sizes: Edge Lengths: 20mm, 25mm, 30mm, 40mm, 50mm.
It takes two copies to make a whole regular icosahedron, of course.
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