On Feb 27, 9:21 pm, Frederick Williams <freddywilli...@btinternet.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? > -- > When a true genius appears in the world, you may know him by > this sign, that the dunces are all in confederacy against him. > Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
The question can be formalized in the following manner : Define a 'semi-Platonic' solid as a solid with equal number of edges per face , same number of faces as a regular counterpart , all edges of the same length . It's the same as a Platonic solid , just drop the condition of equal angles per face , and the 'planar' nature of faces .
The question is : Is a 'semi-Platonic' solid necessarily platonic? That means , for a Platonic solid , does there exist a solid with the same properties except different angles? As pointed out , solids with triangular faces are not deformable . (being an equilateral triangle uniquely determines angles , as opposed to having a higher number of sides)
As calculated , all Platonic solids with non-triangular faces are deformable .