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 rhombic dodecahedron is a

convex polyhedron A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a

Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan so ...

, and 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 oth ...

of the

cuboctahedron A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...

Properties

The rhombic dodecahedron is a zonohedron. Its polyhedral dual is the

. The long face-diagonal length is exactly times the short face-diagonal length; thus, the

acute Acute may refer to: Science and technology * Acute angle ** Acute triangle ** Acute, a leaf shape in the glossary of leaf morphology * Acute (medicine), a disease that it is of short duration and of recent onset. ** Acute toxicity, the adverse ef ...

angles on each face measure arccos(), or approximately 70.53°. Being the dual of an Archimedean polyhedron, the rhombic dodecahedron 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 faces are the same. More specifically, all faces must be not merely congrue ...

, 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 A and B, there is a

rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...

of the solid that leaves it occupying the same region of space while moving face A to face B. The rhombic dodecahedron can be viewed as the convex hull of the union of the vertices of a cube and an octahedron. The 6 vertices where 4 rhombi meet correspond to the vertices of the

octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

, while the 8 vertices where 3 rhombi meet correspond to the vertices of the

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

. The rhombic dodecahedron is one of the nine

edge-transitive In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given t ...

convex polyhedra, the others being the five

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...

s, the

, the

icosidodecahedron In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 i ...

, and the

rhombic triacontahedron In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Ca ...

. The rhombic dodecahedron can be used to

tessellate A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...

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'' has ...

s fill a plane. This polyhedron in a space-filling tessellation can be seen as the Voronoi tessellation of the

face-centered cubic lattice In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties ...

. It is the Brillouin zone of body centered cubic (bcc) crystals. Some minerals such as

garnet Garnets () are a group of silicate minerals that have been used since the Bronze Age as gemstones and abrasives. All species of garnets possess similar physical properties and crystal forms, but differ in chemical composition. The different ...

form a rhombic dodecahedral

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

. As

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

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 Afro-Eurasia. After bees spread naturally throughout Africa and Eurasia, humans became responsible for the current cosm ...

s use the geometry of rhombic dodecahedra to form

honeycomb A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen. Beekeepers may remove the entire honeycomb to harvest honey. Honey bees consume about of honey ...

s from a tessellation of cells each of which is a hexagonal prism capped with half a rhombic dodecahedron. The rhombic dodecahedron also appears in the unit cells of

diamond Diamond is a solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Another solid form of carbon known as graphite is the chemically stable form of carbon at room temperature and pressure, b ...

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 The graph of the rhombic dodecahedron is nonhamiltonian. A rhombic dodecahedron can be dissected into 4 obtuse trigonal trapezohedra around its center. These

rhombohedra In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) is a three-dimensional figure with six faces which are rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be use ...

are the cells of a trigonal trapezohedral honeycomb. Analogy: a

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

can be dissected into 3

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 ...

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

Louvre The Louvre ( ), or the Louvre Museum ( ), is the world's most-visited museum, and an historic landmark in Paris, France. It is the home of some of the best-known works of art, including the ''Mona Lisa'' and the '' Venus de Milo''. A central ...

include a die in the shape of a rhombic dodecahedron dating from Ptolemaic Egypt. 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. File:Rhombic dodecahedron 4color.png, Rhombic dodecahedron dissected into 4 rhombohedra File:Rhombic_dissected_hexagon_3color.svg, Hexagon dissected into 3 rhombi File:Grenat_pyrope_1.jpg, A

crystal File:R1-cube.gif, This animation shows the construction of a rhombic dodecahedron from a cube, by inverting the center-face-pyramids of a cube.

Dimensions

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

radius In classical geometry, a radius (plural, : radii) 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 name comes from the latin ''radius'', ...

of its inscribed sphere (

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

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

r_\mathrm = \frac~a \approx 0.816\,496\,5809~a\quad

(), *the radius of its midsphere is :

r_\mathrm = \frac~a \approx 0.942\,809\,041\,58~a\quad

(), *the radius of the sphere passing through the six order 4 vertices, but not through the eight order 3 vertices, is :

r_\mathrm = \frac~a \approx 1.154\,700\,538~a\quad

(), *the radius of the sphere passing through the eight order 3 vertices is exactly equal to the length of the sides :

r_\mathrm = a

Area and volume

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

A = 8\sqrt~a^2 \approx 11.313\,7085~a^2

V = \frac~a^3 \approx 3.079\,201\,44~a^3

Orthogonal projections

The ''rhombic dodecahedron'' has four special

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...

s along its axes of symmetry, centered on a face, an edge, and the two types of vertex, threefold and fourfold. The last two correspond to the B₂ and A₂ Coxeter planes.

