Dodecahedron Information
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 
In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") or duodecahedron^{ [1]} is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, 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.
Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry, while the tetartoid has tetrahedral symmetry.
The rhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it has octahedral symmetry. The elongated dodecahedron and trapezorhombic dodecahedron variations, along with the rhombic dodecahedra, are spacefilling. There are numerous other dodecahedra.
While the regular dodecahedron shares many features with other Platonic solids, one unique property of it is that one can start at a corner of the surface and draw an infinite number of straight lines across the figure that return to the original point without crossing over any other corner.^{ [2]}
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  

(See here for a rotating model.)  
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).^{ [3]} 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
The name crystal pyrite comes from one of the two common crystal habits shown by pyrite (the other one being the cube). In pyritohedral pyrite, the faces have a Miller index of (210), which means that the dihedral angle is 2·arctan(2) ≈ 126.87° and each pentagonal face has one angle of approximately 121.6° in between two angles of approximately 106.6° and opposite two angles of approximately 102.6°. The following formulas show the measurements for the face of a perfect crystal (which is rarely found in nature).

Cartesian coordinates
The eight vertices of a cube have the coordinates (±1, ±1, ±1).
The coordinates of the 12 additional vertices are (0, ±(1 + h), ±(1 − h^{2})), (±(1 + h), ±(1 − h^{2}), 0) and (±(1 − h^{2}), 0, ±(1 + h)).
h is the height of the wedgeshaped "roof" above the faces of that cube with edge length 2.
An important case is h = 1/2 (a quarter of the cube edge length) for perfect natural pyrite (also the pyritohedron in the Weaire–Phelan structure).
Another one is h = 1/ φ = 0.618... for the regular dodecahedron. See section Geometric freedom for other cases.
Two pyritohedra with swapped nonzero coordinates are in dual positions to each other like the dodecahedra in the compound of two dodecahedra.


Animations  

Honeycomb of alternating convex and concave pyritohedra with heights between ±1/ φ  Heights between 0 (cube) and 1 (rhombic dodecahedron) 
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.
Special cases of the pyritohedron  

Versions with equal absolute values and opposing signs form a honeycomb together. (Compare
this animation.) The ratio shown is that of edge lengths, namely those in a set of 24 (touching cube vertices) to those in a set of 6 (corresponding to cube faces).  
Ratio  1 : 1  0 : 1  1 : 1  2 : 1  1 : 1  0 : 1  1 : 1 
h  −√5 + 1/2  −1  −√5 + 1/2  0  √5 − 1/2  1  √5 + 1/2 
−1.618...  −0.618...  0.618...  1.618...  
Image 
Regular star, great stellated dodecahedron, with regular pentagram faces 
Degenerate, 12 vertices in the center 
The concave equilateral dodecahedron, called an endododecahedron.^{[ clarification needed]} 
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  

(See here for a rotating model.)  
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.^{ [4]} The mineral cobaltite can have this symmetry form.^{ [5]}
Abstractions sharing the solid's topology and symmetry can be created from the cube and the tetrahedron. In the cube each face is bisected by a slanted edge. In the tetrahedron each edge is trisected, and each of the new vertices connected to a face center. (In Conway polyhedron notation this is a gyro tetrahedron.)


Relationship to the dyakis dodecahedron  

A tetartoid can be created by enlarging 12 of the 24 faces of a dyakis dodecahedron. (The tetartoid shown here is based on one that is itself created by enlarging 24 of the 48 faces of the disdyakis dodecahedron.)
The crystal model on the right shows a tetartoid created by enlarging the blue faces of the dyakis dodecahedral core. Therefore, the edges between the blue faces are covered by the red skeleton edges. 
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:^{ [6]}
 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.
Geometric freedom
The regular dodecahedron is a tetartoid with more than the required symmetry. The triakis tetrahedron is a degenerate case with 12 zerolength edges. (In terms of the colors used above this means, that the white vertices and green edges are absorbed by the green vertices.)
Tetartoid variations from regular dodecahedron to triakis 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.^{ [7]} This figure is another spacefiller, and can also occur in nonperiodic spacefillings along with the rhombic triacontahedron, the rhombic icosahedron and rhombic hexahedra.^{ [8]}
Other dodecahedra
There are 6,384,634 topologically distinct convex dodecahedra, excluding mirror images—the number of vertices ranges from 8 to 20.^{ [9]} (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
Practical usage
Armand Spitz used a dodecahedron as the "globe" equivalent for his Digital Dome planetarium projector.^{ [10]} based upon a suggestion from Albert Einstein.
See also
 120cell − A regular polychoron (4D polytope) whose surface consists of 120 dodecahedral cells.
 Braarudosphaera bigelowii − A dodecahedron shaped coccolithophore (a unicellular phytoplankton algae).
 Pentakis dodecahedron
 Roman dodecahedron
 Snub dodecahedron
 Truncated dodecahedron
References
 ^ 1908 Chambers's Twentieth Century Dictionary of the English Language, 1913 Webster's Revised Unabridged Dictionary
 ^ Athreya, Jayadev S.; Aulicino, David; Hooper, W. Patrick (May 27, 2020). "Platonic Solids and High Genus Covers of Lattice Surfaces". Experimental Mathematics: 1–31. arXiv: 1811.04131. doi: 10.1080/10586458.2020.1712564. S2CID 119318080.
 ^ 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.
 ^ Ley, Willy (February 1965). "Forerunners of the Planetarium". For Your Information. Galaxy Science Fiction. pp. 87–98.
External links
Wikimedia Commons has media related to Polyhedra with 12 faces. 
 Plato's Fourth Solid and the "Pyritohedron", by Paul Stephenson, 1993, The Mathematical Gazette, Vol. 77, No. 479 (Jul., 1993), pp. 220–226 [1]
 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
 Virtual Reality Polyhedra The Encyclopedia of Polyhedra
 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