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A dodecahedron (Greek d?de??ed???, from d?de?a 'twelve' + ed??? 'base', 'seat' or 'face') is any polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
 composed of twelve regular pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
al faces, with three meeting at each vertex. It has twenty (20) vertices and thirty (30) edges. Its dual polyhedron
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 is the icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
. If one were to make every one of the Platonic solids with edges of the same length, the dodecahedron would be the largest.

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
Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller....
 = (1+v5)/2 is the golden ratio
Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller....
 (also written t).






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A dodecahedron (Greek d?de??ed???, from d?de?a 'twelve' + ed??? 'base', 'seat' or 'face') is any polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
 composed of twelve regular pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
al faces, with three meeting at each vertex. It has twenty (20) vertices and thirty (30) edges. Its dual polyhedron
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 is the icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
. 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 surface area A and the volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
 V of a regular dodecahedron of edge length a are:

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
Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller....
 = (1+v5)/2 is the golden ratio
Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller....
 (also written t). The edge length is 2/f = v5−1. The containing sphere has a radius of v3.

The dihedral angle
Dihedral angle

In geometry, the angle between two Plane s is called their dihedral or torsion angle.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection....
 of a dodecahedron is 2arctan(f) 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 formed by a n-agonal trapezohedron with n-agonal pyramids truncated from its 2 polar axis vertices....
 which can be constructed by truncating the two axial vertices of a pentagonal trapezohedron
Pentagonal trapezohedron

The pentagonal trapezohedron or deltohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedra to the antiprisms....
.

The stellation
Stellation

Stellation is a process of constructing new polygons , new polyhedron in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again....
s of the dodecahedron make up three of the four Kepler-Poinsot polyhedra.

A rectified
Rectification (geometry)

In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points....
 dodecahedron forms an icosidodecahedron
Icosidodecahedron

An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
.

The regular dodecahedron has 120 symmetries, forming the group .

Vertex arrangement


The dodecahedron shares its vertex arrangement
Vertex arrangement

In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes....
 with four nonconvex uniform polyhedron
Uniform polyhedron

A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
s and three uniform compounds
Polyhedral compound

A polyhedral compound is a polyhedron that is itself composed of several other polyhedra sharing a common centre. They are the three-dimensional analogs of star polygon#Star figuress such as the hexagram....
.

Five cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
s fit within, 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 that is itself composed of several other polyhedra sharing a common centre. They are the three-dimensional analogs of star polygon#Star figuress such as the hexagram....
 of five cubes. Since two tetrahedra can fit on alternate cube vertices, five and ten tetrahedra can also fit in a dodecahedron.

Great Stellated Dodecahedron

Great stellated dodecahedron
Great stellated dodecahedron

In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra.It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex....
Small Ditrigonal Icosidodecahedron

Small ditrigonal icosidodecahedron
Small ditrigonal icosidodecahedron

In geometry, the small ditrigonal icosidodecahedron is a nonconvex uniform polyhedron, indexed as U30.It shares the vertex arrangement with the regular dodecahedron....
Ditrigonal Dodecadodecahedron

Ditrigonal dodecadodecahedron
Ditrigonal dodecadodecahedron

In geometry, the Ditrigonal dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U41.It shares its vertex arrangement with the regular dodecahedron....
Great Ditrigonal Icosidodecahedron

Great ditrigonal icosidodecahedron
Great ditrigonal icosidodecahedron

In geometry, the great ditrigonal icosidodecahedron is a nonconvex uniform polyhedron, indexed as U47.It shares the vertex arrangement with the regular dodecahedron, which is therefore its convex hull....
Compound of Five Cubes

Compound of five cubes
Compound of five cubes

This polyhedral compound is a symmetric arrangement of five cubes. This compound was first described by Edmund Hess in 1876.It is one of five Polyhedral_compound#Regular_compounds, and dual to the compound of five octahedra....
Compound of Five Tetrahedra

Compound of five tetrahedra
Compound of five tetrahedra

This Polyhedron compound polyhedron is also a stellation of the regular icosahedron. It was first described by Edmund Hess in 1876....
Compound of Ten Tetrahedra

Compound of ten tetrahedra
Compound of ten tetrahedra

This polyhedron can be seen as either a polyhedral stellation or a Polyhedron compound. This compound was first described by Edmund Hess in 1876....


