Somehow we have managed a heck of a miscommunication. Augmenting one face of a dodecahedron IS a Johnson solid. The only reason it's not J93 is that it's already in the catalog as J58. J58 through J61 are the four permissible ways to stick equilateral pyramids on the faces of a dodecahedron. On Mon, Mar 11, 2019, 10:09 PM Bill Gosper <billgosper@gmail.com> wrote:
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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun