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Which archimedean solids can't be generated with triangle tilings?

Polyhedra are generated by reflecting a triangle across each of its edges, over and over until a closed polyhedron is made. Since all of the angles of the generating triangle are nice (are 180/n degrees for some whole number n), the number of times the triangle is repeated around each vertex is always an even (360/[180/n] = 2n times).


That means that each edge of the generating triangle is a part of a line that goes around the entire generated polyhedron, dividing it into two symmetrical parts. The plane dividing the two halves is called the plane of symmetry.

Since the snub cube and the snub dodecahedron have no planes of symmetry, they cannot be generated this way.



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