Dodecahedron
In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.
I_{h}, order 120  

Regular  Small stellated  Great  Great stellated 
T_{h}, order 24  T, order 12  O_{h}, order 48  Johnson (J_{84}) 
Pyritohedron  Tetartoid  Rhombic  Triangular 
D_{4h}, order 16  D_{3h}, order 12  
Rhombohexagonal  Rhombosquare  Trapezorhombic  Rhombotriangular 
The pyritohedron, a common crystal form in pyrite, is an irregular pentagonal dodecahedron, having the same topology as the regular one but pyritohedral symmetry while the tetartoid has tetrahedral symmetry. The rhombic dodecahedron, seen as a limiting case of the pyritohedron, has octahedral symmetry. The elongated dodecahedron and trapezorhombic dodecahedron variations, along with the rhombic dodecahedra, are spacefilling. There are numerous other dodecahedra.
Regular dodecahedra
The convex regular dodecahedron is one of the five regular Platonic solids and can be represented by its Schläfli symbol {5, 3}.
The dual polyhedron is the regular icosahedron {3, 5}, having five equilateral triangles around each vertex.
Convex regular dodecahedron 
Small stellated dodecahedron 
Great dodecahedron 
Great stellated dodecahedron 
The convex regular dodecahedron also has three stellations, all of which are regular star dodecahedra. They form three of the four Kepler–Poinsot polyhedra. They are the small stellated dodecahedron {5/2, 5}, the great dodecahedron {5, 5/2}, and the great stellated dodecahedron {5/2, 3}. The small stellated dodecahedron and great dodecahedron are dual to each other; the great stellated dodecahedron is dual to the great icosahedron {3, 5/2}. All of these regular star dodecahedra have regular pentagonal or pentagrammic faces. The convex regular dodecahedron and great stellated dodecahedron are different realisations of the same abstract regular polyhedron; the small stellated dodecahedron and great dodecahedron are different realisations of another abstract regular polyhedron.
Other pentagonal dodecahedra
In crystallography, two important dodecahedra can occur as crystal forms in some symmetry classes of the cubic crystal system that are topologically equivalent to the regular dodecahedron but less symmetrical: the pyritohedron with pyritohedral symmetry, and the tetartoid with tetrahedral symmetry:
Pyritohedron
Pyritohedron  

A pyritohedron has 30 edges: 6 corresponding to cube faces, and 24 touching cube vertices.  
Face polygon  irregular pentagon 
Coxeter diagrams  
Faces  12 
Edges  30 (6 + 24) 
Vertices  20 (8 + 12) 
Symmetry group  T_{h}, [4,3^{+}], (3*2), order 24 
Rotation group  T, [3,3]^{+}, (332), order 12 
Dual polyhedron  Pseudoicosahedron 
Properties  face transitive 
Net 
A pyritohedron is a dodecahedron with pyritohedral (T_{h}) symmetry. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices (see figure).[1] However, the pentagons are not constrained to be regular, and the underlying atomic arrangement has no true fivefold symmetry axis. Its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes of rotational symmetry are three mutually perpendicular twofold axes and four threefold axes.
Although regular dodecahedra do not exist in crystals, the pyritohedron form occurs in the crystals of the mineral pyrite, and it may be an inspiration for the discovery of the regular Platonic solid form. The true regular dodecahedron can occur as a shape for quasicrystals (such as holmium–magnesium–zinc quasicrystal) with icosahedral symmetry, which includes true fivefold rotation axes.


Crystal pyrite
Its name comes from one of the two common crystal habits shown by pyrite, the other one being the cube.
Cubic pyrite 
Pyritohedral pyrite 
Cartesian coordinates
The coordinates of the eight vertices of the original cube are:
 (±1, ±1, ±1)
The coordinates of the 12 vertices of the crossedges are:
 (0, ±(1 + h), ±(1 − h^{2}))
 (±(1 + h), ±(1 − h^{2}), 0)
 (±(1 − h^{2}), 0, ±(1 + h))
where h is the height of the wedgeshaped "roof" above the faces of the cube. When h = 1, the six crossedges degenerate to points and a rhombic dodecahedron is formed. When h = 0, the crossedges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h = −1 + √5/2, the multiplicative inverse of the golden ratio, the result is a regular dodecahedron. When h = −1 − √5/2, the conjugate of this value, the result is a regular great stellated dodecahedron.
A reflected pyritohedron is made by swapping the nonzero coordinates above. The two pyritohedra can be superimposed to give the compound of two dodecahedra. The image to the left shows the case where the pyritohedra are convex regular dodecahedra.
Geometric freedom
The pyritohedron has a geometric degree of freedom with limiting cases of a cubic convex hull at one limit of collinear edges, and a rhombic dodecahedron as the other limit as 6 edges are degenerated to length zero. The regular dodecahedron represents a special intermediate case where all edges and angles are equal.
It is possible to go past these limiting cases, creating concave or nonconvex pyritohedra. The endododecahedron is concave and equilateral; it can tessellate space with the convex regular dodecahedron. Continuing from there in that direction, we pass through a degenerate case where twelve vertices coincide in the centre, and on to the regular great stellated dodecahedron where all edges and angles are equal again, and the faces have been distorted into regular pentagrams. On the other side, past the rhombic dodecahedron, we get a nonconvex equilateral dodecahedron with fishshaped selfintersecting equilateral pentagonal faces.
1 : 1  0 : 1  1 : 1  2 : 1  1 : 1  0 : 1  1 : 1 

