Dodecahedron
Regular Dodecahedron  

(Click here for rotating model) 

Type  Platonic solid 
Elements  F = 12, E = 30 V = 20 (χ = 2) 
Faces by sides  12{5} 
Schläfli symbols  {5,3} 
Wythoff symbol  3  2 5 
Coxeter diagram  
Symmetry  I_{h}, H_{3}, [5,3], (*532) 
Rotation group  I, [5,3]^{+}, (532) 
References  U_{23}, C_{26}, W_{5} 
Properties  Regular convex 
Dihedral angle  116.56505° = arccos(1/√5) 
5.5.5 (Vertex figure) 
Icosahedron (dual polyhedron) 
Net 
In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα, dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces, but usually a regular dodecahedron is meant: a Platonic solid. It is composed of 12 regular pentagonal faces, with three meeting at each vertex, and is represented by the Schläfli symbol {5,3}. It has 20 vertices, 30 edges and 160 diagonals. Its dual polyhedron is the icosahedron, with Schläfli symbol {3,5}.
A large number of other (irregular) polyhedra also have twelve faces, most notably the topologically identical pyritohedron with Pyritohedral symmetry, and the rhombic dodecahedron with octahedral symmetry.
Contents
Dimensions
If the edge length of a regular dodecahedron is a, the radius of a circumscribed sphere (one that touches the dodecahedron at all vertices) is
and the radius of an inscribed sphere (tangent to each of the dodecahedron's faces) is
while the midradius, which touches the middle of each edge, is
These quantities may also be expressed as
where φ is the golden ratio.
Note that, given a regular pentagonal dodecahedron of edge length one, r_{u} is the radius of a circumscribing sphere about a cube of edge length φ, and r_{i} is the apothem of a regular pentagon of edge length φ.
Area and volume
The surface area A and the volume V of a regular dodecahedron of edge length a are:
Twodimensional symmetry projections
The dodecahedron has two special orthogonal projections, centered, on vertices and pentagonal faces, correspond to the A_{2} and H_{2} Coxeter planes.
Centered by  Vertex  Edge  Face 

Image  
Projective symmetry 
[[3]] = [6]  [2]  [[5]] = [10] 
In perspective projection, viewed above a pentagonal face, the dodecahedron can be seen as a linearedged schlegel diagram, or stereographic projection as a spherical polyhedron. These projections are also used in showing the fourdimensional 120cell, a regular 4dimensional polytope, constructed from 120 dodecahedra, projecting it down to 3dimensions.
Projection  Orthogonal projection  Perspective projection  

Schlegel diagram  Stereographic projection  
Dodecahedron  
Dodecaplex 
Cartesian coordinates
The following Cartesian coordinates define the vertices of a dodecahedron centered at the origin and suitably scaled and oriented:^{1}
 (±1, ±1, ±1)
 (0, ±1/φ, ±φ)
 (±1/φ, ±φ, 0)
 (±φ, 0, ±1/φ)
where φ = (1 + √5) / 2 is the golden ratio (also written τ) ≈ 1.618. The edge length is 2/φ = √5 – 1. The containing sphere has a radius of √3.
Properties
 The dihedral angle of a dodecahedron is 2 arctan(φ) or approximately 116.5650512 degrees (where again φ = (1 + √5) / 2, the golden ratio). A137218
 If the original dodecahedron has edge length 1, its dual icosahedron has edge length φ.
 If the five Platonic solids are built with same volume, the dodecahedron has the shortest edges.
 It has 43,380 nets.
 The mapcoloring number of a regular dodecahedron's faces is 4.
 The distance between the vertices on the same face not connected by an edge is φ times the edge length,
Geometric relations
The regular dodecahedron is the third in an infinite set of truncated trapezohedra which can be constructed by truncating the two axial vertices of a pentagonal trapezohedron.
The stellations of the dodecahedron make up three of the four KeplerPoinsot polyhedra.
A rectified dodecahedron forms an icosidodecahedron.
The regular dodecahedron has icosahedral symmetry I_{h}, Coxeter group [5,3], order 120, with an abstract group structure of A_{5} × Z_{2}.
Icosahedron vs dodecahedron
When a dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.54%).
A regular dodecahedron with edge length 1 has more than three and a half times the volume of an icosahedron with the same length edges (7.663... compared with 2.181...).
A regular dodecahedron has 12 faces and 20 vertices, whereas a regular icosahedron has 20 faces and 12 vertices. Both have 30 edges.
Pyritohedron
Pyritohedron  

A pyritohedron has 30 edges, divided into two lengths: 24 and 6 in each group. 

