View source for Rhombic dodecahedron

{{Short description|Catalan solid with 12 faces}}
{{infobox polyhedron
 | name = Rhombic dodecahedron
 | image = Rhombicdodecahedron.jpg
 | type = [[Catalan solid]],<br>[[Parallelohedron]]
 | faces = 12 [[rhombus|rhombi]] | edges = 24 | vertices = 14
 | symmetry = [[octahedral symmetry]] <math> \mathrm{O}_h </math>
 | properties = [[Convex set|convex]], [[edge-transitive]], [[face-transitive]]
 | dual = [[cuboctahedron]]
 | angle = 120°
 | net = Rhombicdodecahedron net.svg
}}
[[File:Rhombic dodecahedron.stl|thumb|3D model of a rhombic dodecahedron]]
In [[geometry]], the '''rhombic dodecahedron''' is a [[Polyhedron#Convex_polyhedra|convex polyhedron]] with 12 [[congruence (geometry)|congruent]] [[rhombus|rhombic]] [[face (geometry)|faces]]. It has 24 [[edge (geometry)|edges]], and 14 [[vertex (geometry)|vertices]] of 2 types. As a [[Catalan solid]], it is the [[dual polyhedron]] of the [[cuboctahedron]]. As a [[parallelohedron]], the rhombic dodecahedron can be used to [[Honeycomb (geometry)|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]]. There are some stellations of the rhombic dodecahedron, one of which is the [[Escher's solid]]. The rhombic dodecahedron may also appear in the [[garnet]] crystal, the architectural philosophies, practical usages, and toys.

== As a Catalan solid ==
=== Metric properties ===
The rhombic dodecahedron is a polyhedron with twelve [[rhombus|rhombi]], each of which long face-diagonal length is exactly <math> \sqrt{2} </math> times the short face-diagonal length{{r|cromwell}} and the acute angle measurement is <math display="inline"> \arccos(1/3) \approx 70.53^\circ </math>. Its [[dihedral angle]] between two rhombi is 120°.{{r|williams}}

The rhombic dodecahedron is a [[Catalan solid]], meaning the [[dual polyhedron]] of an [[Archimedean solid]], the [[cuboctahedron]]; they share the same symmetry, the [[octahedral symmetry]].{{r|williams}} It is [[face-transitive]], meaning the [[symmetry group]] of the solid acts [[Transitive action|transitively]] on its set of faces. In elementary terms, this means that for any two faces, there is a [[rotation]] or [[Reflection (mathematics)|reflection]] of the solid that leaves it occupying the same region of space while moving a face to another one.{{r|diudea}} Other than [[rhombic triacontahedron]], it is one of two Catalan solids that each have the property that their isometry groups are [[edge-transitive]]; the other convex polyhedron classes being the five [[Platonic solid]]s and the other two Archimedean solids: its dual polyhedron and [[icosidodecahedron]].

Denoting by ''a'' the edge length of a rhombic dodecahedron,
* the [[radius]] of its [[inscribed sphere]] ([[tangent]] to each of the rhombic dodecahedron's faces) is: ({{OEIS2C|A157697}}) <math display="block"> r_\mathrm{i} = \frac{\sqrt{6}}{3}a \approx 0.817a,</math>
* the radius of its [[midsphere]] is: {{OEIS2C|A179587}}) <math display="block"> r_\mathrm{m} = \frac{2\sqrt{2}}{3}a \approx 0.943a,</math>
* the radius of the sphere passing through the six order four vertices, but not through the eight order 3 vertices, is: ({{OEIS2C|A020832}}) <math display="block"> r_\mathrm{o} = \frac{2\sqrt{3}}{3}a \approx 1.155a, </math>
* the radius of the sphere passing through the eight order three vertices is exactly equal to the length of the sides: <math display="inline"> r_\mathrm{t} = a </math>

The surface area ''A'' and the volume ''V'' of the rhombic dodecahedron with edge length ''a'' are:{{r|berman}}
<math display="block"> \begin{align}
 A &= 8\sqrt{2}a^2 &\approx 11.314a^2, \\
 V &= \frac{16\sqrt{3}}{9}a^3 &\approx 3.079a^3.
\end{align} </math>

The rhombic dodecahedron can be viewed as the [[convex hull]] 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]], while the eight vertices where three rhombi meet correspond to the vertices of the [[cube]].