h = −√5 + 1/2  h = −1  h = −√5 + 1/2  h = 0  h = √5 − 1/2  h = 1  h = √5 + 1/2 
Regular star, great stellated dodecahedron, with regular pentagram faces 
Degenerate, 12 vertices in the center 
The concave equilateral dodecahedron, called an endododecahedron. 
A cube can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions. 
A regular dodecahedron is an intermediate case with equal edge lengths. 
A rhombic dodecahedron is a degenerate case with the 6 crossedges reduced to length zero. 
Selfintersecting equilateral dodecahedron 
Tetartoid
Tetartoid Tetragonal pentagonal dodecahedron  

Face polygon  irregular pentagon 
Conway notation  gT 
Faces  12 
Edges  30 (6+12+12) 
Vertices  20 (4+4+12) 
Symmetry group  T, [3,3]^{+}, (332), order 12 
Properties  convex, face transitive 
A tetartoid (also tetragonal pentagonal dodecahedron, pentagontritetrahedron, and tetrahedric pentagon dodecahedron) is a dodecahedron with chiral tetrahedral symmetry (T). Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes.
Although regular dodecahedra do not exist in crystals, the tetartoid form does. The name tetartoid comes from the Greek root for onefourth because it has one fourth of full octahedral symmetry, and half of pyritohedral symmetry.[2] The mineral cobaltite can have this symmetry form.[3]
cobaltite 