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  convex 
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. However, the pentagons are not necessarily regular, so the structure normally has no fivefold symmetry axes. Its 30 edges are divided into two sets  containing 24 and 6 edges of the same length.
Although regular dodecahedra do not exist in crystals, the distorted, pyritohedron form occurs in the crystal pyrite, and it may be an inspiration for the discovery of the regular Platonic solid form.
Crystal pyrite
Its name comes from one of the two common crystal forms of pyrite, the other one being cubical.
Cubic pyrite 
Pyritohedral 
HoMgZn quasicrystal 
Cartesian coordinates
The coordinates of the 8 vertices:
 (±1, ±1, ±1)
The coordinates of the 12 vertices are the permutations of:
 (0, 1+h, 1−h^{2})
where h is the height of the wedge roof above the faces of the cube. When h=1, the 6 edges degenerate to points and rhombic dodecahedron is formed. For the regular dodecahedron, h=(√5−1)/2, the golden ratio.
Geometric freedom
The pyritohedron has a geometric degree of freedom with limiting cases of a cubic convex hull at one limit of colinear 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.
1 : 1  1 : 1  2 : 1  1.3092... : 1  1 : 1  0 : 1 

h=0  h=(√5−1)/2  h=1  
Regular star, great stellated dodecahedron, with pentagons distorted into regular pentagrams 
Concave pyritohedral dodecahedron 
A cube can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions. 
The geometric proportions of the pyritohedron in the Weaire–Phelan structure 
A regular dodecahedron is an intermediate case with equal edge lengths. 
A rhombic dodecahedron is the limiting case with the 6 crossedges reducing to length zero. 
A regular dodecahedron can be formed from a cube in the following way: The top square in the cube is replaced by a "roof" composed of two pentagons, joined along the top of the roof. The diagonals in the pentagons parallel to the top of the roof coincide with two opposite sides of the square. The other five squares are replaced by a pair of pentagons in a similar way. The pyritohedron is constructed by changing the slope of these "roofs".
Other dodecahedra
The truncated pentagonal trapezohedron has D_{5d} dihedral symmetry. 
The pyritohedron has T_{h} tetrahedral symmetry. 
The term dodecahedron is also used for other polyhedra with twelve faces, most notably the rhombic dodecahedron which is dual to the cuboctahedron (an Archimedean solid) and occurs in nature as a crystal form.^{3} The Platonic solid dodecahedron can be called a pentagonal dodecahedron or a regular dodecahedron to distinguish it. The pyritohedron is an irregular pentagonal dodecahedron.
There are 6,384,634 topologically distinct convex dodecahedra, excluding mirror images, having at least 8 vertices.^{4} (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 include:
 Pentagonal dodecahedra:
 Regular dodecahedron, 12 pentagonal faces, I_{h} symmetry, order 120
 Pentagonal truncated trapezohedron  same topology as regular form, but D_{5d} symmetry, order 20
 "Pyritohedron"  same topology as regular form, but T_{h} symmetry, order 12
 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
 Rhombic dodecahedron (mentioned above) – 12 rhombi, dual of cuboctahedron, O_{h} symmetry, order 48
 Other irregular faced:
 Hendecagonal pyramid – 11 isosceles triangles and 1 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
History and uses
Dodecahedral objects have found some practical applications, and have also played a role in the visual arts and in philosophy.
The dodecahedron was the last of the Platonic solids to be discovered. Iamblichus states that Hippasus, a Pythagorean, perished in the sea, because he boasted that he first divulged "the sphere with the twelve pentagons." ^{5} Plato's dialogue Timaeus (c. 360 B.C.) associates the other four platonic solids with the four classical elements, adding that there is a fifth figure (which is made out of twelve pentagons), the dodecahedron—"this God used in the delineation of the universe."^{6} Aristotle postulated that the heavens were made of a fifth element, aithêr (aether in Latin, ether in American English), but he had no interest in matching it with Plato's fifth solid.
A few centuries later, small, hollow bronze Roman dodecahedra were made and have been found in various Roman ruins in Europe. Their purpose is not certain.
In 20thcentury art, dodecahedra appear in the work of M.C. Escher, such as his lithograph Reptiles (1943), and in his Gravitation. In Salvador Dalí's painting The Sacrament of the Last Supper (1955), the room is a hollow dodecahedron.
In modern roleplaying games, the dodecahedron is often used as a twelvesided die, one of the more common polyhedral dice. Some quasicrystals have dodecahedral shape (see figure). Some regular crystals such as garnet and diamond are also said to exhibit "dodecahedral" habit, but this statement actually refers to the rhombic dodecahedron shape.^{3}
Immersive media, a Camera manufacturing company, has made the Dodeca 2360 camera, the world's first 360°, full motion camera which captures highresolution video from every direction simultaneously at more than 100 million pixels per second or 30 frames per second. It is based on dodecahedron.
The popular puzzle game Megaminx is in the shape of a dodecahedron.
In the children's novel The Phantom Tollbooth, the Dodecahedron appears as a character in the land of Mathematics. Each of his faces wears a different expression—e.g. happy, angry, sad—which he swivels to the front as required to match his mood.
Dodecahedron is the name of avantgarde black metal band from Netherlands.^{7}
Shape of the universe
Various models have been proposed for the global geometry of the universe. In addition to the primitive geometries, these proposals include the Poincaré dodecahedral space, a positively curved space consisting of a dodecahedron whose opposite faces correspond (with a small twist). This was proposed by JeanPierre Luminet and colleagues in 2003^{8}^{9} and an optimal orientation on the sky for the model was estimated in 2008.^{10}
In Bertrand Russell's 1954 short story "THE MATHEMATICIAN'S NIGHTMARE: The Vision of Professor Squarepunt," the number 5 said: "I am the number of fingers on a hand. I make pentagons and pentagrams. And but for me dodecahedra could not exist; and, as everyone knows, the universe is a dodecahedron. So, but for me, there could be no universe."
As a graph
The skeleton of the dodecahedron—the vertices and edges—form a graph. This graph can also be constructed as the generalized Petersen graph G(10, 2). The high degree of symmetry of the polygon is replicated in the properties of this graph, which is distancetransitive, distanceregular, and symmetric. The automorphism group has order 120. The vertices can be colored with 3 colors, as can the edges, and the diameter is 5.^{11}
The dodecahedral graph is Hamiltonian—there is a cycle containing all the vertices. Indeed, this name derives from a mathematical game invented in 1857 by William Rowan Hamilton, the icosian game. The game's object was to find a Hamiltonian cycle along the edges of a dodecahedron.
Related polyhedra and tilings
The regular dodecahedron is topologically related to a series of tilings by vertex figure n^{3}.
Spherical Polyhedra 
Polyhedra  Euclidean  Hyperbolic tilings  

