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Dodecahedron

A dodecahedron is any polyhedron Polyhedron

A polyhedron is a geometric shape which in mathematics [i] is defined by three related meanings. ... 

 with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid Platonic solid

In geometry [i], a Platonic solid is a convex [i] regular polyhedron [i]. ... 

 composed of twelve regular pentagon Pentagon

In geometry [i], a pentagon is any five-sided polygon [i]. ... 

al faces, with three meeting at each vertex. It has twenty vertices and thirty edges. Its dual polyhedron Dual polyhedron

In geometry [i], polyhedra [i] are associated into pairs called duals, where the vertices [i] ... 

 is the icosahedron Icosahedron

An icosahedron noun is ... 

.

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Encyclopedia

A dodecahedron is any polyhedron Polyhedron

A polyhedron is a geometric shape which in mathematics [i] is defined by three related meanings. ... 

 with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid Platonic solid

In geometry [i], a Platonic solid is a convex [i] regular polyhedron [i]. ... 

 composed of twelve regular pentagon Pentagon

In geometry [i], a pentagon is any five-sided polygon [i].
... 

al faces, with three meeting at each vertex. It has twenty vertices and thirty edges. Its dual polyhedron Dual polyhedron

In geometry [i], polyhedra [i] are associated into pairs called duals, where the vertices [i] ... 

 is the icosahedron Icosahedron

An icosahedron noun is
... 

.

Area and volume


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

Cartesian coordinates

The following Cartesian coordinates Cartesian coordinate system

In mathematics [i], the Cartesian coordinate system is used to uniquely determine each point [i]... 

 define the vertices of a dodecahedron centered at the origin:


where φ = /2 is the golden ratio Golden ratio

The golden ratio, usually denoted , expresses the relationship that the sum of two quantities is to the ... 

 . The side length is 2/φ = -1 + √5.

The dihedral angle Dihedral angle

In geometry [i], the angle [i] between two plane [i]s is called their dihedral angle.
... 

 of a dodecahedron is 2arctan or approximately 116.565 degrees.

Geometric relations


The regular dodecahedron is the third in an infinite set of truncated trapezohedra Truncated trapezohedron

An n-agonal truncated trapezohedron is a polyhedron [i] formed by a n-agonal trapezohedron [i] w ... 

 which can be constructed by truncating the two axial vertices of a pentagonal trapezohedron Pentagonal trapezohedron

The pentagonal trapezohedron [i] or deltohedron is the third in an infinite series of face-uniform ... 

.

Five cube Cube

A cube is a three-dimensional [i] Platonic solid [i] composed of six square [i] ... 

s can be made from these, with their edges as diagonals of the dodecahedron's faces, and together these make up the regular polyhedral compound Polyhedral compound

A polyhedral compound is a polyhedron [i] which is itself composed of several other polyhedra sharing a ... 

 of five cubes. Since two tetrahedra can fit on alternate cube vertices, five and ten tetrahedra can also fit in a dodecahedron.





The stellation Stellation

Stellation is a process of constructing new polygon [i]s, new polyhedra [i] in three dimensions, or, in ... 

s of the dodecahedron make up three of the four Kepler-Poinsot solid Kepler-Poinsot solid

A Kepler-Poinsot solid is a regular non-convex polyhedron [i], all the faces of which are identical regu ... 

s.

Icosahedron vs dodecahedron


Despite appearances, when a dodecahedron is inscribed in a sphere Sphere

A sphere is a perfectly symmetrical [i] geometrical [i] object. ... 

, it occupies more of the sphere's volume than an icosahedron Icosahedron

An icosahedron noun is
... 

 inscribed in the same sphere .

A regular dodecahedron with edges length 1 has more than three and a half times the volume of an icosahedron Icosahedron

An icosahedron noun is
... 

 with the same length edges .

Other dodecahedra


The term dodecahedron is also used for other polyhedra Polyhedron

A polyhedron is a geometric shape which in mathematics [i] is defined by three related meanings. ... 

 with twelve faces, most notably the rhombic dodecahedron Rhombic dodecahedron

The rhombic dodecahedron is a convex [i] polyhedron [i] with 12 rhombic [i] faces. ... 

 which is dual to the cuboctahedron Cuboctahedron

A cuboctahedron is a polyhedron [i] with eight triangular faces and six square faces.... 

  and occurs in nature as a crystal form. The Platonic solid Platonic solid

In geometry [i], a Platonic solid is a convex [i] regular polyhedron [i]. ... 

 dodecahedron can be called a pentagonal dodecahedron or a regular dodecahedron to distinguish it.

Other dodecahedra include:
  • Uniform polyhedra Uniform polyhedron

    A uniform [i] polyhedron is a polyhedron [i] with regular polygon [i]s as faces and ide... 

    :
    1. Pentagonal antiprism - 10 equilateral triangles, 2 pentagons
    2. Decagonal prism - 10 squares, 2 decagons
  • Johnson solid Johnson solid

    In geometry [i], a Johnson solid is a convex [i] polyhedron [i], each face of which is a regular polygon [i] ... 

    s :
    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 Metabidiminished icosahedron

      In geometry [i], the Metabidiminished icosahedron is one of the Johnson solid [i]s.


... 

 - 10 triangles and 2 pentagons
  • Congruent nonregular faced:
    1. Hexagonal bipyramid - 12 isosceles triangle Triangle

      A triangle is one of the basic shape [i]s of geometry [i]: a polygon [i] with three vertices [i] ... 

      s, dual of hexagonal prism
    2. Hexagonal trapezohedron Hexagonal trapezohedron

      The hexagonal trapezohedron [i] or deltohedron is the fourth in an infinite series of face-uniform ... 

