Date: Feb 28, 2013 1:27 AM
Subject: Re: Deformable platonic "solids"
On Feb 27, 9:21 pm, Frederick Williams <freddywilli...@btinternet.com>
> 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