{2,3} 
{3,3} 
{4,3} 
{5,3} 
{6,3} 
{7,3} 
{8,3} 
...  (∞,3} 
The dodecahedron can be transformed by a truncation sequence into its dual, the icosahedron:
Symmetry: [5,3], (*532)  [5,3]^{+}, (532)  

{5,3}  t{5,3}  r{5,3}  2t{5,3}=t{3,5}  2r{5,3}={3,5}  rr{5,3}  tr{5,3}  sr{5,3} 
Duals to uniform polyhedra  
V5.5.5  V3.10.10  V3.5.3.5  V5.6.6  V3.3.3.3.3  V3.4.5.4  V4.6.10  V3.3.3.3.5 
Symmetry: [4,3], (*432)  [4,3]^{+} (432) 
[1^{+},4,3] = [3,3] (*332) 
[3^{+},4] (3*2) 


{4,3}  t{4,3}  r{4,3} r{3^{1,1}} 
t{3,4} t{3^{1,1}} 
{3,4} {3^{1,1}} 
rr{4,3} s_{2}{3,4} 
tr{4,3}  sr{4,3}  h{4,3} {3,3} 
h_{2}{4,3} t{3,3} 
s{4,3} s{3^{1,1}} 
= 
= 
= 
= or 
= or 
= 

Duals to uniform polyhedra  
V4^{3}  V3.8^{2}  V(3.4)^{2}  V4.6^{2}  V3^{4}  V3.4^{3}  V4.6.8  V3^{4}.4  V3^{3}  V3.6^{2}  V3^{5} 
The regular dodecahedron is a member of a sequence of otherwise nonuniform polyhedra and tilings, composed of pentagons with face configurations (V3.3.3.3.n). (For n > 6, the sequence consists of tilings of the hyperbolic plane.) These facetransitive figures have (n32) rotational symmetry.
Symmetry n32 [n,3]^{+} 
Spherical  Euclidean  Compact hyperbolic  Paracompact  

232 [2,3]^{+} D_{3} 
332 [3,3]^{+} T 
432 [4,3]^{+} O 
532 [5,3]^{+} I 
632 [6,3]^{+} P6 
732 [7,3]^{+} 
832 [8,3]^{+}... 
∞32 [∞,3]^{+} 

Snub figure 
3.3.3.3.2 
3.3.3.3.3 
3.3.3.3.4 
3.3.3.3.5 
3.3.3.3.6 
3.3.3.3.7 
3.3.3.3.8 
3.3.3.3.∞ 
Coxeter Schläfli 
sr{2,3} 
sr{3,3} 
sr{4,3} 
sr{5,3} 
sr{6,3} 
sr{7,3} 
sr{8,3} 
sr{∞,3} 
Snub dual figure 
V3.3.3.3.2 
V3.3.3.3.3 
V3.3.3.3.4 
V3.3.3.3.5 
V3.3.3.3.6 
V3.3.3.3.7 
V3.3.3.3.8  V3.3.3.3.∞ 
Coxeter 
Vertex arrangement
The dodecahedron shares its vertex arrangement with four nonconvex uniform polyhedra and three uniform polyhedron compounds.
Five cubes fit within, with their edges as diagonals of the dodecahedron's faces, and together these make up the regular polyhedral compound of five cubes. Since two tetrahedra can fit on alternate cube vertices, five and ten tetrahedra can also fit in a dodecahedron.
Stellations
The 3 stellations of the dodecahedron are all regular (nonconvex) polyhedra: (Kepler–Poinsot polyhedra)
0  1  2  3  