Cartesian coordinates

The eight vertices where three faces meet at their obtuse angles have

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...

: :(±1, ±1, ±1) The coordinates of the six vertices where four faces meet at their acute angles 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 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 pentago ...

, with permutation of coordinates and with parameter ''h'' = 1.

Topologically equivalent forms

Parallelohedron

The ''rhombic dodecahedron'' is a parallelohedron, 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 ...

, dodecahedrille, being the dual to the ''tetroctahedrille'' or half cubic honeycomb, and described by two

Coxeter diagram Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

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

trigonal trapezohedron In geometry, a trigonal trapezohedron is a rhombohedron (a polyhedron with six rhombus-shaped faces) in which, additionally, all six faces are congruent. Alternative names for the same shape are the ''trigonal deltohedron'' or ''isohedral rh ...

Dihedral rhombic dodecahedron

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, order 16. It can be seen as a

with

square pyramid In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral square pyrami ...

s augmented on the top and bottom.

Bilinski dodecahedron

In 1960 Stanko Bilinski discovered a second rhombic dodecahedron with 12 congruent rhombus faces, the Bilinski dodecahedron. 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 sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

Deltoidal dodecahedron

Another topologically equivalent variation, sometimes called a deltoidal dodecahedron or trapezoidal dodecahedron, is isohedral with

tetrahedral symmetry 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 combine a reflection ...

order 24, distorting rhombic faces into

kites A kite is a tethered heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create lift and drag forces. A kite consists of wings, tethers and anchors. Kites often have a bridle and tail to guide the face ...

(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. Always holds + = 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'')

Related polyhedra

When projected onto a sphere (see right), it can be seen that the edges make up the edges of two tetrahedra arranged in their dual positions (the stella octangula). This trend continues on with the

deltoidal icositetrahedron In geometry, the deltoidal icositetrahedron (or trapezoidal icositetrahedron, tetragonal icosikaitetrahedron, tetragonal trisoctahedron, strombic icositetrahedron) is a Catalan solid. Its 24 faces are congruent kites. The deltoidal icosit ...

and

deltoidal hexecontahedron In geometry, a deltoidal hexecontahedron (also sometimes called a ''trapezoidal hexecontahedron'', a ''strombic hexecontahedron'', or a ''tetragonal hexacontahedron'') is a Catalan solid which is the dual polyhedron of the rhombicosidodecahedron, ...

for the dual pairings of the other regular polyhedra (alongside the

triangular bipyramid In geometry, the triangular bipyramid (or dipyramid) is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces. As the name suggests, ...

if improper tilings are to be considered), giving this shape the alternative systematic name of ''deltoidal dodecahedron''. This polyhedron is a part of a sequence of rhombic polyhedra and tilings with 'n'',3

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...

symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares. Similarly it relates to the infinite series of tilings with the

face configuration In geometry, a vertex configurationCrystallography ...

s V3.2''n''.3.2''n'', the first in the Euclidean plane, and the rest in the hyperbolic plane.

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. R1-stel

The first stellation, often simply called the stellated rhombic dodecahedron, is well known. It 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: Three flattened octahedra compound.png, The first stellation of the rhombic dodecahedron first_rhombic_dodecahedron_stellation.stl, 3D model of decomposition into 12 pyramids and 4 half-cubes 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 polytopes

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

tesseract In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of e ...

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 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 "octahedral complex"), icosatetrahedroid, o ...

, 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 rhombuses. The 24-cell may also be constructed in an analogous way using two

Practical usage

In spacecraft

reaction wheel A reaction wheel (RW) is used primarily by spacecraft for three-axis attitude control, and does not require rockets or external applicators of torque. They provide a high pointing accuracy, and are particularly useful when the spacecraft must be ...

layout, a

tetrahedral In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

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 Attitude control is the process of controlling the orientation of an aerospace vehicle with respect to an inertial frame of reference or another entity such as the celestial sphere, certain fields, and nearby objects, etc. Controlling vehicle ...

system (considering idealized actuators) are given by projecting the

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).

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.

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