Icosahedron vs dodecahedron


When a dodecahedron is inscribed in a sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
, it occupies more of the sphere's volume (66.49%) than an icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
 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
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
 with the same length edges (7.663... compared with 2.181...).

Also, as these are duals, it is possible to transform one into the other.(See below)
Uniform Polyhedron 53 T0

Dodecahedron
Uniform Polyhedron 53 T01

Truncated dodecahedron
Truncated dodecahedron

In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangle faces, 60 vertices and 90 edges....
Uniform Polyhedron 53 T1

Icosidodecahedron
Icosidodecahedron

An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
Uniform Polyhedron 53 T12

Truncated icosahedron
Truncated icosahedron

The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....
Uniform Polyhedron 53 T2

Icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....


Stellations


The 3 stellation
Stellation

Stellation is a process of constructing new polygons , new polyhedron in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again....
s of the dodecahedron are all regular (nonconvex
List of regular polytopes

This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....
) polyhedra: (Kepler-Poinsot polyhedra)

0123
Stellation
Dodecahedron

Dodecahedron
Small Stellated Dodecahedron

Small stellated dodecahedron
Small stellated dodecahedron

In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex....
Great Dodecahedron

Great dodecahedron
Great dodecahedron

In geometry, the great dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces , with five pentagons meeting at each vertex, intersecting each other making a pentagrammic path....
Great Stellated Dodecahedron

Great stellated dodecahedron
Great stellated dodecahedron

In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra.It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex....
Facet diagram
Zeroth Stellation of Dodecahedron Facets
First Stellation of Dodecahedron Facets
Second Stellation of Dodecahedron Facets


Other dodecahedra


The term dodecahedron is also used for other polyhedra
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 with twelve faces, most notably the rhombic dodecahedron
Rhombic dodecahedron

The rhombic dodecahedron is a convex set polyhedron with 12 rhombus faces. It is an Archimedean solid solid, or a Catalan solid. Its dual is the cuboctahedron....
 which is dual to the cuboctahedron
Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
 (an Archimedean solid
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
) and occurs in nature as a crystal form. The Platonic solid
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
 dodecahedron can be called a pentagonal dodecahedron or a regular dodecahedron to distinguish it. The pyritohedron
Pyritohedron

In geometry, a pyritohedron is an irregular dodecahedron. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 corners....
 is an irregular pentagonal dodecahedron.

Other dodecahedra include:
  • Uniform polyhedra
    Uniform polyhedron

    A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
    :
    1. Pentagonal antiprism
      Pentagonal antiprism

      In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps....
       - 10 equilateral triangles, 2 pentagons
    2. Decagonal prism
      Decagonal prism

      In geometry, the decagonal prism is the eighth in an infinite set of Prism formed by square sides and two regular polygon caps.If faces are all regular, it is a semiregular polyhedron....
       - 10 squares, 2 decagons
  • Johnson solid
    Johnson solid

    In geometry, a Johnson solid is a strictly convex set polyhedron, each face of which is a regular polygon, but which is not uniform polyhedron, i.e., not a Platonic solid, Archimedean solid, prism or antiprism....
    s (regular faced):
    1. Pentagonal cupola
      Pentagonal cupola

      In geometry, the pentagonal cupola is one of the Johnson solids . It can be obtained as a slice of the rhombicosidodecahedron.The 92 Johnson solids were named and described by Norman Johnson in 1966....
       - 5 triangles, 5 squares, 1 pentagon, 1 decagon
    2. Snub disphenoid
      Snub disphenoid

      In geometry, the snub disphenoid is one of the Johnson solids . It is a three-dimensional solid that has only equilateral triangles as faces, and is therefore a deltahedron....
       - 12 triangles
    3. Elongated square dipyramid
      Elongated square dipyramid

      In geometry, the elongated square dipyramid is one of the Johnson solids . As the name suggests, it can be constructed by elongating an octahedron by inserting a square prism between its congruent halves....
       - 8 triangles and 4 squares
    4. Metabidiminished icosahedron
      Metabidiminished icosahedron

      In geometry, the metabidiminished icosahedron is one of the Johnson solids .The 92 Johnson solids were named and described by Norman Johnson in 1966....
       - 10 triangles and 2 pentagons
  • Congruent nonregular faced: (face-transitive)
    1. Hexagonal bipyramid
      Hexagonal bipyramid

      A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramid joined at their bases. The resulting solid has 12 triangular face , 8 vertex and 18 edges....
       - 12 isosceles triangle
      Triangle

