geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the rhombic dodecahedron is a

convex polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...

with 12

congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...

rhombic faces. It has 24 edges, and 14 vertices of 2 types. As a

Catalan solid The Catalan solids are the dual polyhedron, dual polyhedra of Archimedean solids. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and symmetric vertices. The faces of the Catalan solids correspond by duality to ...

, it is the

dual polyhedron In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other ...

of the

cuboctahedron A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertex (geometry), vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edge (geometry), edges, each separating a tr ...

. As a

parallelohedron In geometry, a parallelohedron or Fedorov polyhedron is a convex polyhedron that can be Translation (geometry), translated without rotations to fill Euclidean space, producing a Honeycomb (geometry), honeycomb in which all copies of the polyhed ...

, the rhombic dodecahedron can be used to tesselate its copies in space creating a rhombic dodecahedral honeycomb. There are some variations of the rhombic dodecahedron, one of which is the

Bilinski dodecahedron In geometry, the Bilinski dodecahedron is a Convex set, convex polyhedron with twelve Congruence (geometry), congruent golden rhombus faces. It has the same topology as the face-transitive rhombic dodecahedron, but a different geometry. It is a ...

. There are some stellations of the rhombic dodecahedron, one of which is the

Escher's solid In geometry, the first stellation of the rhombic dodecahedron is a self-intersecting polyhedron with 12 faces, each of which is a non-convex Hexagon. It is a stellation of the rhombic dodecahedron and has the same outer shell and the same visual ...

. The rhombic dodecahedron may also appear in nature (such as in the

garnet Garnets () are a group of silicate minerals that have been used since the Bronze Age as gemstones and abrasives. Garnet minerals, while sharing similar physical and crystallographic properties, exhibit a wide range of chemical compositions, de ...

crystal), the architectural philosophies, practical usages, and toys.

As a Catalan solid

Metric properties

The rhombic dodecahedron is a polyhedron with twelve

rhombi In plane Euclidean geometry, a rhombus (: 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 rhom ...

, each of which long face-diagonal length is exactly

\sqrt

times the short face-diagonal length and the acute angle measurement is

\arccos(1/3) \approx 70.53^\circ

. Its dihedral angle between two rhombi is 120°. The rhombic dodecahedron is a

, meaning the

of an

Archimedean solid The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygon and are vertex-transitive, although they aren't face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They ...

, the

; they share the same symmetry, the

octahedral symmetry A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...

. It is

face-transitive In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its Face (geometry), faces are the same. More specifically, all faces must be not ...

, meaning the

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

of the solid acts transitively on its set of faces. In elementary terms, this means that for any two faces, there is a

rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...

or reflection of the solid that leaves it occupying the same region of space while moving a face to another one. Other than

rhombic triacontahedron The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombus, rhombic face (geometry), faces. It has 60 edge (geometry), edges and 32 vertex ...

, it is one of two Catalan solids that each have the property that their isometry groups are

edge-transitive In geometry, a polytope (for example, a polygon or a polyhedron) or a Tessellation, tiling is isotoxal () or edge-transitive if its Symmetry, symmetries act Transitive group action, transitively on its Edge (geometry), edges. Informally, this mea ...

; the other convex polyhedron classes being the five

Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...

s and the other two Archimedean solids: its dual polyhedron and

icosidodecahedron In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (''icosi-'') triangular faces and twelve (''dodeca-'') pentagonal faces. An icosidodecahedron has 30 identical Vertex (geometry), vertices, with two triang ...

. Denoting by ''a'' the edge length of a rhombic dodecahedron, * the

radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

of its

inscribed sphere image:Circumcentre.svg, An inscribed triangle of a circle In geometry, an inscribed plane (geometry), planar shape or solid (geometry), solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figu ...

