Elongated dodecahedron

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Elongated dodecahedron
Rhombo-hexagonal dodecahedron.png
Type Plesiohedron
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 D4h, [4,2], (*422), order 16
Rotation group D4, [4,2]+, (422), order 8
Dual polyhedron -
Properties convex, parallelohedron
Net
Elongated dodecahedron net.png

In geometry, the elongated dodecahedron,[1] extended rhombic dodecahedron, rhombo-hexagonal dodecahedron[2] or hexarhombic dodecahedron[3] is a convex dodecahedron with 8 rhombic and 4 hexagonal faces. The hexagons can be made equilateral, or regular depending on the shape of the rhombi. It can be seen as constructed from a rhombic dodecahedron elongated by a square prism. Along with the rhombic dodecahedron, it is a space-filling polyhedron.

Tessellation[edit]

Rhombo-hexagonal dodecahedron tessellation.png

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[edit]

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.

Elongated dodecahedron flat.png
Coplanar polyhedron
Elongated dodecahedron flat net.png
Net
Elongated dodecahedron flat honeycomb.png
Honeycomb
Elongated dodecahedron concave.png
Coplanar polyhedron
Elongated dodecahedron concave net.png
Net
Elongated dodecahedron concave honeycomb.png
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.png
Contracted truncated octahedron
Contracted truncated octahedron net.png
Net
Contracted truncated octahedron honeycomb.png
Honeycomb

See also[edit]

References[edit]

External links[edit]