The graph of the rhombic dodecahedron is [[Hamiltonian path|nonhamiltonian]].

=== Construction ===
{{multiple image
| image1            = Pyritohedron animation.gif
| caption1          = [[Pyritohedron]] variations between a cube and a rhombic dodecahedron. Halfway in the transition, the polyhedron becomes a Regular Dodecahedron.
| image2            = R1-cube.gif
| caption2          = Construction of rhombic dodecahedron by inverting the center-face-pyramids of a [[cube]]
| total_width       = 300
| align             = left
}}
For edge length {{sqrt|3}}, the eight vertices where three faces meet at their obtuse angles have [[Cartesian coordinates]] (±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#Cartesian coordinates 2|pyritohedron]], with permutation of coordinates {{nowrap|(±1, ±1, ±1)}} and {{nowrap|(0, 1 + ''h'', 1 − ''h''<sup>2</sup>)}} with parameter ''h''&nbsp;=&nbsp;1.

These coordinates illustrate that a rhombic dodecahedron can be seen as a [[cube]] with six [[square pyramid]]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.{{r|cardil}} Alternatively, the rhombic dodecahedron can be constructed by inverting six square pyramids until their [[Apex (geometry)|apices]] meet at the cube's center.{{r|cundy}}
{{-}}

== As a space-filling polyhedron ==
{{multiple image
| image1            = Parallelohedron edges rhombic dodecahedron.png
| caption1          = The rhombic dodecahedron as a parallelohedron
| image2            = Rhombic dodecahedra.png
| caption2          = The rhombic dodecahedron can [[Rhombic dodecahedral honeycomb|tessellate space by translational copies of itself]]
| image3            = Grenat pyrope 1.jpg
| caption3          = A [[garnet]] crystal
| total_width       = 500
}}

The rhombic dodecahedron is a [[space-filling polyhedron]], meaning it can be applied to [[Tessellation|tessellate]] three-dimensional space: it can be stacked to fill a space, much like [[hexagon]]s fill a plane. It is a [[parallelohedron]] because it can be space-filling a [[Honeycomb (geometry)|honeycomb]] in which all of its copies meet face-to-face.{{r|alexandrov}} More generally, every parellelohedron is [[zonohedron]], a [[centrally symmetric]] polyhedron with [[Zonogon|centrally symmetric faces]].{{r|eppstein}} As a parallelohedron, the rhombic dodecahedron can be constructed with four sets of six parallel edges.{{r|alexandrov}}

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]]s: {{CDD|node_f1|3|node|split1-43|nodes}} and {{CDD|node_f1|split1|nodes|split2|node}}. With D<sub>3d</sub> symmetry, it can be seen as an [[Elongation (geometry)|elongated]] [[trigonal trapezohedron]]. It can be seen as the [[Voronoi tessellation]] of the [[Crystal structure|face-centered cubic lattice]]. It is the Brillouin zone of body-centered cubic (bcc) crystals. Some minerals such as [[garnet]] form a rhombic dodecahedral [[crystal habit]]. As [[Johannes Kepler]] noted in his 1611 book on snowflakes (''Strena seu de Nive Sexangula''), [[honey bee]]s use the geometry of rhombic dodecahedra to form [[honeycomb]]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]] and [[diamondoid]]s. In these cases, four vertices (alternate threefold ones) are absent, but the chemical bonds lie on the remaining edges.<ref name=cryst1>[http://www.khulsey.com/jewelry/crystal_habit.html#h-5. Dodecahedral Crystal Habit] {{webarchive|url=https://web.archive.org/web/20090412012420/http://www.khulsey.com/jewelry/crystal_habit.html |date=2009-04-12 }}. khulsey.com</ref>

{{multiple image
 | image1 = Rhombic dodecahedron 4color.png
 | image2 = Rhombic_dissected_hexagon_3color.svg
 | footer = Dissection of the rhombic dodecahedron analoguous to that of the hexagon
 | total_width = 300
}}
A rhombic dodecahedron can be [[Dissection (geometry)|dissected]] into four congruent, obtuse [[Trigonal trapezohedron|trigonal trapezohedra]] around its center. These [[Rhombohedron|rhombohedra]] are the cells of a [[trigonal trapezohedral honeycomb]]. Analogously, a [[regular hexagon]] can be dissected into 3 [[rhombi]] around its center. These rhombi are the tiles of a [[rhombille]].{{cn|date=September 2024}}

As seen in the rightmost image displaying a garnet crystal, this polyhedron can be seen in crystal structures of various gemstones, such as Garnet (As seen in the image), and [[Almandine]].