      A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
      s, dual of hexagonal prism
      Hexagonal prism

      In geometry, the hexagonal prism is a Prism with hexagonal base.It is an octahedron. However, the term octahedron is mainly used with "regular" in front or implied, hence not meaning a hexagonal prism; in the general meaning the term octahedron it is not much used because there are different types which have not much in common exce...
    2. Hexagonal trapezohedron
      Hexagonal trapezohedron

      The hexagonal trapezohedron or deltohedron is the fourth in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms....
       - 12 kite
      Kite (geometry)

      In geometry a kite, or deltoid, is a quadrilateral with two disjoint sets pairs of congruent adjacent sides, in contrast to a parallelogram, where the congruent sides are opposite....
      s, dual of hexagonal antiprism
      Hexagonal antiprism

      In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by 2 polygon caps....
    3. Triakis tetrahedron
      Triakis tetrahedron

      A triakis tetrahedron is an Archimedean solid solid, or a Catalan solid. Its dual is the truncated tetrahedron.It can be seen as a tetrahedron with Tetrahedron added to each face....
       - 12 isosceles triangle
      Triangle

      A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
      s, dual of truncated tetrahedron
      Truncated tetrahedron

      The truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangle faces, 12 vertices and 18 edges....
    4. Rhombic dodecahedron
      Rhombic dodecahedron

      The rhombic dodecahedron is a convex set polyhedron with 12 rhombus faces. It is an Archimedean solid solid, or a Catalan solid. Its dual is the cuboctahedron....
       (mentioned above) - 12 rhombi
      Rhombus

      In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length....
      , dual of cuboctahedron
      Cuboctahedron

      In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
  • Other nonregular faced:
    1. Hendecagon
      Hendecagon

      In geometry, a hendecagon is an 11-sided polygon.The name "undecagon" is often seen as incorrect, but the matter is up for debate. The Greek language prefix 'hen', is preferable to the Latin 'uni' or 'un' ....
      al pyramid
      Pyramid (geometry)

      In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex . Each base edge and apex form a triangle....
       - 11 isosceles triangles and 1 hendecagon
      Polygon

      In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
    2. Trapezo-rhombic dodecahedron
      Trapezo-rhombic dodecahedron

      The Trapezo-rhombic dodecahedron is a convex polygon polyhedron with 6 rhombus and 6 trapezoid faces.This shape could be constructed by taking a tall uniform hexagonal prism, and making 3 angled cuts on the top and bottom....
       - 6 rhombi, 6 trapezoid
      Trapezoid

      In geometry, a trapezoid or trapezium is a quadrilateral with twoparallel sides. The term “trapezoid” is used in North America, while the term “trapezium” is prevalent in Britain....
      s - dual of Triangular orthobicupola
      Triangular orthobicupola

      In geometry, the triangular orthobicupola is one of the Johnson solids . As the name suggests, it can be constructed by attaching two triangular cupolas along their bases....
    3. Rhombo-hexagonal dodecahedron
      Rhombo-hexagonal dodecahedron

      The rhombo-hexagonal dodecahedron is a convex polyhedron with 8 rhombic and 4 equilateral hexagonal faces.It is also called an elongated dodecahedron and extended rhombic dodecahedron because it is related to the rhombic dodecahedron by expanding four rhombic faces of the rhombic dodecahedron into hexagons....
       or Elongated Dodecahedron - 8 rhombi and 4 equilateral hexagon
      Hexagon

      In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
      s.


In all there are 6,384,634 topologically distinct dodecahedra.

History and uses


Dodecahedral objects have found some practical applications, and have also played a role in the visual arts and in philosophy.

Plato
Plato

Plato , was a Classical Greece Greeks philosopher, mathematician, writer of philosophical dialogues, and founder of the Platonic Academy in Ancient Athens, the first institution of higher learning in the western world....
's dialogue Timaeus
Timaeus

Timaeus is a Greek name, meaning "Honour". It may refer to:*Timaeus , a Socratic dialogue by Plato*Timaeus of Locri, the 5th-century Pythagorean philosopher, appearing in Plato's dialogue...
 (c. 360 B.C.) associates the other four platonic solid
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s with the four classical element
Classical element

Many ancient philosophy used a set of archetype classical elements to explain patterns in nature. In this context, the word element refers to a chemical substance that is either a chemical compound or a mixture of chemical compounds , rather than a chemical element of modern physical science....
s; Aristotle
Aristotle

Aristotle was a Greeks philosopher, a student of Plato and teacher of Alexander the Great. He wrote on many subjects, including physics, metaphysics, Poetics , theater, music, logic, rhetoric, politics, government, ethics, biology and zoology....
 postulated that the heavens were made of a fifth element, aithęr
Aether (classical element)

According to ancient and History of science in the Middle Ages, aether , also spelled ?ther or ether, is the material that fills the region of the Universe above the Sublunary sphere....
 (aether in Latin, ether in American English), but he had no interest in matching it with Plato's fifth solid.

