Dodecahedron
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A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. It has twenty vertices and thirty edges. Its dual polyhedron is the icosahedron.
Area and volume
The area A and the volume V of a regular dodecahedron of edge length a are:
- [A=3\sqrta^2]
- [V=\begin\end(15+7\sqrt5)a^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)
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.
Five cubes can be made from these, 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.
The stellations of the dodecahedron make up three of the four Kepler-Poinsot solids.
Icosahedron vs dodecahedron
Despite appearances, 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 edges 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 normal dodecahedron is sometimes called the pentagonal dodecahedron to distinguish it.
Other dodecahedra include:
- Congruent faces: (Face-uniform)
- # 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
- Mixed faces:
- # hendecagonal pyramid - 11 isosceles triangles and 1 hendecagon
- # pentagonal antiprism - 10 equilateral triangles, 2 pentagons
- # decagonal prism - 10 squares, 2 decagons
- # pentagonal cupola - 5 triangles, 5 squares, 1 pentagon, 1 decagon
- # trapezo-rhombic dodecahedron - 6 rhombi, 6 trapezoids - dual of Triangular orthobicupola
- # rhombo-hexagonal dodecahedron or Elongated Dodecahedron - 8 rhombi and 4 equilateral hexagons.
Uses
- If each edge of a dodecahedron is a one-ohm resistor, the resistance between adjacent vertices is 19/30 ohm, and that between opposite vertices is 7/6 ohm.
- 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.
Regular dodecahedra in the arts and sciences
- A dodecahedron sits on the table in M. C. Escher's lithograph print "Reptiles" (1943).
- In Salvador Dalí's painting of the Last Supper (1955), the room is a hollow dodecahedron.
- A Dodecahedron is a character in Norton Juster's book The Phantom Tollbooth (1961).
- The 20 vertices and 30 edges of a dodecahedron form the map for an early computer game, Hunt the Wumpus.
- The Dodecahedron was the mysterious power source for an underground city in the Doctor Who episode Meglos (1980).
- The shape of the Machine in Carl Sagan's novel Contact (1985) is a dodecahedron.
- Used as the special quantum object in Roger Penrose's book Shadows of the Mind (1994).
- In The Simpsons, Lisa tried to teach Maggie to say a number of words including "dodecahedron".
- Mentioned in the Clutch song "Mice & Gods".
- "Dodecahedron" is the title of a song by Aphex Twin.
- In the PC RPG "Planescape: Torment", the main character's previous incarnation recorded journal information within a "dodecahedron", which contained numerous traps and rubix-like points of articulation.
- 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 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.
See also
- [Spinning dodecahedron]
- Truncated dodecahedron
- Hamiltonian path
- Hunt the Wumpus
- Dodecaeder
- 120-cell: a regular polychoron (4D polytope) whose surface consists of 120 dodecahedral cells.
External links
- [The Uniform Polyhedra]
- [Spinning Dodecahedron] Math Is Fun
- [Dodecahedron calendar]
- [Paper Models of Polyhedra] Many links
- [12 sided calendar] Create, print and fold a 12 sided calendar as either a dodecahedron or a rhombic dodecahedron
- [Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
- * VRML models
- # [Regular dodecahedron] regular
- # [Rhombic dodecahedron] quasiregular
- # [Decagonal prism] vertex-uniform
- # [Pentagonal antiprism] vertex-uniform
- # [Hexagonal dipyramid] face uniform
- # [Triakis tetrahedron] face-uniform
- # [hexagonal trapezohedron] face-uniform
- # [Pentagonal cupola] regular faces
- , [Dodecahedron] at MathWorld.
- , [Elongated Dodecahedron] at MathWorld.
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- , [Elongated Dodecahedron] at MathWorld.