== Appearances ==
=== Practical usage ===
In spacecraft [[reaction wheel]] layout, a [[Tetrahedron|tetrahedral]] 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 [[Spacecraft attitude control|attitude control]] system (considering idealized actuators) are given by projecting the [[tesseract]] 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.<ref>{{citation|last=Markley|first=F. Landis|date=September 2010|title=Maximum Torque and Momentum Envelopes for Reaction-Wheel Arrays|url=https://ntrs.nasa.gov/citations/20110015369|access-date=2020-08-20|website=ntrs.nasa.gov}}</ref>  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]] grid, used in cosmology for storing and manipulating maps of the [[cosmic microwave background]], and in computer graphics for storing [[environment mapping#HEALPix mapping|environment maps]].

=== Miscellaneous ===
The collections of the [[Louvre]] 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.{{r|perdrizet}}

== Other related figures ==
=== Topologically equivalent forms ===
{{multiple image
 | image1 = Squared rhombic dodecahedron.png
 | caption1 = A rhombic dodecahedron but with square and rhombic faces
 | image2 = Bilinski dodecahedron (gray).gif
 | caption2 = The [[Bilinski dodecahedron]]
 | total_width = 300
}}
Other symmetry constructions of the rhombic dodecahedron are also space-filling, and as [[Parallelohedron#Parallelotope|parallelotope]]s they are similar to variations of space-filling [[truncated octahedra]].<ref name=Critchlow>Order in Space: A design source book, Keith Critchlow, p.56–57</ref> For example, with 4 square faces, and 60-degree rhombic faces, and D<sub>4h</sub> [[Dihedral symmetry in three dimensions|dihedral symmetry]], order 16. It can be seen as a cuboctahedron with square pyramids attached on the top and bottom.

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]].{{r|grunbaum|coxeter}}

{{multiple image
 | align = right
 | total_width = 300
 | image1 = Meyers b10 s0232 b5.png
 | image2 = Modell, Kristallform Deltoiddodekaeders -Krantz- (2).jpg
 | footer = Drawing and [[crystal model]] of deltoidal dodecahedron
}}
{{anchor|Deltoidal dodecahedron}}The ''deltoidal dodecahedron'' is another topological equivalence of a rhombic dodecahedron form.<ref>[https://archive.org/details/economicmineral00croogoog/page/n24 <!-- pg=8 quote="deltoidal dodecahedron". --> Economic Mineralogy: A Practical Guide to the Study of Useful Minerals], p.8</ref> It is [[isohedron|isohedral]] with [[tetrahedral symmetry]] order 24, distorting rhombic faces into [[kite (geometry)|kites]] (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 {{sfrac|1|2}}, ''b'' approaches infinity. It always holds that {{sfrac|1|''a''}} + {{sfrac|1|''b''}} = 2, with ''a'', ''b'' > {{sfrac|1|2}}.
:(±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'')

{| class=wikitable
!(1,1)
!({{sfrac|7|8}},{{sfrac|7|6}})
!({{sfrac|3|4}},{{sfrac|3|2}})
!({{sfrac|2|3}},2)
!({{sfrac|5|8}},{{sfrac|5|2}})
!({{sfrac|9|16}},{{sfrac|9|2}})
<!--!({{sfrac|1|2}},∞)-->
|-
|[[File:Rhombic dodecahedron.png|100px]]
|[[File:Skew_rhombic_dodecahedron-116.png|100px]]
|[[File:Skew_rhombic_dodecahedron-150.png|100px]]
|[[File:Skew_rhombic_dodecahedron-200.png|100px]]
|[[File:Skew_rhombic_dodecahedron-250.png|100px]]
|[[File:Skew_rhombic_dodecahedron-450.png|100px]]
<!--|[[File:Skew_rhombic_dodecahedron-infin.png|100px]]-->
|}

