Elongated dodecahedron

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.

Elongated dodecahedron
TypeParallelohedron
Faces8 rhombi
4 hexagons
Edges28
Vertices18
Vertex configuration(8) 4.6.6
(8) 4.4.6
(2) 4.4.4.4
Symmetry groupD4h, [4,2], (*422), order 16
Rotation groupD4, [4,2]+, (422), order 8
Net

Along with the rhombic dodecahedron, it is a space-filling polyhedron, one of the five types of parallelohedron identified by Evgraf Fedorov that tile space face-to-face by translations.

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
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.