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Rhombic dodecahedron

 
Rhombic Dodecahedron

 
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Rhombic dodecahedron
 
 
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The rhombic dodecahedron (Hexagonal Trapezehedron) is a convex
Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object....
 polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 with 12 rhombic
Rhombus

In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length....
 faces. It is an Archimedean dual
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
 solid, or a Catalan solid
Catalan solid

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgium mathematician, Eug?ne Catalan, who first described them in 1865....
. Its dual is the cuboctahedron
Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
.

g the dual of an Archimedean polyhedron
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
, the rhombic dodecahedron is face-transitive, meaning the symmetry group
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
 of the solid acts transitively on the set of faces. In elementary terms, this means that for any 2 faces A and B there is a rotation
Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
 or reflection
Reflection (mathematics)

In mathematics, a reflection is a function that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q....
 of the solid that leaves it occupying the same region of space while moving face A to face B.

The rhombic dodecahedron is 1 of the 9 edge-transitive convex polyhedra, the others being the 5 Platonic solid
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s, the cuboctahedron
Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
, the icosidodecahedron
Icosidodecahedron

An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
 and the rhombic triacontahedron
Rhombic triacontahedron

In geometry, the rhombic triacontahedron is a convex set polyhedron with 30 rhombus faces. It is an Archimedean solid solid, or a Catalan solid....
.

The rhombic dodecahedron can be used to tessellate
Tessellation

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces....
 3-dimensional space.






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Encyclopedia


The rhombic dodecahedron (Hexagonal Trapezehedron) is a convex
Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object....
 polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 with 12 rhombic
Rhombus

In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length....
 faces. It is an Archimedean dual
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
 solid, or a Catalan solid
Catalan solid

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgium mathematician, Eug?ne Catalan, who first described them in 1865....
. Its dual is the cuboctahedron
Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
.

Properties


It is the polyhedral dual
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 of the cuboctahedron
Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
, and a zonohedron
Zonohedron

A zonohedron is a convex set polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180?....
. The long diagonal of each face is exactly v2 times the length of the short diagonal, so that the acute
Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
 angles on each face measure cos−1(1/3), or approximately 70.53°.

Being the dual of an Archimedean polyhedron
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
, the rhombic dodecahedron is face-transitive, meaning the symmetry group
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
 of the solid acts transitively on the set of faces. In elementary terms, this means that for any 2 faces A and B there is a rotation
Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
 or reflection
Reflection (mathematics)

In mathematics, a reflection is a function that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q....
 of the solid that leaves it occupying the same region of space while moving face A to face B.

The rhombic dodecahedron is 1 of the 9 edge-transitive convex polyhedra, the others being the 5 Platonic solid
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s, the cuboctahedron
Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
, the icosidodecahedron
Icosidodecahedron

An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
 and the rhombic triacontahedron
Rhombic triacontahedron

In geometry, the rhombic triacontahedron is a convex set polyhedron with 30 rhombus faces. It is an Archimedean solid solid, or a Catalan solid....
.

Rhombic Dodecahedra
The rhombic dodecahedron can be used to tessellate
Tessellation

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces....
 3-dimensional space. It can be stacked to fill a space much like hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
s fill a plane.

This tessellation can be seen as the Voronoi tessellation of the face-centred cubic lattice
Crystal structure

In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. A crystal structure is composed of a motif, a set of atoms arranged in a particular way, and a lattice....
. Some minerals such as garnet
Garnet

The garnet group includes a group of minerals that have been used since the Bronze Age as gemstones and abrasives. The name "garnet" comes from the Latin language granatus , possibly a reference to the Punica granatum , a plant with red seeds similar in shape, size, and color to some garnet crystals....
 form a rhombic dodecahedral crystal habit
Crystal habit

In mineralogy, shape and size give rise to descriptive terms applied to the typical appearance, or habit of crystals.The many terms used by mineralogists to describe crystal habits are useful in communicating what specimens of a particular mineral often look like....
. Honeybees use the geometry of rhombic dodecahedra to form honeycomb
Honeycomb

A honeycomb is a mass of hexagonal waxcells built by honey bees in their beehive to contain their larva and stores of honey and pollen.beekeeping may remove the entire honeycomb to harvest honey....
 from a tessellation of cells each of which is a hexagonal prism capped with half a rhombic dodecahedron.

