Date: Feb 28, 2013 1:27 AM
Author: dan.ms.chaos@gmail.com
Subject: Re: Deformable platonic "solids"
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 .