Truncated trapezohedron

In geometry, an $n$-gonal truncated trapezohedron is a polyhedron formed by a $n$-gonal trapezohedron with $n$-gonal pyramids truncated from its two polar axis vertices.

The vertices exist as 4 $n$-gons in four parallel planes, with alternating orientation in the middle creating the pentagons.

The regular dodecahedron is the most common polyhedron in this class, being a Platonic solid, with 12 congruent pentagonal faces.

A truncated trapezohedron has all vertices with 3 faces. This means that the dual polyhedra, the set of gyroelongated dipyramids, have all triangular faces. For example, the icosahedron is the dual of the dodecahedron.

Forms

 * Triangular truncated trapezohedron (Dürer's solid) – 6 pentagons, 2 triangles, dual gyroelongated triangular dipyramid
 * Truncated square trapezohedron – 8 pentagons, 2 squares, dual gyroelongated square dipyramid
 * Truncated pentagonal trapezohedron or regular dodecahedron – 12 pentagonal faces, dual icosahedron
 * Truncated hexagonal trapezohedron – 12 pentagons, 2 hexagons, dual gyroelongated hexagonal dipyramid
 * Truncated n-gonal trapezohedron – 2n pentagons, 2 n-gons, dual gyroelongated dipyramids
 * Truncated n-gonal trapezohedron – 2n pentagons, 2 n-gons, dual gyroelongated dipyramids