Elongated dodecahedron

Elongated dodecahedron
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 square biantiprism
Properties convex, parallelohedron
Net

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

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, D2h symmetry, order 8.


Contracted truncated octahedron
Net
Honeycomb

See also

References

  1. ↑ Coxeter (1973) p.257
  2. ↑ Williamson (1979) p169
  3. ↑ Fedorov's five parallelohedra in R³
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. rhombo-hexagonal dodecahedron, p169
  • H.S.M. Coxeter, Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 p. 257
  • Weisstein, Eric W. "Space-filling polyhedron". MathWorld.
  • Weisstein, Eric W. "Elongated dodecahedron". MathWorld.
  • Uniform space-filling using only rhombo-hexagonal dodecahedra
  • Elongated dodecahedron VRML Model
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