Elongated dodecahedron

Elongated dodecahedron
Type	Parallelohedron
Faces	8 rhombi 4 hexagons
Edges	28
Vertices	18
Properties	Convex
Net

3D model of an elongated dodecahedron

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.

Parallelohedron

Main article: Parallelohedron

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.

Elongated dodecahedral honeycomb

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, D_2h symmetry, order 8.

Contracted truncated octahedron

Net

Honeycomb

See also

• Trapezo-rhombic dodecahedron, a space-filling polyhedron with six rhombi and six trapezohedral faces
• Elongated gyrobifastigium, a space-filling polyhedron with four quadrilaterals and four pentagonal faces.

References

1. Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). Dover Publications. p. 257. ISBN   0-486-61480-8.
2. Akiyama, Jin; Kobayashi, Midori; Nakagawa, Hiroshi; Nakamura, Gisaku; Sato, Ikuro (2013). Bárány, Imre; Böröczky, Károly J.; Tóth, Gábor Fejes; Pach, János (eds.). Geometry - Intuitive, Discrete, and Convex. Bolyai Society Mathematical Studies. Vol. 24. Springer. pp. 23–43. doi:10.1007/978-3-642-41498-5. ISBN   978-3-642-41497-8.
3. Ammari, Habib M. (3 October 2022). "A Polyhedral Space Filler Tessellation-Based Approach for Connected ${\displaystyle k}$-Coverage". Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform. Springer. pp. 309–352. ISBN   978-3-031-07823-1.
4. Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 169. ISBN   0-486-23729-X.
5. Dienst, Thilo. "Fedorov's five parallelohedra in ${\displaystyle \mathbb {R} ^{3}}$". University of Dortmund. Archived from the original on 2016-03-04.
6. Hargittai, Istvan (November 1998). "Symmetry in crystallography" (PDF). Acta Crystallographica, Section A. 54 (6): 697–706. Bibcode:1998AcCrA..54..697H. doi:10.1107/s0108767398006709.
7. Eppstein, David (1996). "Zonohedra and zonotopes". Mathematica in Education and Research. 5 (4): 15–21.

External links

• Weisstein, Eric W. "Space-filling polyhedron". MathWorld .
• Weisstein, Eric W. "Elongated dodecahedron". MathWorld .
• Uniform space-filling using only rhombo-hexagonal dodecahedra
• Elongated dodecahedron Archived 2017-10-29 at the Wayback Machine VRML Model