On Wed, 27 Feb 2013 19:21:32 +0000, Frederick Williams <email@example.com> wrote:
>Suppose the platonic solids aren't solid at all but are made of rigid >line segments with completely flexible hinges at the vertices. The cube >can be flattened into a... um... non cube. The tetrahedron, octahedron >and icosahedron cannot be deformed at all. But what about the >dodecahedron, can it be deformed?
Yet another flattening procedure:
Consider a line segment, together with the four line segments sprouting from its end points. On a solid dodecahedron the four free end points are the vertices of square and the remaining 25 line segments can be divided in five more of these groups of five in such a way that the resulting six squares are the faces of a cube.
Flexibility makes it possible to gradually shrink all squares to a single point. This deforms the original groups of five line segment to triangles that share just a single point, so they can be folded into a single triangle.