       - 12 kite Kite

      A kite is a flying tethered man-made object.... 

      s, dual of hexagonal antiprism
    3. Triakis tetrahedron Triakis tetrahedron

      A triakis tetrahedron is a catalan solid [i] which looks a bit like an overinflated tetrahedron [i] and ... 

       - 12 isosceles triangle Triangle

      A triangle is one of the basic shape [i]s of geometry [i]: a polygon [i] with three vertices [i] ... 

      s, dual of truncated tetrahedron Truncated tetrahedron

      The truncated tetrahedron is an Archimedean solid [i]. ... 

    4. Rhombic dodecahedron Rhombic dodecahedron

      The rhombic dodecahedron is a convex [i] polyhedron [i] with 12 rhombic [i] faces. ... 

        - 12 rhombi Rhombus

      In geometry [i], a rhombus is a quadrilateral [i] in which all of the sides are of equal length, i.e., i ... 

      , dual of cuboctahedron Cuboctahedron

      A cuboctahedron is a polyhedron [i] with eight triangular faces and six square faces.... 


  • Other nonregular faced:
    1. Hendecagon Hendecagon

      In geometry [i], a hendecagon is an 11-sided polygon [i].


... 

al pyramid Pyramid

Pyramids are among the largest man-made constructions as well as one of the great Wonders of the ancient world... 

 - 11 isosceles triangles and 1 hendecagon Polygon

A polygon is a closed [i] planar [i] path composed of a finite number of sequential ... 


  1. Trapezo-rhombic dodecahedron Trapezo-rhombic dodecahedron

    [i] with 6 [[rhombus|rhombic]... 

     - 6 rhombi, 6 trapezoid Trapezoid

    A trapezoid or trapezium is a quadrilateral [i] two of whose sides are parallel [i] to ea ... 

    s - dual of Triangular orthobicupola
  2. Rhombo-hexagonal dodecahedron Rhombo-hexagonal dodecahedron

    [i] with 8 rhombic and 4 equilateral [[hexagon]... 

     or Elongated Dodecahedron - 8 rhombi and 4 equilateral hexagon Hexagon

    In geometry [i], a hexagon is a polygon [i] with six edge [i]s and six vertices [i]. ... 

    s.

Uses

  • If each edge of a dodecahedron is a one-ohm resistor Resistor

    |- align = "center"

|
|width = "25"|
... 

, 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 , one of the more common polyhedral dice Dice

    A die is a small polyhedral [i] object, usually cubical, used for generating random number [i] ... 

    .
  • One of the main characters in the movie and book of The Phantom Tollbooth Is a talking dodecahedron with arms and legs and has 12 different "faces".

Regular dodecahedra in the arts and sciences

  • A dodecahedron sits on the table in M. C. Escher M. C. Escher

    Maurits Cornelis Escher was a Dutch [i] graphic artist [i] known for his often ... 

    's lithograph print "Reptiles" , and a stellated dodecahedron is used in his "Gravitation Gravitation

    In physics [i], gravitation or gravity is the tendency of objects with mass [i] to accelerate [i] ... 

    ".
  • In Salvador Dalí Salvador Dalí

    Salvador Felip Jacint Dal Domnech, Marquess of Pubol , known popularly as Salvador Dal, was a Catalan [i] ... 

    's painting of the Last Supper Last Supper

    According to gospel [i], the Last Supper was the last meal Jesus [i] shared with his apostles [i] before ... 

     , the room is a hollow dodecahedron.
  • The 20 vertices and 30 edges of a dodecahedron form the map Graph theory

    In mathematics [i] and computer science [i], graph theory is the study of graphs [i], mathema ... 

     for an early computer game, Hunt the Wumpus.
  • The Dodecahedron was the mysterious power source for an underground city in the Doctor Who Doctor Who

    Doctor Who is a long-running British [i] science fiction television [i] ... 

     episode Meglos .
  • "Dodecahedron" is the title of a song by Aphex Twin Aphex Twin

    Aphex Twin is an electronic music [i] artist, credited with pushing forward the genres of techno [i] ... 

    .
  • In 2003, an apparent periodicity in the cosmic microwave background Cosmic microwave background radiation

    In cosmology [i], the cosmic microwave background radiation is a form of electromagnetic radiation [i] ... 

     led to the suggestion, by of the Observatoire de Paris Paris Observatory

    The Paris Observatory is the foremost astronomical [i] observatory of France [i], and one of t... 

     and colleagues, that the shape of the Universe Shape of the Universe

    The shape of the Universe is a subject of investigation within physical cosmology [i].... 

     is a finite dodecahedron, attached to itself by each pair of opposite faces to form a Poincaré sphere. During the following year, astronomers searched for more evidence to support this hypothesis but found none.

See also

  • Truncated dodecahedron Truncated dodecahedron

    The truncated dodecahedron is an Archimedean solid [i]. ... 

  • Hamiltonian path Hamiltonian path

    In the mathematical [i] field of graph theory [i], a Hamiltonian path is a path [i] ... 

  • Hunt the Wumpus
  • Dodecaeder
  • 120-cell 120-cell

    [i]

[i]
... 

: a regular polychoron Convex regular 4-polytope

In mathematics [i], a convex regular 4-polytope is 4-dimensional polytope [i] which is both regular [i] ... 

  whose surface consists of 120 dodecahedral cells.

External links

  • Math Is Fun
  • Many links
  • Create, print and fold a 12 sided calendar as either a dodecahedron or a rhombic dodecahedron Rhombic dodecahedron

    The rhombic dodecahedron is a convex [i] polyhedron [i] with 12 rhombic [i] faces. ... 

  • The Encyclopedia of Polyhedra
    • VRML models
      1. regular
      2. quasiregular
      3. vertex-uniform
      4. vertex-uniform
      5. face uniform
      6. face-uniform
      7. face-uniform
      8. regular faces