On Mon, Sep 30, 2013 at 6:42 PM, Bill Gosper <billgosper@gmail.com> wrote:
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
NeilB, again while riding in his Mom's car, today noticed an error on that p 328, which gives an improbable dodecahedron FCC packing density of .9405, making its volume 15.8 times the endododecahedron. Instead Neil gets .904508 = (5 + √5)/8, giving a volume ratio of 9.47 = 5 + 2 √5, which is still pretty surprising. --rwg This argues for giving known constants exactly, or at least to high precision.
Actually the concave figure in http://en.wikipedia.org/wiki/Pyritohedronis 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: [shortened to]
ListAnimate[
Table[Graphics3D[ Polygon[Join[#, -#] &@ Join[#, Map[RotateLeft, #, {2}], Map[RotateRight, #, {2}]] &@{#, {-1, 1, -1}*# & /@ #} &@{{-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, -π/2, π/2, π/30}], AnimationRunning -> True, AnimationDirection -> ForwardBackward] The mgif: [gosper.org/pyrominia.gif <http://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.