geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the elongated dodecahedron, extended rhombic dodecahedron, rhombo-hexagonal dodecahedron or hexarhombic dodecahedronFedorov's five parallelohedra in R³
/ref> is a convex

dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...

with 8 rhombic and 4

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...

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 In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahed ...

elongated by a square prism.

Parallelohedron

Along with the rhombic dodecahedron, it is a

space-filling polyhedron In geometry, a space-filling polyhedron is a polyhedron that can be used to fill all of three-dimensional space via translations, rotations and/or reflections, where ''filling'' means that, taken together, all the instances of the polyhedron const ...

, 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. : Parallelohedron edges elongated rhombic dodecahedron

Parallelohedron edges elongated rhombic dodecahedron

Tessellation

* It can tesselate all space by translations. * It is the

Wigner–Seitz cell The Wigner–Seitz cell, named after Eugene Wigner and Frederick Seitz, is a primitive cell which has been constructed by applying Voronoi decomposition to a crystal lattice. It is used in the study of crystalline materials in crystallography. ...

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 In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille. The internal angle of th ...

with the

rhombi In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...

projected into

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

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. 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.

References

* ''rhombo-hexagonal dodecahedron'', p169 * H.S.M. Coxeter, ''Regular Polytopes'', Third edition, (1973), Dover edition, p. 257

External links

* * {{mathworld , urlname = ElongatedDodecahedron , title = Elongated dodecahedron
Uniform space-filling using only rhombo-hexagonal dodecahedra

Elongated dodecahedron

VRML VRML (Virtual Reality Modeling Language, pronounced ''vermal'' or by its initials, originally—before 1995—known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graph ...

Model Space-filling polyhedra Zonohedra

Parallelohedron

Tessellation

Variations

See also

References

External links