Date: Feb 27, 2013 7:27 PM
Author: David Hartley
Subject: Re: Deformable platonic "solids"
In message <Pfbr9KkEloLRFwGE@212648.invalid>, David Hartley

<me9@privacy.net> writes

>onsider 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.

Oops! Serious visualisation failure there. The two halves can't both be

squashed that way as they join to different points on the equator. And I

used the wrong value for the angle interior angle of a pentagon.

But the basic idea still works. Squash the top and bottom towards each

other until the equator is planer, and then twist them until all is

flat.

--

David Hartley