Its topology can be as a cube with square faces bisected into 2 rectangles like the pyritohedron, and then the bisection lines are slanted retaining 3fold rotation at the 8 corners.
Cartesian coordinates
The following points are vertices of a tetartoid pentagon under tetrahedral symmetry:
 (a, b, c); (−a, −b, c); (−n/d_{1}, −n/d_{1}, n/d_{1}); (−c, −a, b); (−n/d_{2}, n/d_{2}, n/d_{2}),
under the following conditions:[4]
 0 ≤ a ≤ b ≤ c,
 n = a^{2}c − bc^{2},
 d_{1} = a^{2} − ab + b^{2} + ac − 2bc,
 d_{2} = a^{2} + ab + b^{2} − ac − 2bc,
 nd_{1}d_{2} ≠ 0.
Variations
It can be seen as a tetrahedron, with edges divided into 3 segments, along with a center point of each triangular face. In Conway polyhedron notation it can be seen as gT, a gyro tetrahedron.
Dual of triangular gyrobianticupola
A lower symmetry form of the regular dodecahedron can be constructed as the dual of a polyhedra constructed from two triangular anticupola connected basetobase, called a triangular gyrobianticupola. It has D_{3d} symmetry, order 12. It has 2 sets of 3 identical pentagons on the top and bottom, connected 6 pentagons around the sides which alternate upwards and downwards. This form has a hexagonal crosssection and identical copies can be connected as a partial hexagonal honeycomb, but all vertices will not match.
Rhombic dodecahedron
The rhombic dodecahedron is a zonohedron with twelve rhombic faces and octahedral symmetry. It is dual to the quasiregular cuboctahedron (an Archimedean solid) and occurs in nature as a crystal form. The rhombic dodecahedron packs together to fill space.
The rhombic dodecahedron can be seen as a degenerate pyritohedron where the 6 special edges have been reduced to zero length, reducing the pentagons into rhombic faces.
The rhombic dodecahedron has several stellations, the first of which is also a parallelohedral spacefiller.
Another important rhombic dodecahedron, the Bilinski dodecahedron, has twelve faces congruent to those of the rhombic triacontahedron, i.e. the diagonals are in the ratio of the golden ratio. It is also a zonohedron and was described by Bilinski in 1960.[5] This figure is another spacefiller, and can also occur in nonperiodic spacefillings along with the rhombic triacontahedron, the rhombic icosahedron and rhombic hexahedra.[6]
Other dodecahedra
There are 6,384,634 topologically distinct convex dodecahedra, excluding mirror images—the number of vertices ranges from 8 to 20.[7] (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)
Topologically distinct dodecahedra (excluding pentagonal and rhombic forms)
 Uniform polyhedra:
 Decagonal prism – 10 squares, 2 decagons, D_{10h} symmetry, order 40.
 Pentagonal antiprism – 10 equilateral triangles, 2 pentagons, D_{5d} symmetry, order 20
 Johnson solids (regular faced):
 Pentagonal cupola – 5 triangles, 5 squares, 1 pentagon, 1 decagon, C_{5v} symmetry, order 10
 Snub disphenoid – 12 triangles, D_{2d}, order 8
 Elongated square dipyramid – 8 triangles and 4 squares, D_{4h} symmetry, order 16
 Metabidiminished icosahedron – 10 triangles and 2 pentagons, C_{2v} symmetry, order 4
 Congruent irregular faced: (facetransitive)
 Hexagonal bipyramid – 12 isosceles triangles, dual of hexagonal prism, D_{6h} symmetry, order 24
 Hexagonal trapezohedron – 12 kites, dual of hexagonal antiprism, D_{6d} symmetry, order 24
 Triakis tetrahedron – 12 isosceles triangles, dual of truncated tetrahedron, T_{d} symmetry, order 24
 Other less regular faced:
 Hendecagonal pyramid – 11 isosceles triangles and 1 regular hendecagon, C_{11v}, order 11
 Trapezorhombic dodecahedron – 6 rhombi, 6 trapezoids – dual of triangular orthobicupola, D_{3h} symmetry, order 12
 Rhombohexagonal dodecahedron or elongated Dodecahedron – 8 rhombi and 4 equilateral hexagons, D_{4h} symmetry, order 16
 Truncated pentagonal trapezohedron, D_{5d}, order 20, topologically equivalent to regular dodecahedron
See also
 120cell: a regular polychoron (4D polytope) whose surface consists of 120 dodecahedral cells.
 Pentakis dodecahedron
 Snub dodecahedron
 Truncated dodecahedron
 Roman dodecahedron
References
 Crystal Habit. Galleries.com. Retrieved on 20161202.
 Dutch, Steve. The 48 Special Crystal Forms Archived 20130918 at the Wayback Machine. Natural and Applied Sciences, University of WisconsinGreen Bay, U.S.
 Crystal Habit. Galleries.com. Retrieved on 20161202.
 The Tetartoid. Demonstrations.wolfram.com. Retrieved on 20161202.
 Hafner, I. and Zitko, T. Introduction to golden rhombic polyhedra. Faculty of Electrical Engineering, University of Ljubljana, Slovenia.
 Lord, E. A.; Ranganathan, S.; Kulkarni, U. D. (2000). "Tilings, coverings, clusters and quasicrystals". Curr. Sci. 78: 64–72.
 Counting polyhedra. Numericana.com (20011231). Retrieved on 20161202.
External links
Wikimedia Commons has media related to Polyhedra with 12 faces. 
 Weisstein, Eric W. "Dodecahedron". MathWorld.
 Weisstein, Eric W. "Elongated Dodecahedron". MathWorld.
 Weisstein, Eric W. "Pyritohedron". MathWorld.
 Plato's Fourth Solid and the "Pyritohedron", by Paul Stephenson, 1993, The Mathematical Gazette, Vol. 77, No. 479 (Jul., 1993), pp. 220–226
 THE GREEK ELEMENTS
 Stellation of Pyritohedron VRML models and animations of Pyritohedron and its stellations
 Klitzing, Richard. "3D convex uniform polyhedra o3o5x – doe".
 Editable printable net of a dodecahedron with interactive 3D view
 The Uniform Polyhedra
 Origami Polyhedra – Models made with Modular Origami
 Dodecahedron – 3D model that works in your browser
 Virtual Reality Polyhedra The Encyclopedia of Polyhedra
 Dodecahedra variations
 VRML models
 Regular dodecahedron regular
 Rhombic dodecahedron quasiregular
 Decagonal prism vertextransitive
 Pentagonal antiprism vertextransitive
 Hexagonal dipyramid facetransitive
 Triakis tetrahedron facetransitive
 hexagonal trapezohedron facetransitive
 Pentagonal cupola regular faces
 K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other SemiRegular Polyhedra
 Dodecahedron 3D Visualization
 Stella: Polyhedron Navigator: Software used to create some of the images on this page.
 How to make a dodecahedron from a Styrofoam cube
 Roman dodecahedrons: Mysterious objects that have been found across the territory of the Roman Empire