The roundest polyhedron in PolyhedronData: In[86]:= MaximalBy[# | (# | N@# &@Min@PolyhedronData[#, "DihedralAngles"]) & /@ PolyhedronData[], #[[2, 2]] &] // tim During evaluation of In[86]:= 0.537811 (* seconds *),1 (* winner *) Out[86]= {"DisdyakisTriacontahedron" | (ArcCos[1/241 (-179 - 24 Sqrt[5])] | 2.87783661046122)} {. . . | Sharpest dihedral} In[87]:= Labeled[PolyhedronData@%[[1, 1]], %[[1, 1]]] Out[87]=DisdyakisTriacontahedron <http://gosper.org/D120.png> In[91]:= PolyhedronData@120 Out[91]= {"DisdyakisTriacontahedron", "IcosahedronSixCompound", {"IcosahedronStellation", 3}, {"IcosidodecahedronStellation", 1}} In[92]:= Tally[PolyhedronData[%[[1]], "DihedralAngles"]] Out[92]= {{ArcCos[1/241 (-179 - 24 Sqrt[5])], 180}} claims that all 180 edges (60 short, 60 medium, 60 long) have this same angle. Is this obvious? In[95]:= PolyhedronData[%91[[1]], "AlternateNames"] Out[95]= {"28\[Hyphen]uniform dual polyhedron", "hexakis icosahedron"} —rwg
I think there's at least one degree of freedom there: You can raise (or lower) any class of vertex to change the dihedral angles. But for the "roundest", perhaps they need to be equal? (How is "roundest" defined? Minimum surface area for a fixed volume? Minimum ratio between min and max radius?) For what it's worth, I too chose equal dihedral angles for my polyhedron database: https://www.karzes.com/polyhedra/polyhedron.html?ph=V4.6.10 (If you have a mouse, you can click-and-drag to manually rotate.) Tom Bill Gosper writes:
The roundest polyhedron in PolyhedronData: In[86]:= MaximalBy[# | (# | N@# &@Min@PolyhedronData[#, "DihedralAngles"]) & /@ PolyhedronData[], #[[2, 2]] &] // tim
During evaluation of In[86]:= 0.537811 (* seconds *),1 (* winner *)
Out[86]= {"DisdyakisTriacontahedron" | (ArcCos[1/241 (-179 - 24 Sqrt[5])] | 2.87783661046122)}
{. . . | Sharpest dihedral}
In[87]:= Labeled[PolyhedronData@%[[1, 1]], %[[1, 1]]] Out[87]=DisdyakisTriacontahedron <http://gosper.org/D120.png> In[91]:= PolyhedronData@120
Out[91]= {"DisdyakisTriacontahedron", "IcosahedronSixCompound", {"IcosahedronStellation", 3}, {"IcosidodecahedronStellation", 1}}
In[92]:= Tally[PolyhedronData[%[[1]], "DihedralAngles"]]
Out[92]= {{ArcCos[1/241 (-179 - 24 Sqrt[5])], 180}}
claims that all 180 edges (60 short, 60 medium, 60 long) have this same angle. Is this obvious?
In[95]:= PolyhedronData[%91[[1]], "AlternateNames"]
Out[95]= {"28\[Hyphen]uniform dual polyhedron", "hexakis icosahedron"} —rwg
I think there are two degrees of freedom, but there is a fairly clear "canonical" exemplar, where you start with an equal-edged truncated icosidodecahedron, and "dualize" it by replacing each vertex with a face oriented normal to the line connecting the original vertex to the center. Since all the original vertices are on the same sphere, and all the original edges are the same length, the dihedrals of the result are equal by construction. On Thu, Aug 27, 2020 at 11:48 PM Tom Karzes <karzes@sonic.net> wrote:
I think there's at least one degree of freedom there: You can raise (or lower) any class of vertex to change the dihedral angles. But for the "roundest", perhaps they need to be equal? (How is "roundest" defined? Minimum surface area for a fixed volume? Minimum ratio between min and max radius?)
For what it's worth, I too chose equal dihedral angles for my polyhedron database:
https://www.karzes.com/polyhedra/polyhedron.html?ph=V4.6.10
(If you have a mouse, you can click-and-drag to manually rotate.)
Tom
Bill Gosper writes:
The roundest polyhedron in PolyhedronData: In[86]:= MaximalBy[# | (# | N@# &@Min@PolyhedronData[#, "DihedralAngles"]) & /@ PolyhedronData[], #[[2, 2]] &] // tim
During evaluation of In[86]:= 0.537811 (* seconds *),1 (* winner *)
Out[86]= {"DisdyakisTriacontahedron" | (ArcCos[1/241 (-179 - 24 Sqrt[5])] | 2.87783661046122)}
{. . . | Sharpest dihedral}
In[87]:= Labeled[PolyhedronData@%[[1, 1]], %[[1, 1]]] Out[87]=DisdyakisTriacontahedron <http://gosper.org/D120.png> In[91]:= PolyhedronData@120
Out[91]= {"DisdyakisTriacontahedron", "IcosahedronSixCompound", {"IcosahedronStellation", 3}, {"IcosidodecahedronStellation", 1}}
In[92]:= Tally[PolyhedronData[%[[1]], "DihedralAngles"]]
Out[92]= {{ArcCos[1/241 (-179 - 24 Sqrt[5])], 180}}
claims that all 180 edges (60 short, 60 medium, 60 long) have this same angle. Is this obvious?
