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.

Set of n-gonal truncated trapezohedra
Example: pentagonal truncated trapezohedron (regular dodecahedron)
Faces	2 n-sided polygons, 2n pentagons
Edges	6n
Vertices	4n
Conway notation	t4dA4 t5dA5 t6dA6
Symmetry group	Dnd, [2+,2n], (2*n), order 4n
Rotation group	Dn, [2,n]+, (22n), order 2n
Dual polyhedron	gyroelongated bipyramids
Properties	convex

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

See also

• Diminished trapezohedron

External links

• Conway Notation for Polyhedra Try: "tndAn", where n=4,5,6... example "t5dA5" is a dodecahedron.