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Re: Deformable platonic "solids"
Posted:
Feb 27, 2013 5:31 PM
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In message
<ken.pledger-1E8759.09331928022013@news.eternal-september.org>, Ken
Pledger <ken.pledger@vuw.ac.nz> writes
>Here's an intuitive line of thought, not a complete proof.
>
> Starting from a face of the cube, the four adjacent edges are
>parallel. That seems to be what permits the deformation. None of the
>other regular polyhedra has parallel edges adjacent to a face, so I
>suspect none can be deformed.
and here's an intuitive line of thought leading to the opposite
conclusion.
Among the polygons, only the triangles cannot be deformed. So any
polyhedron with all triangular sides is likely to also be
non-deformable.
Polyhedrons with no triangular sides are probably deformable.
Consider a dodecahedral frame standing on one face. Push down on the top
face. Each of the five surrounding faces pivots around the edge in
common with the top face, widening the angle between the faces. The far
ends of the edges adjoining the common edge move apart, the angle
between the two further edges increases to allow that. The bottom half
mirrors this. The deformation can continue until the angles between the
further edges becomes 180 degrees. The "equator" of the dodecahedron,
which was a non-planar decahedron has become a planar pentagon.
The two further sides "lock straight" when the other angles in the face
become 60 and 120 degrees (two of each). If the deformation could
continue until squashed flat the larger angles would be 144 degrees, so
that is not possible. However, you could now twist the top and bottom
faces which would lower the overall height further. I think that allows
the whole thing to be squashed flat. It will look like a pentagon with
sides of length 2 containing two concentric pentagons of with sides of
length 1 each rotated wrt to the outer pentagon so that its vertices are
each at length one from a linked vertex of the outer pentagon. Rotating
by 36 degrees appears to put that length at a little over 1, so a
slightly smaller rotation should do it.
--
David Hartley
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