Rhombic dodecahedron

The rhombic dodecahedron is a convex Convex set

In Euclidean space [i], an object is convex if for every pair of points within the object, every point o ...

polyhedron Polyhedron

A polyhedron is a geometric shape which in mathematics [i] is defined by three related meanings. ...

with 12 rhombic Rhombus

In geometry [i], a rhombus is a quadrilateral [i] in which all of the sides are of equal length, i.e., i ...

faces. Multiples of it can be stacked to fill a space much like hexagon Hexagon

In geometry [i], a hexagon is a polygon [i] with six edge [i]s and six vertices [i]. ...

s fill a plane; the cells in a honeycomb Honeycomb

A honeycomb is a mass of hexagon [i]al wax [i] cells built by honeybee [i]s in their nests to contain th ...

have a shape similar to the rhombic dodecahedron cut in half. It is the polyhedral dual Dual polyhedron

In geometry [i], polyhedra [i] are associated into pairs called duals, where the vertices [i] ...

of the cuboctahedron Cuboctahedron

A cuboctahedron is a polyhedron [i] with eight triangular faces and six square faces. ...

and a zonohedron. The long diagonal of each face is exactly v2 Square root of 2

The square root [i] of 2, also known as Pythagoras' constant, often denoted by ...

times the length of the short diagonal, so that the acute angles on each face measure 2 tan−1, or approximately 70.53. Being the dual of an Archimedean polyhedron Archimedean solid

In geometry [i] an Archimedean solid or semi-regular solid [i] is a semi-r ...

, the rhombic dodecahedron is face-uniform, meaning the symmetry group Symmetry group

The symmetry [i] group of an object is the group [i] of all isometries [i] under which it is invariant [i] ...

of the solid acts transitively on the set of faces.

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Rhombic dodecahedron
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Type	Catalan Catalan solid In mathematics [i], a Catalan solid, or Archimedean dual, is a dual polyhedron [i] to an Archimedean solid [i] ...
Face polygon	rhombus Rhombus In geometry [i], a rhombus is a quadrilateral [i] in which all of the sides are of equal length, i.e., i ...
Faces	12
Edges	24
Vertices	14 = 6 + 8
Face configuration Face configuration In geometry, a face configuration is notational description of a face-uniform [i] polyhedron [i]. ...	V3.4.3.4
Symmetry group Symmetry group The symmetry [i] group of an object is the group [i] of all isometries [i] under which it is invariant [i] ...	octahedral
Dual polyhedron Dual polyhedron In geometry [i], polyhedra [i] are associated into pairs called *duals*, where the vertices [i] ...	cuboctahedron Cuboctahedron A cuboctahedron is a polyhedron [i] with eight triangular faces and six square faces....
Properties	convex Convex set In Euclidean space [i], an object is convex if for every pair of points within the object, every point o ... , face-uniform, edge-uniform, zonohedron

The rhombic dodecahedron is a convex Convex set

In Euclidean space [i], an object is convex if for every pair of points within the object, every point o ...

polyhedron Polyhedron

A polyhedron is a geometric shape which in mathematics [i] is defined by three related meanings. ...

with 12 rhombic Rhombus

In geometry [i], a rhombus is a quadrilateral [i] in which all of the sides are of equal length, i.e., i ...

faces. Multiples of it can be stacked to fill a space much like hexagon Hexagon

In geometry [i], a hexagon is a polygon [i] with six edge [i]s and six vertices [i]. ...

s fill a plane; the cells in a honeycomb Honeycomb

In geometry [i], polyhedra [i] are associated into pairs called duals, where the vertices [i] ...

of the cuboctahedron Cuboctahedron

A cuboctahedron is a polyhedron [i] with eight triangular faces and six square faces....

and a zonohedron. The long diagonal of each face is exactly v2 Square root of 2

The square root [i] of 2, also known as Pythagoras' constant, often denoted by
...

times the length of the short diagonal, so that the acute angles on each face measure 2 tan⁻¹, or approximately 70.53�.

Being the dual of an Archimedean polyhedron Archimedean solid

In geometry [i] an Archimedean solid or semi-regular solid [i] is a semi-r ...

, the rhombic dodecahedron is face-uniform, meaning the symmetry group Symmetry group

The symmetry [i] group of an object is the group [i] of all isometries [i] under which it is invariant [i] ...

of the solid acts transitively on the set of faces. In elementary terms, this means that for any two faces A and B there is a rotation Rotation

Rotation is the movement of an object in a circular motion....

or reflection of the solid that leaves it occupying the same region of space while moving face A to face B. The rhombic dodecahedron is also somewhat special in being one of the nine edge-uniform convex polyhedra, the others being the five Platonic solid Platonic solid

In geometry [i], a Platonic solid is a convex [i] regular polyhedron [i]. ...

s, the cuboctahedron Cuboctahedron

A cuboctahedron is a polyhedron [i] with eight triangular faces and six square faces....

, the icosidodecahedron Icosidodecahedron

An icosidodecahedron is a polyhedron [i] with twenty triangular faces and twelve pentagonal faces. ...

and the rhombic triacontahedron Rhombic triacontahedron

[i] with [[30|30]...

.

The rhombic dodecahedron can be used to tessellate Tessellation

A tessellation or tiling of the plane [i] is a collection of plane figure [i]s that fills th ...

3-dimensional space. This tessellation can be seen as the Voronoi tessellation Voronoi diagram

In mathematics [i], a Voronoi diagram, named after Georgy Voronoi [i], also called a V ...

of the face-centred cubic lattice Crystal structure

In mineralogy [i] and crystallography [i], a crystal structure is a unique arrangement of atoms in a crystal [i] ...

. Honeybee Honey bee

Honey bees are a subset of bee [i]s which represent a far smaller fraction of bee diversity than most pe ...

s use the geometry of rhombic dodecahedra to form honeycomb Honeycomb

A honeycomb is a mass of hexagon [i]al wax [i] cells built by honeybee [i]s in their nests to contain th ...

from a tessellation of cells each of which is a hexagonal prism capped with half a rhombic dodecahedron.

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

In geometry [i], the tesseract is the 4-dimensional [i] analog [i] of the cube [i], wh ...

to 3 dimensions. There are exactly two ways of decomposing a rhombic dodecahedron into 4 congruent parallelepiped Parallelepiped

In geometry [i], a parallelepiped or parallelopipedon is a three-dimensional figure like a cube [i] ...

s, giving 8 possible parallelepipeds. The 8 cells of the tesseract under this projection map precisely to these 8 parallelepipeds.

Cartesian coordinates

The eight vertices where three faces meet at their obtuse angles have Cartesian coordinates Cartesian coordinate system

In mathematics [i], the Cartesian coordinate system is used to uniquely determine each point [i]...

The six vertices where four faces meet at their acute angles are given by the permutations of

External links

– from MathWorld
– The Encyclopedia of Polyhedra
– make a rhombic dodecahedron calendar without glue

Rhombic dodecahedron

Discussions

Encyclopedia

Cartesian coordinates

See also

External links