On 2013-09-02 15:15, Adam P. Goucher wrote:
The legendary Bill Gosper wrote: I'd rather be young.
is made by replacing ten of the regular pentagons of a dodecahedron with concave equilateral pentagons, preserving the topology: http://gosper.org/dodohedron.png ( !#%&*@! was deleted again. Apologies to any of you who looked for it.)
http://en.wikipedia.org/wiki/Dodecahedron glaringly omits it:
Even worse, the Wikipedia article omits the much more well-known _endo-dodecahedron_ (the Symmetries of Things, page 328), which has concave regular pentagonal faces and is *face-transitive*.
And it's equilateral!
Its symmetry group is pyritohedral (3*2), and it is defined as:
"The spaces left over by positioning dodecahedra of maximum size in the natural orientation, centred on the points of the FCC lattice"
The faces are different from the concave pentagons of the tympanohedron, having internal angles of {7pi/5, pi/5, 3pi/5, 3pi/5, pi/5} in cyclic order.
Sincerely,
Adam P. Goucher
http://cp4space.wordpress.com Actually the concave figure in http://en.wikipedia.org/wiki/Pyritohedron is probably your endododecahedron, but they don't say. It would appear in their stupid throbbing animation, if the author wasn't so hung up on convexiity. Here's my stupid throbbing animation: ListAnimate[ Table[Graphics3D[ Polygon[Join[#, -#] &@ Join[#, Map[RotateLeft, #, {2}], Map[RotateRight, #, {2}]] &@{{{-(1/2) + 2 z^2, -(1/2) + z, 0}, {-(1/2), -(1/2), -(1/2)}, {0, -(1/2) + 2 z^2, -(1/2) + z}, {1/2, -(1/2), -(1/2)}, {1/2 - 2 z^2, -(1/2) + z, 0}}, {{1/2 - 2 z^2, -(1/2) + z, 0}, {1/2, -(1/2), 1/ 2}, {0, -(1/2) + 2 z^2, 1/2 - z}, {-(1/2), -(1/2), 1/ 2}, {-(1/2) + 2 z^2, -(1/2) + z, 0}}}]] /. z -> (5*Sin[t] + Sin[5*t])/12, {t, -\[Pi]/2, \[Pi]/2, \[Pi]/30}], AnimationRunning -> True, AnimationDirection -> ForwardBackward]
The mgif: gosper.org/dodex.gif --rwg BTW, yesterday Neil predictively sketched the key frames of this on a small piece of paper while riding in a car in the Santa Cruz Mountains.