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{{Short description|Polyhedron with 8 rhombic and 4 hexagonal faces}}
{{Infobox Polyhedron |
{{Infobox polyhedron
Image_File=Rhombo-hexagonal dodecahedron.png|
|image=Rhombo-hexagonal dodecahedron.png
Polyhedron_Type=[[Dodecahedron#Other_dodecahedra|Dodecahedron]]|
|type=[[Parallelohedron]]
Face_List=8 [[rhombus|rhombi]]<br>4 [[hexagon]]s|
|faces=8 [[rhombus|rhombi]]<br>4 [[hexagon]]s
Edge_Count=28|
|edges=28
Vertex_Count=18|
|vertices=18
Vertex_List=(8) 4.6.6<BR>(8) 4.4.6<BR>(2) 4.4.4.4|
|vertex_config={{math|(8) 4.6.6<br>(8) 4.4.6<br>(2) 4.4.4.4}}
Symmetry_Group=[[Dihedral symmetry in three dimensions|D<sub>4h</sub>]], [4,2], (*422), order 16|
Rotation_Group=D<sub>4</sub>, [4,2]<sup>+</sup>, (422), order 8|
|symmetry=[[Dihedral symmetry in three dimensions|Dihedral]] {{math|(D{{sub|4h}}), [4,2], (*422),}} order 16
|net=Elongated_dodecahedron_net.png
Dual=-|
|rotation_group={{math|D{{sub|4}}, [4,2]{{sup|+}}, (422),}} order 8
Property_List=[[convex set|convex]], [[Zonohedron]]
|dual=
|properties= [[Convex polytope|Convex]]
}}
}}
[[File:Elongated Dodecahedron.stl|thumb|alt=3D model of a elongated dodecahedron|3D model of a elongated dodecahedron]]


In [[geometry]], the '''elongated dodecahedron'''<ref>Coxeter (1973) p.257</ref>, '''extended rhombic dodecahedron''', or '''rhombo-hexagonal dodecahedron'''<ref>Williamson (1979) p169</ref> is a convex [[polyhedron]] with 8 rhombic and 4 [[hexagon]]al faces. The hexagons may be be equilateral, but not [[regular hexagon|regular]]. It can be seen as constructed from a [[rhombic dodecahedron]] [[elongation (geometry)|elongated]] by a [[square prism]]. Along with the rhombic dodecahedron, it is a [[space-filling polyhedron]].
In [[geometry]], the '''elongated dodecahedron''',<ref>Coxeter (1973) p.257</ref> '''extended rhombic dodecahedron''', '''rhombo-hexagonal dodecahedron'''<ref>Williamson (1979) p169</ref> or '''hexarhombic dodecahedron'''<ref>[https://web.archive.org/web/20080413143937/http://www.matha.mathematik.uni-dortmund.de/~thilo/contents/fedorov.htm Fedorov's five parallelohedra in R³]</ref> is a convex [[dodecahedron]] with 8 [[Rhombus|rhombic]] and 4 [[hexagon]]al 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]] [[elongation (geometry)|elongated]] by a [[square prism]].
== Parallelohedron==
Along with the rhombic dodecahedron, it is a [[space-filling polyhedron]], one of the five types of [[parallelohedron]] identified by [[Evgraf Fedorov]] that [[Honeycomb (geometry)|tile space]] face-to-face by translations. It has 5 sets of parallel edges, called zones or belts.
: [[File:Parallelohedron edges elongated rhombic dodecahedron.png|160px]]


== Tessellation ==
== Tessellation ==
* It can [[tesselate]] all space by translations.
* It can [[tesselate]] all space by translations.
* It is the [[Wigner-Seitz cell]] for certain [[Bravais lattice|body-centered tetragonal lattices]].
* It is the [[Wigner–Seitz cell]] for certain [[Bravais lattice|body-centered tetragonal lattices]].


{| class=wikitable
[[Image:Rhombo-hexagonal dodecahedron tessellation.png|240px|Part of a tessellation of space using rhombo-hexagonal dodecahedra]]
|[[File:Rhombo-hexagonal dodecahedron tessellation.png|240px]]
|}


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]].
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 [[square]]s.

== 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.
{| class=wikitable
|- align=center
|[[File:Elongated_dodecahedron_flat.png|140px]]<BR>Coplanar polyhedron
|[[File:elongated_dodecahedron_flat_net.png|200px]]<BR>Net
|[[File:Elongated_dodecahedron_flat_honeycomb.png|160px]]<BR>Honeycomb
|- align=center
|[[File:Elongated_dodecahedron_concave.png|140px]]<BR>Concave
|[[File:elongated_dodecahedron_concave_net.png|200px]]<BR>Net
|[[File:Elongated_dodecahedron_concave_honeycomb.png|160px]]<BR>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<sub>2h</sub> symmetry, order 8.

{| class=wikitable width=480
|- align=center
|[[File:Contracted_truncated_octahedron.png|160px]]<BR>Contracted truncated octahedron
|[[File:Contracted_truncated_octahedron_net.png|160px]]<BR>Net
|[[File:Contracted_truncated_octahedron_honeycomb.png|160px]]<BR>Honeycomb
|}

== See also==
* [[Trapezo-rhombic dodecahedron]]
* [[Elongated octahedron]]
* [[Elongated gyrobifastigium]]


==References==
==References==
{{reflist}}
{{reflist}}
*{{The Geometrical Foundation of Natural Structure (book)}} ''rhombo-hexagonal dodecahedron'', p169
*{{The Geometrical Foundation of Natural Structure (book)}} ''rhombo-hexagonal dodecahedron'', p169
*[[H.S.M. Coxeter]], [[Regular Polytopes (book)|''Regular Polytopes'']], Third edition, (1973), Dover edition, ISBN 0-486-61480-8 p.257
*[[H.S.M. Coxeter]], [[Regular Polytopes (book)|''Regular Polytopes'']], Third edition, (1973), Dover edition, {{ISBN|0-486-61480-8}} p.&nbsp;257


== External links ==
== External links ==
* {{mathworld | urlname = Space-FillingPolyhedron | title = Space-filling polyhedron}}
* {{mathworld | urlname = Space-FillingPolyhedron | title = Space-filling polyhedron}}
* {{mathworld | urlname = ElongatedDodecahedron | title = Elongated dodecahedron}}
* {{mathworld | urlname = ElongatedDodecahedron | title = Elongated dodecahedron}}
*[http://www.nanomedicine.com/NMI/Figures/5.7.jpg] Uniform space-filling using only rhombo-hexagonal dodecahedra
* [http://www.nanomedicine.com/NMI/Figures/5.7.jpg Uniform space-filling using only rhombo-hexagonal dodecahedra]
*[[VRML]] Model [http://www.georgehart.com/virtual-polyhedra/vrml/elongated_dodecahedron.wrl]
* [http://www.georgehart.com/virtual-polyhedra/vrml/elongated_dodecahedron.wrl Elongated dodecahedron] [[VRML]] Model


[[Category:Space-filling polyhedra]]
[[Category:Space-filling polyhedra]]

Latest revision as of 13:38, 25 December 2022

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 groupDihedral (D4h), [4,2], (*422), order 16
Rotation groupD4, [4,2]+, (422), order 8
PropertiesConvex
Net
3D model of a elongated dodecahedron
3D model of a 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.

Parallelohedron

[edit]

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. It has 5 sets of parallel edges, called zones or belts.

Tessellation

[edit]

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.


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

[edit]

References

[edit]
  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
[edit]
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