Elongated dodecahedron: Difference between revisions

Elongated dodecahedron
Elongated dodecahedron
Type	Parallelohedron
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	Dihedral (D4h), [4,2], (*422), order 16
Rotation group	D4, [4,2]+, (422), order 8
Properties	Convex
Net

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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

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

It can tesselate all space by translations.
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, D_2h symmetry, order 8.

Contracted truncated octahedron	Net	Honeycomb

References

^ Coxeter (1973) p.257
^ Williamson (1979) p169
^ 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

External links

[1] Coxeter (1973) p.257

[2] Williamson (1979) p169

[3] Fedorov's five parallelohedra in R³

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
+{{Short description|Polyhedron with 8 rhombic and 4 hexagonal faces}}
 {{Infobox polyhedron
 |image=Rhombo-hexagonal dodecahedron.png
+|type=[[Parallelohedron]]
-|type=[[Dodecahedron#Other dodecahedra|Dodecahedron]]
 |faces=8 [[rhombus|rhombi]]<br>4 [[hexagon]]s
 |edges=28
 |vertices=18
-|vertex_config=(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=[[Dihedral symmetry in three dimensions|D<sub>4h</sub>]], [4,2], (*422), order 16
+|symmetry=[[Dihedral symmetry in three dimensions|Dihedral]] {{math|(D{{sub|4h}}), [4,2], (*422),}} order 16
 |net=Elongated_dodecahedron_net.png
-|rotation_group=D<sub>4</sub>, [4,2]<sup>+</sup>, (422), order 8
+|rotation_group={{math|D{{sub|4}}, [4,2]{{sup|+}}, (422),}} order 8
-|dual=-
+|dual=
-|properties=[[convex set|convex]], [[parallelohedron]]
+|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''', '''rhombo-hexagonal dodecahedron'''<ref>Williamson (1979) p169</ref> or '''hexarhombic dodecahedron'''<ref>http://www.matha.mathematik.uni-dortmund.de/~thilo/contents/fedorov.htm</ref> is a convex [[polyhedron]] with 8 rhombic and 4 [[hexagon]]al faces. The hexagons can be made equilateral, or [[regular hexagon|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]]. 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]]
-== Variations==
-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. This construction has half the symmetry, D<sub>2h</sub> symmetry, order 8.
-:[[File:Contracted_truncated_octahedron.png|160px]]
 == Tessellation ==
 * 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
 |[[File:Rhombo-hexagonal dodecahedron tessellation.png|240px]]
-|[[File:Contracted_truncated_octahedron_honeycomb.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]] 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==
 {{reflist}}
 *{{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.&nbsp;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 ==