Triaugmented truncated dodecahedron

In geometry, the triaugmented truncated dodecahedron is one of the Johnson solids (J₇₁); of them, it has the greatest volume in proportion to the cube of the side length, as well as the greatest number of edges. As its name suggests, it is created by attaching three pentagonal cupolas (J₅) onto three nonadjacent decagonal faces of a truncated dodecahedron.

Triaugmented truncated dodecahedron
Type	Johnson J70 – J71 – J72
Faces	2+3x3+4x6 triangles 3+2x6 squares 3 pentagons 3x3 decagons
Edges	135
Vertices	75
Vertex configuration	4x3+3x6(3.102) 3+2x6(3.4.5.4) 5x6(3.4.3.10)
Symmetry group	C3v
Dual polyhedron	-
Properties	convex
Net

A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.^[1]