In message <Pfbr9KkEloLRFwGE@212648.invalid>, David Hartley <firstname.lastname@example.org> 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.