Stellation  Dodecahedron 
Small stellated dodecahedron 
Great dodecahedron 
Great stellated dodecahedron 
Facet diagram 
See also
 120cell: a regular polychoron (4D polytope) whose surface consists of 120 dodecahedral cells.
 Pentakis dodecahedron
 Snub dodecahedron
 Truncated dodecahedron
References
 ^ Weisstein, Eric W., "Icosahedral group", MathWorld.
 ^ Polyhedra are "topologically identical" if they have the same intrinsic arrangement of faces and vertices, such that one can be distorted into the other simply by changing the lengths of edges or the angles between edges or faces.
 ^ ^{a} ^{b} Dodecahedral Crystal Habit
 ^ Counting polyhedra
 ^ Florian Cajori, A History of Mathematics (1893)
 ^ Plato, Timaeus, Jowett translation [line 13178]; the Greek word translated as delineation is 'diazographein', painting in semblance of life.
 ^ "Dodecahedron on Metal Archives".
 ^ "Is the universe a dodecahedron?", article at PhysicsWeb.
 ^ Luminet, JeanPierre; Jeff Weeks, Alain Riazuelo, Roland Lehoucq, JeanPhillipe Uzan (20031009). "Dodecahedral space topology as an explanation for weak wideangle temperature correlations in the cosmic microwave background". Nature 425 (6958): 593–5. arXiv:astroph/0310253. Bibcode:2003Natur.425..593L. doi:10.1038/nature01944. PMID 14534579.
 ^ Roukema, Boudewijn; Zbigniew Buliński; Agnieszka Szaniewska; Nicolas E. Gaudin (2008). "A test of the Poincaré dodecahedral space topology hypothesis with the WMAP CMB data". Astronomy and Astrophysics 482 (3): 747. arXiv:0801.0006. Bibcode:2008A&A...482..747L. doi:10.1051/00046361:20078777.
 ^ Weisstein, Eric W., "Dodecahedral Graph", MathWorld.
 Plato's Fourth Solid and the "Pyritohedron", by Paul Stephenson, 1993, The Mathematical Gazette, Vol. 77, No. 479 (Jul., 1993), pp. 220–226 [1]
 THE GREEK ELEMENTS
External links
Wikimedia Commons has media related to Dodecahedron. 
 Weisstein, Eric W., "Dodecahedron", MathWorld.
 Weisstein, Eric W., "Elongated Dodecahedron", MathWorld.
 Weisstein, Eric W., "Pyritohedron", MathWorld.
 Stellation of Pyritohedron VRML models and animations of Pyritohedron and its stellations.
 Richard Klitzing, 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
 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.


Fundamental convex regular and uniform polytopes in dimensions 2–10  

Family  A_{n}  BC_{n}  I_{2}(p) / D_{n}  E_{6} / E_{7} / E_{8} / F_{4} / G_{2}  H_{n}  
Regular polygon  Triangle  Square  pgon  Hexagon  Pentagon  
Uniform polyhedron  Tetrahedron  Octahedron • Cube  Demicube  Dodecahedron • Icosahedron  
Uniform polychoron  5cell  16cell • Tesseract  Demitesseract  24cell  120cell • 600cell  
Uniform 5polytope  5simplex  5orthoplex • 5cube  5demicube  
Uniform 6polytope  6simplex  6orthoplex • 6cube  6demicube  1_{22} • 2_{21}  
Uniform 7polytope  7simplex  7orthoplex • 7cube  7demicube  1_{32} • 2_{31} • 3_{21}  
Uniform 8polytope  8simplex  8orthoplex • 8cube  8demicube  1_{42} • 2_{41} • 4_{21}  
Uniform 9polytope  9simplex  9orthoplex • 9cube  9demicube  
Uniform 10polytope  10simplex  10orthoplex • 10cube  10demicube  
Uniform npolytope  nsimplex  northoplex • ncube  ndemicube  1_{k2} • 2_{k1} • k_{21}  npentagonal polytope  
Topics: Polytope families • Regular polytope • List of regular polytopes 
HPTS  Area Progetti  EduSoft  JavaEdu  N.Saperi  Ass.Scuola..  TS BCTV  TS VideoRes  TSODP  TRTWE  