(

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

to each of the rhombic dodecahedron's faces) is: ()

r_\mathrm = \fraca \approx 0.817a,

* the radius of its

midsphere In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every Edge (geometry), edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedron, uniform polyhedra, including the reg ...

is: )

r_\mathrm = \fraca \approx 0.943a,

* the radius of the sphere passing through the six order four vertices, but not through the eight order 3 vertices, is: ()

r_\mathrm = \fraca \approx 1.155a,

* the radius of the sphere passing through the eight order three vertices is exactly equal to the length of the sides:

r_\mathrm = a

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

\begin
 A &= 8\sqrta^2 &\approx 11.314a^2, \\
 V &= \fraca^3 &\approx 3.079a^3.
\end

The rhombic dodecahedron can be viewed as the

convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...

of the union of the vertices of a cube and an octahedron where the edges intersect perpendicularly. The six vertices where four rhombi meet correspond to the vertices of the

octahedron In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...

, while the eight vertices where three rhombi meet correspond to the vertices of the

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

. The

skeleton A skeleton is the structural frame that supports the body of most animals. There are several types of skeletons, including the exoskeleton, which is a rigid outer shell that holds up an organism's shape; the endoskeleton, a rigid internal fra ...

of a rhombic dodecahedron is called a rhombic dodecahedral graph, with 14 vertices and 24 edges. It is the Levi graph of the Miquel configuration (8₃ 6₄).

Construction

For edge length , the eight vertices where three faces meet at their obtuse angles have

Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

(±1, ±1, ±1). In the case of the coordinates of the six vertices where four faces meet at their acute angles, they are (±2, 0, 0), (0, ±2, 0) and (0, 0, ±2). The rhombic dodecahedron can be seen as a degenerate limiting case of a pyritohedron, with permutation of coordinates and with parameter ''h'' = 1. These coordinates illustrate that a rhombic dodecahedron can be seen as a

with six

square pyramid In geometry, a square pyramid is a Pyramid (geometry), pyramid with a square base and four triangles, having a total of five faces. If the Apex (geometry), apex of the pyramid is directly above the center of the square, it is a ''right square p ...

s attached to each face, allowing them to fit together into a cube. Therefore, the rhombic dodecahedron has twice the volume of the inscribed cube with edges equal to the short diagonals of the rhombi. Alternatively, the rhombic dodecahedron can be constructed by inverting six square pyramids until their apices meet at the cube's center.

As a space-filling polyhedron

The rhombic dodecahedron 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 c ...

, meaning it can be applied to tessellate three-dimensional space: it can be stacked to fill a space, much like

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 is de ...

s fill a plane. It is a

because it can be space-filling a

honeycomb A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic cells built from beeswax by honey bees in their beehive, nests to contain their brood (eggs, larvae, and pupae) and stores of honey and pol ...

in which all of its copies meet face-to-face. More generally, every parellelohedron is

zonohedron In geometry, a zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...

, a centrally symmetric polyhedron with centrally symmetric faces. As a parallelohedron, the rhombic dodecahedron can be constructed with four sets of six parallel edges. The rhombic dodecahedral honeycomb (or ''dodecahedrille'') is an example of a honeycomb constructed by filling all rhombic dodecahedrons. It is dual to the ''tetroctahedrille'' or half cubic honeycomb, and it is described by two

Coxeter diagram Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

s: and . With D_3d symmetry, it can be seen as an elongated

trigonal trapezohedron In geometry, a trigonal trapezohedron is a polyhedron with six congruent quadrilateral faces, which may be scalene or rhomboid. The variety with rhombus-shaped faces faces is a rhombohedron. An alternative name for the same shape is the ''trig ...

. It can be seen as the Voronoi tessellation of the face-centered cubic lattice. It is the Brillouin zone of body-centered cubic (bcc) crystals. Some minerals such as

form a rhombic dodecahedral

crystal habit In mineralogy, crystal habit is the characteristic external shape of an individual crystal or aggregate of crystals. The habit of a crystal is dependent on its crystallographic form and growth conditions, which generally creates irregularities d ...

