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.
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.
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, D2h symmetry, order 8.
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