On 2019-03-11 13:56, Allan Wechsler wrote:
The missing fact, I think, is that if you augment two adjacent pentagons of a dodecahedron, the edge between the two pyramids will be a valley and not a ridge, so the resulting polyhedron won't be convex.
Alan, if you would just visit https://www.wolframcloud.com/objects/810fbe92-6ab3-480b-9f86-e4869990cf46 (and maybe refresh a couple of times) you can tumble the graphic and see that it is manifestly convex. And, were it not, just augmenting one face of a dodecahedron could serve as J93. And if you don't believe your eyes, why not believe In[29]:= PolyhedronData@60 Out[29]= {"CubeTenCompound", "CumulatedDodecahedron", \ "DeltoidalHexecontahedron", "DodecahedronFiveCompound", \ "MathematicaPolyhedron", "PentagonalHexecontahedron", \ "PentakisDodecahedron", "RhombicHexecontahedron", \ "SmallTriambicIcosahedron", "TriakisIcosahedron"} In[30]:= Intersection[%, PolyhedronData@"Johnson"] Out[30]= {} In[2]:= Max@PolyhedronData["PentakisDodecahedron", "DihedralAngles"] Out[2]= \[Pi] - ArcCos[1/109 (80 + 9 Sqrt[5])] In[3]:= %/Degree // N Out[3]= 156.718553726459 This is getting unreal. —rwg