Area and volume

The area A and the volume V of the rhombic dodecahedron of edge length a are:

Cartesian coordinates

The 8 vertices where 3 faces meet at their obtuse angles have Cartesian coordinates
(±1, ±1, ±1)


The 6 vertices where 4 faces meet at their acute angles are given by the permutations of
(0, 0, ±2)


Related polyhedra


This polyhedron is related to an infinite series of tilings with the face configuration
Face configuration

In geometry, a face configuration is notational description of a face-transitive polyhedron. It represents a sequential count of the number of faces that exist at each vertex around a face ....
s V3.2n.3.2n, the first in the Euclidean plane, and the rest in the hyperbolic plane.

V3.4.3.4
(Drawn as a net
Net (polyhedron)

In geometry the net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded to become the faces of the polyhedron....
)
Tile V3636

V3.6.3.6
Euclidean plane tiling
Quasiregular rhombic tiling
Quasiregular rhombic tiling

In geometry, the quasiregular rhombic tiling is a tiling of identical 60° rhombi polygons on the Euclidean plane. There are two types of vertices, one with three rhombi and one with six rhombi....

V3.8.3.8
Hyperbolic plane tiling
(Drawn in a Poincaré disk model
Poincaré disk model

In geometry, the Poincar? disk model, also called the conformal disk model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or ball , and the straight lines of the hyperbolic geometry are segments of circles contained in the disk orthogonal to the boundary of the disk...
)


Related polytopes


The rhombic dodecahedron forms the hull of the vertex-first projection of a tesseract
Tesseract

In geometry, the tesseract, also called an 8-cell or regular octachoron, is the Fourth dimension analog of the cube. The tesseract is to the cube as the cube is to the square ....
 to 3 dimensions. There are exactly 2 ways of decomposing a rhombic dodecahedron into 4 congruent parallelepiped
Parallelepiped

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. It is to a parallelogram as a cube is to a square : Euclidean geometry supports all four notions but affine geometry admits only parallelograms and parallelepipeds....
s, giving 8 possible parallelepipeds. The 8 cells of the tesseract under this projection map precisely to these 8 parallelepipeds.

The rhombic dodecahedron forms the maximal cross-section of a 24-cell
24-cell

In geometry, the 24-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . It is also called an octaplex and polyoctahedron, being constructed of Octahedron Cell ....
, and also forms the hull of its vertex-first parallel projection into 3 dimensions. The rhombic dodecahedron can be decomposed into 6 congruent (but non-regular) square dipyramids meeting at a single vertex in the center; these form the images of 6 pairs of the 24-cell's octahedral cells. The remaining 12 octahedral cells project onto the faces of the rhombic dodecahedron. The non-regularity of these images are due to projective distortion; the facets of the 24-cell are regular octahedra in 4-space.

This decomposition gives an interesting method for constructing the rhombic dodecahedron: cut a cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
 into 6 congruent square pyramids, and attach them to the faces of a 2nd cube. The triangular faces of each pair of adjacent pyramids lie on the same plane, and so merge into rhombuses. The 24-cell may also be constructed in an analogous way using 2 tesseract
Tesseract

In geometry, the tesseract, also called an 8-cell or regular octachoron, is the Fourth dimension analog of the cube. The tesseract is to the cube as the cube is to the square ....
s.

See also

  • Dodecahedron
    Dodecahedron

    A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
  • Rhombic triacontahedron
    Rhombic triacontahedron

    In geometry, the rhombic triacontahedron is a convex set polyhedron with 30 rhombus faces. It is an Archimedean solid solid, or a Catalan solid....
  • Quasiregular rhombic tiling
    Quasiregular rhombic tiling

    In geometry, the quasiregular rhombic tiling is a tiling of identical 60° rhombi polygons on the Euclidean plane. There are two types of vertices, one with three rhombi and one with six rhombi....
  • Truncated rhombic dodecahedron
    Truncated rhombic dodecahedron

    The truncated rhombic dodecahedron is a convex polygon polyhedron constructed from the rhombic dodecahedron by Truncation the 6 vertices.The 6 vertices are truncated such that all edges are equal length....
  • 24-cell
    24-cell

    In geometry, the 24-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . It is also called an octaplex and polyoctahedron, being constructed of Octahedron Cell ....
     - 4D analog of rhombic dodecahedron
  • Rhombic dodecahedral honeycomb
    Rhombic dodecahedral honeycomb

    The rhombic dodecahedra honeycomb is a space-filling tessellation in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which is believed to be the densest possible packing of equal spheres in ordinary space ....


External links

  • – The Encyclopedia of Polyhedra


Computer models

  • -- interactive 3-d model
  • , and by Sándor Kabai, The Wolfram Demonstrations Project.


Paper projects

  • – make a rhombic dodecahedron calendar without glue
  • – made by plaiting paper strips


Practical applications

  • Examples of actual housing construction projects using this geometry