Elongated dodecahedron

In geometry, the elongated dodecahedron,^[1] elongated rhombic dodecahedron,^[2] extended rhombic dodecahedron,^[3] rhombo-hexagonal dodecahedron^[4] or hexarhombic dodecahedron^[5] is a convex dodecahedron with eight rhombic and four hexagonal faces.

Elongated dodecahedron
Type	Parallelohedron
Faces	8 rhombi 4 hexagons
Edges	28
Vertices	18
Properties	Convex
Net

3D model of an elongated dodecahedron

Parallelohedron

Main article: Parallelohedron

The elongated dodecahedron can be constructed by elongating a rhombic dodecahedron – i.e., slicing it into two congruent concave polyhedra and covering the bases of a square prism.^[2] As a result, it has eighteen vertices, twenty-eight edges, and twelve faces (which include eight rhombi and four hexagons).^[5]

Both the rhombic dodecahedron and the elongated dodecahedron are two of the five types of parallelohedron identified by Evgraf Fedorov. In other words, it is a space-filling polyhedron, meaning the elongated dodecahedron and its copy can tile space face-to-face by translations periodically.^[6] For the elongated dodecahedron, it has five sets of parallel edges called zones or belts.^[7] This produces an elongated dodecahedral honeycomb.^[4] It is the Wigner–Seitz cell for certain body-centered tetragonal lattices.

Elongated dodecahedral honeycomb

This is related to the rhombic dodecahedral honeycomb with an elongation of zero. Projected normal to the elongation direction, the honeycomb looks like a square tiling with the rhombi projected into squares.

Variations

The expanded dodecahedra can be distorted into cubic volumes, with the honeycomb as a half-offset stacking of cubes. It can also be made concave by adjusting the 8 corners downward by the same amount as the centers are moved up.

Coplanar polyhedron	Net	Honeycomb
Concave	Net	Honeycomb

The elongated dodecahedron can be constructed as a contraction of a uniform truncated octahedron, where square faces are reduced to single edges and regular hexagonal faces are reduced to 60-degree rhombic faces (or pairs of equilateral triangles). This construction alternates square and rhombi on the 4-valence vertices, and has half the symmetry, D_2h symmetry, order 8.

Contracted truncated octahedron

Net

Honeycomb

See also

• Trapezo-rhombic dodecahedron, a space-filling polyhedron with six rhombi and six trapezohedral faces
• Elongated gyrobifastigium, a space-filling polyhedron with four quadrilaterals and four pentagonal faces.