. As

Johannes Kepler Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best know ...

noted in his 1611 book on snowflakes (''Strena seu de Nive Sexangula''),

honey bee A honey bee (also spelled honeybee) is a eusocial flying insect within the genus ''Apis'' of the bee clade, all native to mainland Afro-Eurasia. After bees spread naturally throughout Africa and Eurasia, humans became responsible for the ...

s use the geometry of rhombic dodecahedra to form

s from a tessellation of cells each of which is a

hexagonal prism In geometry, the hexagonal prism is a Prism (geometry), prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 face (geometry), faces, 18 Edge (geometry), edges, and 12 vertex (geometry), vertices.. As a semiregular polyhedro ...

capped with half a rhombic dodecahedron. The rhombic dodecahedron also appears in the unit cells of

diamond Diamond is a Allotropes of carbon, solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Diamond is tasteless, odourless, strong, brittle solid, colourless in pure form, a poor conductor of e ...

and diamondoids. In these cases, four vertices (alternate threefold ones) are absent, but the chemical bonds lie on the remaining edges.Dodecahedral Crystal Habit
. khulsey.com A rhombic dodecahedron can be dissected into four congruent, obtuse trigonal trapezohedra around its center. These

rhombohedra In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) is a special case of a parallelepiped in which all six faces are congruent rhombus, rhombi. It can be used to define the rhombohedral lattice system, a Ho ...

are the cells of a trigonal trapezohedral honeycomb. Analogously, a regular hexagon can be dissected into 3

around its center. These rhombi are the tiles of a rhombille.

Applications

Practical usage

In spacecraft

reaction wheel A reaction wheel (RW) is an electric motor attached to a flywheel, which, when its rotation speed is changed, causes a counter-rotation proportionately through conservation of angular momentum. A reaction wheel can rotate only around its center ...

layout, a

tetrahedral In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

configuration of four wheels is commonly used. For wheels that perform equally (from a peak torque and max angular momentum standpoint) in both spin directions and across all four wheels, the maximum torque and maximum momentum envelopes for the 3-axis

attitude control Spacecraft attitude control is the process of controlling the orientation of a spacecraft (vehicle or satellite) with respect to an inertial frame of reference or another entity such as the celestial sphere, certain fields, and nearby objects, ...

system (considering idealized actuators) are given by projecting the

tesseract In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six ...

representing the limits of each wheel's torque or momentum into 3D space via the 3 × 4 matrix of wheel axes; the resulting 3D polyhedron is a rhombic dodecahedron. Such an arrangement of reaction wheels is not the only possible configuration (a simpler arrangement consists of three wheels mounted to spin about orthogonal axes), but it is advantageous in providing redundancy to mitigate the failure of one of the four wheels (with degraded overall performance available from the remaining three active wheels) and in providing a more convex envelope than a cube, which leads to less agility dependence on axis direction (from an actuator/plant standpoint). Spacecraft mass properties influence overall system momentum and agility, so decreased variance in envelope boundary does not necessarily lead to increased uniformity in preferred axis biases (that is, even with a perfectly distributed performance limit within the actuator subsystem, preferred rotation axes are not necessarily arbitrary at the system level). The polyhedron is also the basis for the

HEALPix HEALPix (sometimes written as Healpix), an acronym for Hierarchical Equal Area isoLatitude Pixelisation of a 2-sphere, is an algorithm for pixelisation of the 2-sphere based on subdivision of a distorted rhombic dodecahedron, and the associate ...

grid, used in cosmology for storing and manipulating maps of the

cosmic microwave background The cosmic microwave background (CMB, CMBR), or relic radiation, is microwave radiation that fills all space in the observable universe. With a standard optical telescope, the background space between stars and galaxies is almost completely dar ...

, and in computer graphics for storing environment maps.

Miscellaneous

The collections of the

Louvre The Louvre ( ), or the Louvre Museum ( ), is a national art museum in Paris, France, and one of the most famous museums in the world. It is located on the Rive Droite, Right Bank of the Seine in the city's 1st arrondissement of Paris, 1st arron ...

include a die in the shape of a rhombic dodecahedron dating from

Ptolemaic Egypt Ptolemaic is the adjective formed from the name Ptolemy, and may refer to: Pertaining to the Ptolemaic dynasty * Ptolemaic dynasty, the Macedonian Greek dynasty that ruled Egypt founded in 305 BC by Ptolemy I Soter *Ptolemaic Kingdom Pertaining ...

. The faces are inscribed with Greek letters representing the numbers 1 through 12: Α Β Γ Δ Ε Ϛ Z Η Θ Ι ΙΑ ΙΒ. The function of the die is unknown.

