Elongated dodecahedron | |
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Type | Parallelohedron |
Faces | 8 rhombi 4 hexagons |
Edges | 28 |
Vertices | 18 |
Vertex configuration | (8) 4.6.6 (8) 4.4.6 (2) 4.4.4.4 |
Symmetry group | Dihedral (D4h), [4,2], (*422), order 16 |
Rotation group | D4, [4,2]+, (422), order 8 |
Properties | Convex |
Net | |
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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.
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.
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.
![]() Contracted truncated octahedron | ![]() Net | ![]() Honeycomb |