Elongated rhombic dodecahedron

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Elongated rhombic dodecahedron

The elongated rhombic dodecahedron is a convex dodecahedron with eight rhombuses and four hexagons as sides, 18 corners and 28 edges.

It is one of five so-called paralleloeders, i.e. convex polyhedra, which parquet the space through translation . They were found by Yevgraf Stepanowitsch Fyodorov in 1885. The other four are parallelepiped , hexagonal prism , rhombic dodecahedron , and truncated octahedron .

Room tiling with elongated rhombic dodecahedra

Other names are hexarhombic dodecahedron and stretched dodecahedron and in English elongated dodecahedron (or rhombo-hexagonal dodecahedron ). The name lengthened rhombic dodecahedron comes from the fact that it arises from the deformation of a (lengthened) rhombic dodecahedron with a cube. It can also be obtained by contracting a truncated octahedron.

It is the Wigner-Seitz cell of some body-centered tetragonal crystal lattice. The symmetry group of the polyhedron is in Schoenflies symbology , that is, the dihedral group of a regular quadrilateral in the plane (four-fold axis) with an additional mirror plane perpendicular to the axis of rotation (symbol h).

A picture of the elongated rhombic dodecahedron can be found carved on the tombstone of Arthur Moritz Schoenflies , one of the early classifiers of the space groups next to Fyodorov.

Web links

Individual evidence

  1. The term is used, for example, in JJ Burckhardt , Symmetrie der Kristalle, Birkhäuser, 1988, p. 76, when discussing Fjodorow's paralleloeder, or in Erhard Scholz , Symmetrie, Gruppe, Dualität, Birkhäuser, 1989, p. 118
  2. Harry Nyman, Sten Andersson, The elongated rhombic dodecahedron in alloy structures, Acta Crystallographica, A, Volume 35, 1979, pp. 305-308, for example, and assign this structure. You have a tetragonal crystal lattice with space group I4 / mmm. Other structures discussed there correspond more to a truncated version of the elongated rhombic dodecahedron as with a cubic structure.
  3. Grave of Arthur Schoenflies in the Frankfurt main cemetery (Gewann XIV, grave 403 UG, location )