=== Stellations ===
<!--[[File:R1-stel.gif|thumb|This animation shows the construction of a [[stellated rhombic dodecahedron]] by inverting the center-face pyramids of a rhombic dodecahedron.]]-->
{{multiple image
 | image1 = Three flattened octahedra compound.png
 | image2 = Great rhombic dodecahedron.png
 | image3 = Stellated rhombic dodecahedron.png
 | footer = Some stellated rhombic dodecahedrons
 | total_width = 400
}}
Like many convex polyhedra, the rhombic dodecahedron can be [[stellation|stellated]] by extending the faces or edges until they meet to form a new polyhedron. Several such stellations have been described by Dorman Luke.{{r|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 ===
[[Image:Hypercubeorder.svg|thumb|In a perfect vertex-first projection two of the [[tesseract]]'s vertices (marked in pale green) are projected exactly in the center of the rhombic dodecahedron]]
The rhombic dodecahedron forms the hull of the vertex-first projection of a [[tesseract]] to three dimensions. There are exactly two ways of decomposing a rhombic dodecahedron into four congruent [[rhombohedra]], 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]], 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 dipyramid]]s 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 [[cube]] 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 [[tesseract]]s.<ref>Archived at [https://ghostarchive.org/varchive/youtube/20211205/oJ7uOj2LRso Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20181009034310/https://www.youtube.com/watch?disable_polymer=true&bpctr=9999999999&has_verified=1&hl=en&gl=US&v=oJ7uOj2LRso Wayback Machine]{{cbignore}}: {{citation| url = https://www.youtube.com/watch?v=oJ7uOj2LRso| title = There are SIX Platonic Solids | website=[[YouTube]]| date = 30 November 2015 }}{{cbignore}}</ref>

== See also ==
* [[Truncated rhombic dodecahedron]]
* [[Archimede construction systems]]

==References==
{{reflist|refs=

<ref name="alexandrov">{{citation
 | last = Alexandrov | first = A. D. | author-link = Aleksandr Danilovich Aleksandrov
 | contribution = 8.1 Parallelohedra
 | pages = 349–359
 | publisher = Springer
 | title = Convex Polyhedra | title-link = Convex Polyhedra (book)
 | year = 2005
}}</ref>

<ref name="berman">{{citation
 | last = Berman | first = Martin
 | doi = 10.1016/0016-0032(71)90071-8
 | journal = Journal of the Franklin Institute
 | mr = 290245
 | pages = 329–352
 | title = Regular-faced convex polyhedra
 | volume = 291
 | year = 1971| issue = 5
}}</ref>

<ref name="cromwell">{{citation
 | last = Cromwell | first = Peter R.
 | year = 1997
 | title = Polyhedra
 | publisher = Cambridge University Press
 | url = https://archive.org/details/polyhedra0000crom/page/151
 | page = 151&ndash;152
 | isbn = 978-0-521-55432-9
}}</ref>

<ref name="cardil">{{citation
 | last = Cardil | first = Roberto
 | url = https://old.maa.org/press/periodicals/convergence/kepler-and-the-rhombic-dodecahedron-the-rhombic-dodecahedron-as-a-cube-with-pyramids-some-basic
 | title = The Rhombic Dodecahedron as a Cube with Pyramids - Some Basic Measurement
 | website = Mathematical Association of America
}}</ref>

<ref name="cundy">{{citation
 | last = Cundy | first = H. Martyn
 | year = 1956
 | title = 2642. Unitary Construction of Certain Polyhedra
 | journal = [[The Mathematical Gazette]]
 | volume = 40 | issue = 234 | pages = 280&ndash;282
 | jstor = 3609622
 | doi = 10.2307/3609622
}}</ref>

<ref name="coxeter">{{citation
 | last = Coxeter | first = Harold | author-link = H. S. M. Coxeter
 | title = Regular polytopes
 | publisher = [[Dover Publications]]
 | year = 1973
}}</ref>

<ref name="diudea">{{citation
 | last = Diudea | first = M. V.
 | year = 2018
 | title = Multi-shell Polyhedral Clusters
 | series = Carbon Materials: Chemistry and Physics
 | volume = 10
 | publisher = [[Springer Science+Business Media|Springer]]
 | isbn = 978-3-319-64123-2
 | doi = 10.1007/978-3-319-64123-2
 | url = https://books.google.com/books?id=p_06DwAAQBAJ
}}</ref>