Other related figures

Topologically equivalent forms

Other symmetry constructions of the rhombic dodecahedron are also space-filling, and as parallelotopes they are similar to variations of space-filling truncated octahedra.Order in Space: A design source book, Keith Critchlow, p.56–57 For example, with 4 square faces, and 60-degree rhombic faces, and D_4h

dihedral symmetry In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, g ...

, order 16. It can be seen as a cuboctahedron with square pyramids attached on the top and bottom. In 1960,

Stanko Bilinski Stanko Bilinski (22 April 1909 in Našice – 6 April 1998 in Zagreb) was a Croatian mathematician and academician. He was a professor at the University of Zagreb and a fellow of the Croatian Academy of Sciences and Arts. In 1960, he discovered ...

discovered a second rhombic dodecahedron with 12 congruent rhombus faces, the

. It has the same topology but different geometry. The rhombic faces in this form have the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...

. The ''deltoidal dodecahedron'' is another topological equivalence of a rhombic dodecahedron form. It is isohedral with

tetrahedral symmetry image:tetrahedron.svg, 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that co ...

order 24, distorting rhombic faces into

kites A kite is a tethered heavier than air flight, heavier-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. Kites often have ...

(deltoids). It has 8 vertices adjusted in or out in alternate sets of 4, with the limiting case a tetrahedral envelope. Variations can be parametrized by (''a'',''b''), where ''b'' and ''a'' depend on each other such that the tetrahedron defined by the four vertices of a face has volume zero, i.e. is a planar face. (1,1) is the rhombic solution. As ''a'' approaches , ''b'' approaches infinity. It always holds that + = 2, with ''a'', ''b'' > . :(±2, 0, 0), (0, ±2, 0), (0, 0, ±2) :(''a'', ''a'', ''a''), (−''a'', −''a'', ''a''), (−''a'', ''a'', −''a''), (''a'', −''a'', −''a'') :(−''b'', −''b'', −''b''), (−''b'', ''b'', ''b''), (''b'', −''b'', ''b''), (''b'', ''b'', −''b'')

Stellations

Like many convex polyhedra, the rhombic dodecahedron can be stellated by extending the faces or edges until they meet to form a new polyhedron. Several such stellations have been described by Dorman Luke. The first stellation, often called the stellated rhombic dodecahedron, can be seen as a rhombic dodecahedron with each face augmented by attaching a rhombic-based pyramid to it, with a pyramid height such that the sides lie in the face planes of the neighbouring faces. Luke describes four more stellations: the second and third stellations (expanding outwards), one formed by removing the second from the third, and another by adding the original rhombic dodecahedron back to the previous one.

Related polytope

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

to three dimensions. There are exactly two ways of decomposing a rhombic dodecahedron into four congruent

, giving eight possible rhombohedra as projections of the tesseracts 8 cubic cells. One set of projective vectors are: ''u'' = (1,1,−1,−1), ''v'' = (−1,1,−1,1), ''w'' = (1,−1,−1,1). The rhombic dodecahedron forms the maximal cross-section of a

24-cell In four-dimensional space, four-dimensional geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octa ...

, and also forms the hull of its vertex-first parallel projection into three dimensions. The rhombic dodecahedron can be decomposed into six congruent (but non-regular) square dipyramids meeting at a single vertex in the center; these form the images of six 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

into six congruent square pyramids, and attach them to the faces of a second cube. The triangular faces of each pair of adjacent pyramids lie on the same plane, and so merge into rhombi. The 24-cell may also be constructed in an analogous way using two

s.Archived a
Ghostarchive
and th
Wayback Machine

References

External links

*

– The Encyclopedia of Polyhedra

Computer models

Relating a Rhombic Triacontahedron and a Rhombic DodecahedronRhombic Dodecahedron 5-Compound
an
Rhombic Dodecahedron 5-Compound
by Sándor Kabai,

The Wolfram Demonstrations Project The Wolfram Demonstrations Project is an open-source collection of interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown to over 10,000. The site won a Pa ...

Paper projects

Rhombic Dodecahedron Calendar
– make a rhombic dodecahedron calendar without glue
Another Rhombic Dodecahedron Calendar
– made by plaiting paper strips

Practical applications

Archimede Institute
Examples of actual housing construction projects using this geometry {{Polyhedron navigator Catalan solids Quasiregular polyhedra Space-filling polyhedra Zonohedra Golden ratio