Re: [math-fun] Julian's dodecahedron
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.
Oops, gosper.org/dodex.gif was already something else. Mgif renamed to http://gosper.org/pyritodex.gif . Also, the animator can be further condensed: 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] --rwg
Running the z parameter past 1/2 (vanishing volume and area) out to 𝟇/2 indicates that the stellated icosahedron is an extreme pyritohedron: http://gosper.org/pyredohedra.gif --rwg On Mon, Sep 30, 2013 at 11:01 PM, Bill Gosper <billgosper@gmail.com> wrote:
Oops, gosper.org/dodex.gif was already something else. Mgif renamed to http://gosper.org/pyritodex.gif . Also, the animator can be further condensed:
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] --rwg
Oops, https://en.wikipedia.org/wiki/Great_stellated_dodecahedron says this *is* one. Too pointy for a stellated icosahedron. --rwg On Thu, Oct 31, 2013 at 6:12 AM, Bill Gosper <billgosper@gmail.com> wrote:
Running the z parameter past 1/2 (vanishing volume and area) out to 𝟇/2 indicates that the stellated icosahedron is an extreme pyritohedron: http://gosper.org/pyredohedra.gif --rwg
On Mon, Sep 30, 2013 at 11:01 PM, Bill Gosper <billgosper@gmail.com>wrote:
Oops, gosper.org/dodex.gif was already something else. Mgif renamed to http://gosper.org/pyritodex.gif . Also, the animator can be further condensed:
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] --rwg
http://gosper.org/pyrominia.gif . Please let me know if you strongly like or dislike the slight rotation. --rwg On Thu, Oct 31, 2013 at 7:07 AM, Bill Gosper <billgosper@gmail.com> wrote:
Oops, https://en.wikipedia.org/wiki/Great_stellated_dodecahedron says this *is* one. Too pointy for a stellated icosahedron. --rwg
On Thu, Oct 31, 2013 at 6:12 AM, Bill Gosper <billgosper@gmail.com> wrote:
Running the z parameter past 1/2 (vanishing volume and area) out to 𝟇/2 indicates that the stellated icosahedron is an extreme pyritohedron: http://gosper.org/pyredohedra.gif --rwg
On Mon, Sep 30, 2013 at 11:01 PM, Bill Gosper <billgosper@gmail.com>wrote:
Oops, gosper.org/dodex.gif was already something else. Mgif renamed to http://gosper.org/pyritodex.gif . Also, the animator can be further condensed:
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] --rwg
* Bill Gosper <billgosper@gmail.com> [Nov 01. 2013 08:56]:
http://gosper.org/pyrominia.gif . Please let me know if you strongly like or dislike the slight rotation. --rwg
[...]
Like it (shtronkly, jawoll!).
Much improved! WFL On 11/1/13, Joerg Arndt <arndt@jjj.de> wrote:
* Bill Gosper <billgosper@gmail.com> [Nov 01. 2013 08:56]:
http://gosper.org/pyrominia.gif . Please let me know if you strongly like or dislike the slight rotation. --rwg
[...]
Like it (shtronkly, jawoll!).
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RWG, it's a lovely and informative animation! Just one thing: I wish it were a bit slower, and that the words remained visible a bit longer. (I never took an Evelyn Wood course.) --Dan On 2013-10-31, at 10:57 PM, Bill Gosper wrote:
http://gosper.org/pyrominia.gif . Please let me know if you strongly like or dislike the slight rotation. --rwg
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.
participants (4)
-
Bill Gosper -
Dan Asimov -
Fred Lunnon -
Joerg Arndt