http://gosper.org/dodohedron.png (equilateral dodecahedron topologically equivalent to the regular one) has two kinds of vertices, ten convex and ten nonconvex. I just computed the solid angles to be ArcCos[27 - 42 √(2/5) - 62/(45 √2 + 29 √5)] ~ .514544425310092 π (slightly > than a cube corner), and 2 π + ArcCos[1/27 (-167 + 118 √2 - 70 √5 + 58 √10)] ~ 2.04206803084996 π (slightly > than a half-space). (Which you used to be able to type by holding down the space bar.) —rwg (Could I have neglected to propose this figure as a G4G12 decoration, or was it deemed too ugly?)