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Encyclopedia > Dodecahedron
Regular Dodecahedron

(Click here for rotating model)
Type Platonic solid
Elements F = 12, E = 30, V = 20 (χ = 2)
Faces by sides 12{5}
Schläfli symbol {5,3}
Wythoff symbol 3 | 2 5
Coxeter-Dynkin
Symmetry Ih
References U23, C26, W5
Properties Regular convex
Dihedral angle 116.56505° = arccos(-1/√5) 5.5.5
(Vertex figure)

Icosahedron
(dual polyhedron) Net

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. Image File history File links This is a lossless scalable vector image. ... Spinning dodecahedron, first 17 frames accidentally rendered with flashiness=1, while last 43 are rendered with flashiness=0. ... In geometry, a Platonic solid is a convex regular polyhedron. ... It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ... In mathematics, the SchlÃ¤fli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, a Wythoff construction, named after mathematician Willem Abraham Wijthoff, is a method for constructing a uniform polyhedron or plane tiling. ... Coxeter groups in the plane with equivalent diagrams. ... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... // List of symmetry groups on the sphere Spherical symmetry groups are also called point groups (in 3D). ... The icosahedral rotation group I with fundamental domain Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... H.S.M. Coxeter. ... This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus J. Wenninger. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... Look up Convex set in Wiktionary, the free dictionary. ... In Aerospace engineering, the dihedral is the angle that the two wings make with each other. ... Image File history File links Dodecahedron_vertfig. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ... Image File history File links Size of this preview: 600 Ã— 600 pixel Image in higher resolution (1000 Ã— 1000 pixel, file size: 252 KB, MIME type: image/png)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... Image File history File links Dodecahedron_flat. ... Categories: Polyhedra | Stub ... A polyhedron (plural polyhedra or polyhedrons) is a geometric object with flat faces and straight edges. ... In geometry, a Platonic solid is a convex regular polyhedron. ... Look up pentagon in Wiktionary, the free dictionary. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ...

## Area and volume GA_googleFillSlot("encyclopedia_square");

The area A and the volume V of a regular dodecahedron of edge length a are: The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ... $A = (3sqrt{25+10sqrt{5}}) a^2 approx 20.64572888a^2$ $V = frac{1}{4} (15+7sqrt{5}) a^3 approx 7.66311896a^3$

## Cartesian coordinates

The following Cartesian coordinates define the vertices of a dodecahedron centered at the origin: Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

(±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 = −1+√5. // Articles with similar titles include Golden mean (philosophy), the felicitous middle between two extremes, and Golden numbers, an indicator of years in astronomy and calendar studies. ... // Articles with similar titles include Golden mean (philosophy), the felicitous middle between two extremes, and Golden numbers, an indicator of years in astronomy and calendar studies. ...

The dihedral angle of a dodecahedron is 2arctan(φ) or approximately 116.565 degrees. In Aerospace engineering, the dihedral is the angle that the two wings make with each other. ...

## 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. An n-agonal truncated trapezohedron is a polyhedron formed by a n-agonal trapezohedron with n-agonal pyramids truncated from its two polar axis vertices. ... The pentagonal trapezohedron or deltohedron is the third in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms. ...

The stellations of the dodecahedron make up three of the four Kepler-Poinsot polyhedra. Stellation is a process of constructing new polygons (in two dimensions), new polyhedra in three dimensions, or in general new polytopes in n dimensions. ... A single face is colored yellow and outlined in red to help identify the faces. ...

### Vertex arrangement

The dodecahedron shares its vertex arrangement with four nonconvex uniform polyhedrons and three uniform compounds. A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... A polyhedral compound is a polyhedron which is itself composed of several other polyhedra sharing a common centre, the three-dimensional analogs of polygonal compounds such as the hexagram. ...

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. A cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... A polyhedral compound is a polyhedron which is itself composed of several other polyhedra sharing a common centre, the three-dimensional analogs of polygonal compounds such as the hexagram. ...