A few centuries later, small, hollow bronze Roman dodecahedra were made and have been found in various Roman ruins in Europe. Their purpose is not certain.

In twentieth century art, dodecahedra appear in the work of M. C. Escher
M. C. Escher

Maurits Cornelis Escher , usually referred to as M.C. Escher , was a Netherlands Graphic arts. He is known for his often mathematically-inspired woodcuts, lithography, and mezzotints....
, such as his lithograph Reptiles
Reptiles (M. C. Escher)

Reptiles is a Lithography printmaking by the Netherlands artist M. C. Escher which was first printed in March, 1943.It depicts a desk, on which is a drawing of a Tessellation pattern of reptiles....
 (1943), and in his Gravitation
Gravitation (M. C. Escher)

Gravitation is a mixed media work by the Netherlands artist M. C. Escher which was completed in June, 1952. It was first printed as a black-and-white lithograph and then coloured by hand in watercolour....
. In Salvador Dalí
Salvador Dalí

Salvador Domingo Felipe Jacinto Dal? i Dom?nech, 1st Marquis of P?bol was a Spain Catalonia surrealist painter born in Figueres.Dal? was a skilled Technical drawing, best known for the striking and bizarre images in his surrealism work....
's painting The Sacrament of the Last Supper
The Sacrament of the Last Supper

Completed in 1955 after nine months of work, Salvador Dal?s painting The Sacrament of the Last Supper has remained one of his most popular compositions....
 (1955), the room is a hollow dodecahedron.

In modern role-playing games, the dodecahedron is often used as a twelve-sided die
Dice

A die is a small polyhedron object, usually cubic, used for generating Statistical randomnesss or other symbols. This makes dice suitable as gambling devices, especially for craps or sic bo, or for use in non-gambling tabletop games....
, one of the more common polyhedral dice
Dice

A die is a small polyhedron object, usually cubic, used for generating Statistical randomnesss or other symbols. This makes dice suitable as gambling devices, especially for craps or sic bo, or for use in non-gambling tabletop games....
.

See also

  • Spinning dodecahedron
  • Truncated dodecahedron
    Truncated dodecahedron

    In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangle faces, 60 vertices and 90 edges....
  • Snub dodecahedron
    Snub dodecahedron

    The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid.The snub dodecahedron has 92 faces, of which 12 are pentagons and the other 80 are equilateral triangles....
  • Pentakis dodecahedron
    Pentakis dodecahedron

    A pentakis dodecahedron is a Catalan solid. Its dual is the truncated icosahedron, an Archimedean solid.It can be seen as a dodecahedron with a pentagonal pyramid covering each face....
  • Hamiltonian path
    Hamiltonian path

    In the mathematics field of graph theory, a Hamiltonian path is a path_ in an undirected graph which visits each vertex_ exactly once. A Hamiltonian cycle is a cycle_ in an undirected graph which visits each vertex_ exactly once and also returns to the starting vertex....
  • 120-cell
    120-cell

    In geometry, the 120-cell is the convex regular 4-polytope with Schl?fli symbol .The boundary of the 120-cell is composed of 120 dodecahedral cell with 4 meeting at each vertex....
    : a regular polychoron
    Convex regular 4-polytope

    In mathematics, a convex regular 4-polytope is 4-dimensional polytope which is both regular polytope and convex set. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
     (4D polytope) whose surface consists of 120 dodecahedral cells.


External links

  • - Models made with Modular Origami
  • - 3-d model that works in your browser
  • The Encyclopedia of Polyhedra
    • VRML
      VRML

      VRML is a standard file format for representing 3-D computer graphics interactive vector graphics, designed particularly with the World Wide Web in mind....
       models
      1. regular
      2. quasiregular
      3. vertex-transitive
      4. vertex-transitive
      5. face-transitive
      6. face-transitive
      7. face-transitive
      8. regular faces