Trapezo-rhombic dodecahedron

Trapezo-rhombic dodecahedron
Trapezo-rhombic dodecahedron
Type	Plesiohedron; Johnson solid dual
Faces	6 rhombi; 6 trapezoids
Edges	24
Vertices	14
Vertex configuration	(2) 4.4.4; (6) 4.4.4.4; (6) 4.4.4
Symmetry group	D3h, [3,2], (*322), order 12
Rotation group	D3, [3,2]+, (322), order 6
Dual polyhedron	Triangular orthobicupola
Properties	convex
Net

Concave configuration

In geometry, the trapezo-rhombic dodecahedron or rhombo-trapezoidal dodecahedron is a convex dodecahedron with 6 rhombic and 6 trapezoidal faces. It has $D 3h$ symmetry. A concave form can be constructed with an identical net, seen as excavating trigonal trapezohedra from the top and bottom. It is also called the trapezoidal dodecahedron.^[1]

^ Lagarias, Jeffrey C. (2011). "The Kepler conjecture and its proof". The Kepler Conjecture: The Hales–Ferguson proof. Springer, New York. pp. 3–26. doi:10.1007/978-1-4614-1129-1_1. MR 3050907.; see especially p. 11

[1] Lagarias, Jeffrey C. (2011). "The Kepler conjecture and its proof". The Kepler Conjecture: The Hales–Ferguson proof. Springer, New York. pp. 3–26. doi:10.1007/978-1-4614-1129-1_1. MR 3050907.; see especially p. 11

[1]

Trapezo-rhombic dodecahedron

Type	Plesiohedron Johnson solid dual
Faces	6 rhombi 6 trapezoids
Edges	24
Vertices	14
Vertex configuration	$(2) 4.4.4 (6) 4.4.4.4 (6) 4.4.4$
Symmetry group	$D 3h, [3,2], (*322),$ order 12
Rotation group	$D 3, [3,2] +, (322),$ order 6
Dual polyhedron	Triangular orthobicupola
Properties	convex
Net