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Topic: Deformable platonic "solids"
Replies: 23   Last Post: Mar 12, 2013 8:11 PM

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David Hartley

Posts: 451
Registered: 12/13/04
Re: Deformable platonic "solids"
Posted: Feb 27, 2013 7:27 PM
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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



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