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Animation of a net of a regular (pentagonal) dodecahedron being folded

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. The pyritohedron is an irregular pentagonal dodecahedron, having the same topology as the regular one but pyritohedral symmetry instead. The rhombic dodecahedron has octahedral symmetry. There are a large number of other dodecahedra.

Pentagonal dodecahedron[edit]

Topologically equivalent dodecahedra
Regular dodecahedron
Ih, order 120
Th, order 24.
T, order 12.
Main article: Regular dodecahedron





Face polygon irregular pentagon
Faces 12
Edges 30 (6+24)
Vertices 20 (8+12)
Dual polyhedron Pseudoicosahedron
Properties face transitive
Pyritohedron flat.png



Crystal pyrite[edit]


Cubic pyrite
Pyrite cristal.jpg
Ho-Mg-Zn quasicrystal

Cartesian coordinates[edit]

Pyritohedron animation.gif

The coordinates of the eight vertices of the original cube are:

(±1, ±1, ±1)



where h is the height of the wedge-shaped "roof" above the faces of the cube. When h = 1, the six cross-edges degenerate to points and a rhombic dodecahedron is formed. When h = 0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h = (√5 − 1)/2, the inverse of the golden ratio, the result is a regular dodecahedron.


Geometric freedom[edit]


Special cases of the pyritohedron
h = - (√5 + 1)/2 h = 0 h = (√5 − 1)/2 h = 1
Great stellated dodecahedron.png
Concave pyritohedral dodecahedron.png
Concave pyritohedral dodecahedron
Pyritohedron cube.png
Irregular dodecahedron.png
The geometric proportions of the pyritohedron in the Weaire–Phelan structure
A regular dodecahedron is an intermediate case with equal edge lengths.


Tetragonal pentagonal dodecahedron
Face polygon irregular pentagon
Conway polyhedron notation gT
Faces 12
Edges 30 (6+12+12)
Vertices 20 (4+4+12)
Properties convex, face transitive


Although regular dodecahedra do not exist in crystals, the tetartoid form occurs in the crystals. The name tetartoid comes from the Greek root for one-fourth because it has one forth of full octahedral symmetry, and half that of pyritohedral symmetry.[1] The mineral cobaltite can have this symmetry form.[2]



Cartesian coordinates[edit]

To generate a tetartoid pentagon under the tetrahedral group, choose numbers 0≤a≤b≤c.[3]

Calculate n = a2 c - b c2, d1 = a2 - a b + b2 + a c - 2 b c , d2 = a2 + a b + b2 - a c - 2 b c . If n × d1 × d2 ≠ 0, then the following are vertices of a tetartoid pentagon:

((a, b, c), (-a, -b, c), (-n, -n, n)/d1, (-c, -a, b), (-n, n, n)/d2)



Tetartoid cubic.pngTetartoid tetrahedral.png
Example tetartoid variations

Rhombic dodecahedron[edit]

Rhombic dodecahedron



The rhombic dodecahedron has several stellations, the first of which is also a spacefiller.


Other dodecahedra[edit]


Topologically distinct dodecahedra (excluding pentagonal and rhombic forms)

Dodecahedral graph[edit]

Regular dodecahedron graph
Hamiltonian path.svg
A Hamiltonian cycle in a dodecahedron.
Vertices 20
Edges 30
Radius 5
Diameter 5
Girth 5
Automorphisms 120 (S5)
Chromatic number 3
Properties Hamiltonian, regular, symmetric, distance-regular, distance-transitive, 3-vertex-connected, planar graph

The skeleton of the dodecahedron (the vertices and edges) form a graph. It is one of 5 Platonic graphs, each a skeleton of its Platonic solid.

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 distance-transitive, distance-regular, 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.[8]

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.

Orthogonal projection
Dodecahedron t0 H3.png

See also[edit]



External links[edit]