Half icosahedron, dissection, Platonic Solids

thingiverse

Two young students, a third-grader and a fourth-grader, presented me with an intriguing challenge. They had created two identical polyhedral shapes using snap triangles (polydrons) and believed they had successfully constructed half of a regular icosahedron. However, upon closer inspection, I realized that their creation didn't quite live up to its name. These students were close, but not quite there. The surface area of their polyhedra was indeed accurate, but the underlying structure wasn't exactly what it seemed. This led me to investigate further and examine the dihedral angles of various polyhedra. The dihedral angle of a regular tetrahedron is approximately 70.53 degrees, while that of a regular icosahedron measures about 138.19 degrees. Notably, the former is roughly half the value of the latter, which explains why their models were so convincing. However, this similarity in dihedral angles isn't sufficient to make them identical. To clarify this concept for my young friends, I constructed a genuine half-icosahedron using the same snap triangles. By comparing it with their creation, they were able to see where their model was correct and where it deviated from the actual shape. This exercise turned out to be an enjoyable problem for me as well, and I appreciated the opportunity to explore the intricacies of polyhedral geometry with these inquisitive students.