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