In[95]:= PolyhedronData[%91[[1]], "AlternateNames"]
Out[95]= {"28\[Hyphen]uniform dual polyhedron", "hexakis icosahedron"} —rwg
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I guess that argument applies to all catalan solids (which are all duals of archimedean solids). I checked my database, and each of my catalan solids has a single dihedral angle. I suspect that's true for the Wolfram database as well. Tom Allan Wechsler writes:
I think there are two degrees of freedom, but there is a fairly clear "canonical" exemplar, where you start with an equal-edged truncated icosidodecahedron, and "dualize" it by replacing each vertex with a face oriented normal to the line connecting the original vertex to the center. Since all the original vertices are on the same sphere, and all the original edges are the same length, the dihedrals of the result are equal by construction.
On Thu, Aug 27, 2020 at 11:48 PM Tom Karzes <karzes@sonic.net> wrote:
I think there's at least one degree of freedom there: You can raise (or lower) any class of vertex to change the dihedral angles. But for the "roundest", perhaps they need to be equal? (How is "roundest" defined? Minimum surface area for a fixed volume? Minimum ratio between min and max radius?)
For what it's worth, I too chose equal dihedral angles for my polyhedron database:
https://www.karzes.com/polyhedra/polyhedron.html?ph=V4.6.10
(If you have a mouse, you can click-and-drag to manually rotate.)
Tom
Bill Gosper writes:
The roundest polyhedron in PolyhedronData: In[86]:= MaximalBy[# | (# | N@# &@Min@PolyhedronData[#, "DihedralAngles"]) & /@ PolyhedronData[], #[[2, 2]] &] // tim
During evaluation of In[86]:= 0.537811 (* seconds *),1 (* winner *)
Out[86]= {"DisdyakisTriacontahedron" | (ArcCos[1/241 (-179 - 24 Sqrt[5])] | 2.87783661046122)}
{. . . | Sharpest dihedral}
In[87]:= Labeled[PolyhedronData@%[[1, 1]], %[[1, 1]]] Out[87]=DisdyakisTriacontahedron <http://gosper.org/D120.png> In[91]:= PolyhedronData@120
Out[91]= {"DisdyakisTriacontahedron", "IcosahedronSixCompound", {"IcosahedronStellation", 3}, {"IcosidodecahedronStellation", 1}}
In[92]:= Tally[PolyhedronData[%[[1]], "DihedralAngles"]]
Out[92]= {{ArcCos[1/241 (-179 - 24 Sqrt[5])], 180}}
claims that all 180 edges (60 short, 60 medium, 60 long) have this same angle. Is this obvious?
In[95]:= PolyhedronData[%91[[1]], "AlternateNames"]
Out[95]= {"28\[Hyphen]uniform dual polyhedron", "hexakis icosahedron"} —rwg
@Tom Karzes, nice polyhedral database! (spin a little fast for me). Is it HTML5? On Thu, Aug 27, 2020 at 10:48 PM Tom Karzes <karzes@sonic.net> wrote:
I think there's at least one degree of freedom there: You can raise (or lower) any class of vertex to change the dihedral angles. But for the "roundest", perhaps they need to be equal? (How is "roundest" defined? Minimum surface area for a fixed volume? Minimum ratio between min and max radius?)
For what it's worth, I too chose equal dihedral angles for my polyhedron database:
https://www.karzes.com/polyhedra/polyhedron.html?ph=V4.6.10
(If you have a mouse, you can click-and-drag to manually rotate.)
Tom
Bill Gosper writes:
The roundest polyhedron in PolyhedronData: In[86]:= MaximalBy[# | (# | N@# &@Min@PolyhedronData[#, "DihedralAngles"]) & /@ PolyhedronData[], #[[2, 2]] &] // tim
During evaluation of In[86]:= 0.537811 (* seconds *),1 (* winner *)
Out[86]= {"DisdyakisTriacontahedron" | (ArcCos[1/241 (-179 - 24 Sqrt[5])] | 2.87783661046122)}
{. . . | Sharpest dihedral}
In[87]:= Labeled[PolyhedronData@%[[1, 1]], %[[1, 1]]] Out[87]=DisdyakisTriacontahedron <http://gosper.org/D120.png> In[91]:= PolyhedronData@120
Out[91]= {"DisdyakisTriacontahedron", "IcosahedronSixCompound", {"IcosahedronStellation", 3}, {"IcosidodecahedronStellation", 1}}
In[92]:= Tally[PolyhedronData[%[[1]], "DihedralAngles"]]
Out[92]= {{ArcCos[1/241 (-179 - 24 Sqrt[5])], 180}}
claims that all 180 edges (60 short, 60 medium, 60 long) have this same angle. Is this obvious?
In[95]:= PolyhedronData[%91[[1]], "AlternateNames"]
Out[95]= {"28\[Hyphen]uniform dual polyhedron", "hexakis icosahedron"} —rwg
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Yes, pure HTML5. Tom James Buddenhagen writes:
@Tom Karzes, nice polyhedral database! (spin a little fast for me). Is it HTML5?
On Thu, Aug 27, 2020 at 10:48 PM Tom Karzes <karzes@sonic.net> wrote:
I think there's at least one degree of freedom there: You can raise (or lower) any class of vertex to change the dihedral angles. But for the "roundest", perhaps they need to be equal? (How is "roundest" defined? Minimum surface area for a fixed volume? Minimum ratio between min and max radius?)
For what it's worth, I too chose equal dihedral angles for my polyhedron database:
https://www.karzes.com/polyhedra/polyhedron.html?ph=V4.6.10
(If you have a mouse, you can click-and-drag to manually rotate.)
Tom
participants (4)
-
Allan Wechsler -
Bill Gosper -
James Buddenhagen -
Tom Karzes