Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. It has twenty (20) vertices and thirty (30) edges. Its dual polyhedron is the icosahedron. To the ancient Greeks, the dodecahedron was a symbol of the universe. If one were to make every one of the Platonic solids with edges of the same length, the dodecahedron would be the largest.

Area and volume

The area A and the volume V of a regular dodecahedron of edge length a are:

$A=3{\sqrt {25+10{\sqrt {5}}}}a^{2}\approx 20.64572a^{2}$

$V={\frac {1}{4}}(15+7{\sqrt {5}})a^{3}\approx 7.66311896a^{3}$

Cartesian coordinates

The following Cartesian coordinates define the vertices of a dodecahedron centered at the origin:

(±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). The side length is 2/φ = √5−1. The containing sphere has a radius of √3.

The dihedral angle of a dodecahedron is 2arctan(φ) or approximately 116.565 degrees.

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 Kepler-Poinsot polyhedra.

A rectified dodecahedron forms an icosidodecahedron.

The regular dodecahedron has 120 symmetries, forming the group $A_{5}\times Z_{2}$.

Vertex arrangement

The dodecahedron shares its vertex arrangement with four nonconvex uniform polyhedrons and three uniform 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.

File:Great stellated dodecahedron.png Great stellated dodecahedron	File:Small ditrigonal icosidodecahedron.png Small ditrigonal icosidodecahedron	File:Ditrigonal dodecadodecahedron.png Ditrigonal dodecadodecahedron	File:Great ditrigonal icosidodecahedron.png Great ditrigonal icosidodecahedron
File:Compound of five cubes.png Compound of five cubes	File:Compound of five tetrahedra.png Compound of five tetrahedra	File:Compound of ten tetrahedra.png Compound of ten tetrahedra

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...).

Stellations

The 3 stellations of the dodecahedron are all regular (nonconvex) polyhedra: (Kepler-Poinsot polyhedra polyhedra)

	0	1	2	3
Stellation	File:Dodecahedron.png Dodecahedron	File:Small stellated dodecahedron.png Small stellated dodecahedron	File:Great dodecahedron.png Great dodecahedron	File:Great stellated dodecahedron.png Great stellated dodecahedron
Facet diagram	File:Zeroth stellation of dodecahedron facets.png	File:First stellation of dodecahedron facets.png	File:Second stellation of dodecahedron facets.png	File:Third stellation of dodecahedron facets.png

Other dodecahedra

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. The Platonic solid dodecahedron can be called a pentagonal dodecahedron or a regular dodecahedron to distinguish it. The pyritohedron is an irregular pentagonal dodecahedron.

Other dodecahedra include:

• Uniform polyhedra:
  1. Pentagonal antiprism - 10 equilateral triangles, 2 pentagons
  2. Decagonal prism - 10 squares, 2 decagons
• Johnson solids (regular faced):
  1. Pentagonal cupola - 5 triangles, 5 squares, 1 pentagon, 1 decagon
  2. Snub disphenoid - 12 triangles
  3. Elongated square dipyramid - 8 triangles and 4 squares
  4. Metabidiminished icosahedron - 10 triangles and 2 pentagons
• Congruent nonregular faced: (face-transitive)
  1. Hexagonal bipyramid - 12 isosceles triangles, dual of hexagonal prism
  2. Hexagonal trapezohedron - 12 kites, dual of hexagonal antiprism
  3. Triakis tetrahedron - 12 isosceles triangles, dual of truncated tetrahedron
  4. Rhombic dodecahedron (mentioned above) - 12 rhombi, dual of cuboctahedron
• Other nonregular faced:
  1. Hendecagonal pyramid - 11 isosceles triangles and 1 hendecagon
  2. Trapezo-rhombic dodecahedron - 6 rhombi, 6 trapezoids - dual of Triangular orthobicupola
  3. Rhombo-hexagonal dodecahedron or Elongated Dodecahedron - 8 rhombi and 4 equilateral hexagons.

See also

• Spinning dodecahedron
• Truncated dodecahedron
• Snub dodecahedron
• Pentakis dodecahedron
• Hamiltonian path
• 120-cell: a regular polychoron (4D polytope) whose surface consists of 120 dodecahedral cells.

External links

• Paper models of the dodecahedron
• The Uniform Polyhedra
• Dodecahedron calendar, and another Dodecahedron calendar
• Origami Polyhedra - Models made with Modular Origami
• Dodecahedron - 3-d model that works in your browser
• Paper Models of Polyhedra Many links
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
  • VRML models
  1. Regular dodecahedron regular
  2. Rhombic dodecahedron quasiregular
  3. Decagonal prism vertex-transitive
  4. Pentagonal antiprism vertex-transitive
  5. Hexagonal dipyramid face-transitive
  6. Triakis tetrahedron face-transitive
  7. hexagonal trapezohedron face-transitive
  8. Pentagonal cupola regular faces
• Template:MathWorld
• Template:MathWorld
• K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra