Hexapentakis truncated icosahedron constructed as an augmented truncated icosahedron, adding pyramids to the 20 hexagonal faces, creating 120 new triangular faces. It is visually similar to the chiral snub dodecahedron which has 80 triangles and 12 pentagons. The dual polyhedron can be seen as a "hexatruncated pentakis dodecahedron", a dodecahedron with its faced augmented by pentagonal pyramids (a pentakis dodecahedron), and then its 6-valance vertices truncated. It has similar groups of irregular pentagons as the pentagonal hexecontahedron. Hexapentakis truncated icosahedron The hexapentakis truncated icosahedron is a convex polyhedron constructed as an augmented truncated icosahedron. It is geodesic polyhedron {3,5+}, with pentavalent vertices separated
Regular dodecahedron such a regular dodecahedron is 1 : , or : 1, or (5 + ) : 4. For example, an embedded cube with a volume of 64 (and edge length of 4), will nest within a regular dodecahedron of volume 64 + 32"ϕ" (and edge length of 4"ϕ" − 4). Thus, the difference in volume between the encompassing regular dodecahedron and the enclosed cube is always one half the volume of the cube times "ϕ". From these ratios are derived simple formulas for the volume of a regular dodecahedron with edge length "a" in terms of the golden mean: Golden