Elongated dodecahedron
| Elongated dodecahedron | |
|---|---|
 | |
| Type | Parallelohedron |
| Faces | 8 rhombi 4 hexagons |
| Edges | 28 |
| Vertices | 18 |
| Properties | Convex |
| Net | |
 | |
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
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.
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
- 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
- ^ Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). Dover Publications. p. 257. ISBN 0-486-61480-8.
- ^ a b 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.
- ^ Ammari, Habib M. (3 October 2022). "A Polyhedral Space Filler Tessellation-Based Approach for Connected -Coverage". Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform. Springer. pp. 309–352. ISBN 978-3-031-07823-1.
- ^ a b Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 169. ISBN 0-486-23729-X.
- ^ a b Dienst, Thilo. "Fedorov's five parallelohedra in ". University of Dortmund. Archived from the original on 2016-03-04.
- ^ Hargittai, Istvan (November 1998). "Symmetry in crystallography" (PDF). Acta Crystallographica, Section A. 54 (6): 697–706. Bibcode:1998AcCrA..54..697H. doi:10.1107/s0108767398006709.
- ^ 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 VRML Model
