# Dodecahedron

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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:

- <math>A = 3\sqrt{25+10\sqrt{5}} a^2 \approx 20.64572a^2</math>

- <math>V = \frac{1}{4} (15+7\sqrt{5}) a^3 \approx 7.66311896a^3</math>

## 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 <math>A_5\times Z_2</math>.

### 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.

### 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...).

## 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:
- Pentagonal antiprism - 10 equilateral triangles, 2 pentagons
- Decagonal prism - 10 squares, 2 decagons

- Johnson solids (regular faced):
- Pentagonal cupola - 5 triangles, 5 squares, 1 pentagon, 1 decagon
- Snub disphenoid - 12 triangles
- Elongated square dipyramid - 8 triangles and 4 squares
- Metabidiminished icosahedron - 10 triangles and 2 pentagons

- Congruent nonregular faced: (face-transitive)
- Hexagonal bipyramid - 12 isosceles triangles, dual of hexagonal prism
- Hexagonal trapezohedron - 12 kites, dual of hexagonal antiprism
- Triakis tetrahedron - 12 isosceles triangles, dual of truncated tetrahedron
- Rhombic dodecahedron (mentioned above) - 12 rhombi, dual of cuboctahedron

- Other nonregular faced:
- Hendecagonal pyramid - 11 isosceles triangles and 1 hendecagon
- Trapezo-rhombic dodecahedron - 6 rhombi, 6 trapezoids - dual of Triangular orthobicupola
- Rhombo-hexagonal dodecahedron or
*Elongated Dodecahedron*- 8 rhombi and 4 equilateral hexagons.

## Regular dodecahedra in the arts, sciences, and popular culture

Small, hollow bronze Roman dodecahedra dating from the 3rd century A.D. have been found in various places in Europe. Their purpose is not certain.

A dodecahedron sits on the table in M. C. Escher's lithograph print "Reptiles" (1943), and a stellated dodecahedron is used in his "Gravitation". In Salvador Dalí's painting of *The Sacrament of the Last Supper* (1955), the room is a hollow dodecahedron.

One of the characters in The Phantom Tollbooth, a children's novel from 1961, is named Dodecahedron and is a man with 12 faces.

The Dodecahedron was the mysterious power source for an underground city in the *Doctor Who* episode "Meglos" (1980).

In "Blood Feud", an episode of *The Simpsons*, Lisa attempts to teach Maggie the word *dodecahedron*.

In Carl Sagan's novel *Contact*, the transport device constructed to the plans transmitted by the alien intelligence is dodecahedral.

"Dodecaheedron" (misspelled, possibly intentionally, with an extra "e") is the title of a song by Aphex Twin.

Plato in the dialogue Timaeus c.360 B.C associated platonic solids with the four classical elements. Aristotle added a fifth element, aithêr (aether in Latin, "ether" in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid.

In 2003, an apparent periodicity in the cosmic microwave background led to the suggestion, by Jean-Pierre Luminet of the Observatoire de Paris and colleagues, that the shape of the Universe is a finite dodecahedron, attached to itself by each pair of opposite faces to form a Poincaré homology sphere. ("Is the universe a dodecahedron?", article at PhysicsWeb.) During the following year, astronomers searched for more evidence to support this hypothesis but found none.

The 20 vertices and 30 edges of a dodecahedron form the map for an early computer game, *Hunt the Wumpus*. In the seminal 1980s computer game Elite, the more advanced "Dodec" class space stations took the form of dodecahedra. The save points in the Castlevania games, Castlevania: Symphony of the Night (for the PSX) and Castlevania: Harmony of Dissonance (GBA) are shaped like dodecahedrons. In the Nintendo 64 game Paper Mario, the mountains in the background of Toad Town are dodecahedra. ([1] Image of background)

The regular dodecahedron is often used in role-playing games as a twelve-sided die ("d12" for short), one of the more common polyhedral dice.

Desk calendars are occasionally made in the shape of a dodecahedron, usually from a die-cut folded card, with one month on each face.

## 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.

## References

## External links

Wikimedia Commons has media related to .Dodecahedra |

- 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

- Regular dodecahedron regular
- Rhombic dodecahedron quasiregular
- Decagonal prism vertex-transitive
- Pentagonal antiprism vertex-transitive
- Hexagonal dipyramid face-transitive
- Triakis tetrahedron face-transitive
- hexagonal trapezohedron face-transitive
- 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

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