<ref name="eppstein">{{citation
 | last = Eppstein | first = David | authorlink = David Eppstein
 | year = 1996
 | title = Zonohedra and zonotopes
 | journal = Mathematica in Education and Research
 | volume = 5
 | issue = 4
 | pages = 15–21
 | url = http://www.ics.uci.edu/~eppstein/junkyard/ukraine/ukraine.html
}}</ref>

<ref name="grunbaum">{{citation
 | last = Grünbaum | first = Branko | author-link = Branko Grünbaum
 | doi = 10.1007/s00283-010-9138-7
 | hdl = 1773/15593
 | issue = 4
 | journal = [[The Mathematical Intelligencer]]
 | mr = 2747698
 | pages = 5–15
 | title = The Bilinski dodecahedron and assorted parallelohedra, zonohedra, monohedra, isozonohedra, and otherhedra
 | volume = 32
 | year = 2010| hdl-access = free
}}</ref>

<ref name="luke">{{citation
 | last = Luke | first = Dorman 
 | title = Stellations of the rhombic dodecahedron
 | journal = The Mathematical Gazette
 | volume = 41 | year = 1957 | issue = 337 | pages = 189–194
 | doi = 10.2307/3609190
 | jstor = 3609190
 | s2cid = 126103579
}}</ref>

<ref name=perdrizet>{{citation
 | last = Perdrizet | first =  Paul
 | title =  Le jeu alexandrin de l'icosaèdre
 | journal = Bulletin de l'Institut français d'archéologie orientale
 | volume = 30 | year = 1930 | pages = 1–16
 | doi = 10.3406/bifao.1931.1865
}}</ref>

<ref name="williams">{{citation
 | last = Williams | first = Robert | author-link = Robert Williams (geometer)
 | year = 1979
 | title = The Geometrical Foundation of Natural Structure: A Source Book of Design
 | publisher = Dover Publications, Inc.
 | page = 74&ndash;75
 | isbn = 978-0-486-23729-9
 | url = https://archive.org/details/geometricalfound00will/page/75
}}</ref>

}}

==Further reading==
*{{The Geometrical Foundation of Natural Structure (book)|mode=cs2}} (Section 3-9)
*{{citation |last1=Wenninger |first1=Magnus |author1-link=Magnus Wenninger |title=Dual Models |publisher=Cambridge University Press |url=https://books.google.com/books?id=mfmzUjhs-_8C&pg=PA19 |isbn=978-0-521-54325-5 |doi=10.1017/CBO9780511569371 |mr=730208 |year=1983}} (The thirteen semiregular convex polyhedra and their duals, Page 19, Rhombic dodecahedron)
*[https://books.google.com/books?id=EtQCk0TNafsC&pg=PA345 ''The Symmetries of Things''] 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, {{isbn|978-1-56881-220-5}} (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, p. 285, Rhombic dodecahedron )

==External links==
*{{Mathworld|urlname=RhombicDodecahedron |title=Rhombic dodecahedron|mode=cs2 }}
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] – The Encyclopedia of Polyhedra

===Computer models===
*[http://demonstrations.wolfram.com/RelatingARhombicTriacontahedronAndARhombicDodecahedron/ Relating a Rhombic Triacontahedron and a Rhombic Dodecahedron], [http://demonstrations.wolfram.com/RhombicDodecahedron5Compound/ Rhombic Dodecahedron 5-Compound] and [http://demonstrations.wolfram.com/RhombicDodecahedron5Compound/ Rhombic Dodecahedron 5-Compound] by Sándor Kabai, [[The Wolfram Demonstrations Project]].

===Paper projects===
*[http://www.ii.uib.no/~arntzen/kalender/ Rhombic Dodecahedron Calendar] – make a rhombic dodecahedron calendar without glue
*[http://www.southernct.edu/~fields/PlaitedCalendar/ Another Rhombic Dodecahedron Calendar] – made by plaiting paper strips

===Practical applications===
*[http://archinstitute.blogspot.com Archimede Institute] Examples of actual housing construction projects using this geometry

{{Catalan solids}}
{{Polyhedron navigator}}

[[Category:Catalan solids]]
[[Category:Quasiregular polyhedra]]
[[Category:Space-filling polyhedra]]
[[Category:Zonohedra]]
[[Category:Golden ratio]]