Image File history File links Download high resolution version (640x640, 17 KB) Summary Great stellated dodecahedron, U52 Licensing I, the creator of this work, hereby release it into the public domain. ... In geometry, the great stellated dodecahedron is a Kepler-Poinsot solid. ... Image File history File links Download high resolution version (641x640, 21 KB) Summary Small ditrigonal icosidodecahedron, U30 Licensing I, the creator of this work, hereby release it into the public domain. ... In geometry, the small ditrigonal icosidodecahedron is a nonconvex uniform polyhedron, indexed as U30. ... Image File history File links Download high resolution version (640x640, 24 KB) Summary Ditrigonal dodecadodecahedron, U41 Licensing I, the creator of this work, hereby release it into the public domain. ... In geometry, the Ditrigonal dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U41. ... Image File history File links Download high resolution version (640x640, 23 KB) Summary Image of Great Ditrigonal Icosidodecahedron, U47 Licensing I, the creator of this work, hereby release it into the public domain. ... In geometry, the great ditrigonal icosidodecahedron is a nonconvex uniform polyhedron, indexed as U47. ... Image File history File links Download high resolution version (782x782, 35 KB) Compound of five cubes External link VRML model Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Download high resolution version (868x802, 26 KB) Compound of five tetrahedra, Wenninger model 25 External link VRML model Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Download high resolution version (892x826, 42 KB) Compound of ten tetraahedra, Wenninger model 25 External link VRML model Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

### 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 sphere is a perfectly symmetrical geometrical object. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ...

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...). [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ...

## 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. A polyhedron (plural polyhedra or polyhedrons) is a geometric object with flat faces and straight edges. ... The rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... In geometry, a Platonic solid is a convex regular polyhedron. ... A pyritohedron is an irregular dodecahedron. ...

Other dodecahedra include:

A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... 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. ... In geometry, the decagonal prism is the eighth in an infinite set of prisms formed by square sides and two regular polygon caps. ... The elongated square gyrobicupola (J37), a Johnson solid This 24 square example is not a Johnson solid because it is not strictly convex (has zero-angled dihedral angles. ... In geometry, the Pentagonal cupola is one of the Johnson solids (J5). ... In geometry, the Snub disphenoid is one of the Johnson solids (J84). ... In geometry, the elongated square dipyramid is one of the Johnson solids (J15). ... In geometry, the Metabidiminished icosahedron is one of the Johnson solids (J62). ... In geometry, a polyhedron is isohedral or face-transitive when all its faces are the same. ... A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. ... A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ... In geometry, the hexagonal prism is a prism with hexagonal base. ... The hexagonal trapezohedron or deltohedron is the fourth in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms. ... A kite showing its equal sides and its inscribed circle. ... In geometry, the hexagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. ... A triakis tetrahedron is a catalan solid which looks a bit like an overinflated tetrahedron. ... A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ... The truncated tetrahedron is an Archimedean solid. ... The rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. ... For other uses of the word rhombus, see Rhombus (disambiguation) This shape is a rhombus In geometry, a rhombus (or rhomb; plural rhombi) is a quadrilateral in which all of the sides are of equal length, i. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Categories: Math stubs | Polygons ... This article is about the polyhedron pyramid (a 3-dimensional shape); for other versions including architectural Pyramids, see Pyramid (disambiguation). ... Look up polygon in Wiktionary, the free dictionary. ... The Trapezo-rhombic dodecahedron is a convex polyhedron with 6 rhombic and 6 trapezoidal faces. ... A trapezoid (in North America) or trapezium (in Britain and elsewhere) is a quadrilateral, which is defined as a shape with four sides, which has a pair of parallel sides. ... In geometry, the triangular orthobicupola is one of the Johnson solids (J27). ... The rhombo-hexagonal dodecahedron is a convex polyhedron with 8 rhombic and 4 equilateral hexagonal faces. ... A regular hexagon. ...

## Trivia

• 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.

The ohm (symbol: Î©) is the SI unit of electric resistance. ... Resistor symbols (non-European) Resistor symbols (Europe, IEC) Axial-lead resistors on tape. ... This article is about traditional role-playing games. ... Two standard six-sided pipped dice with rounded corners. ... Two standard six-sided pipped dice with rounded corners. ...

## 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.
• The 20 vertices and 30 edges of a dodecahedron form the map for an early computer game, Hunt the Wumpus.
• One of the characters in The Phantom Tollbooth, a childrens 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 the Carl Sagan book Contact, the transport device constructed to the plans transmitted by the alien intelligence is dodecahedral in shape.
• In the episode Blood Feud of The Simpsons, Lisa attempts to teach Maggie the word dodecahedron.
• "Dodecahedron" is the title of a song by Aphex Twin.
• 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.
• In the Nintendo 64 game Paper Mario, the mountains in the background of Toad Town are dodecahedra. ( Image of background)

Image File history File linksMetadata Size of this preview: 444 Ã— 600 pixelsFull resolution (783 Ã— 1058 pixel, file size: 168 KB, MIME type: image/jpeg) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File linksMetadata Size of this preview: 444 Ã— 600 pixelsFull resolution (783 Ã— 1058 pixel, file size: 168 KB, MIME type: image/jpeg) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Roman dodecahedron found in Germany A Roman dodecahedron is a small roughly-spherical hollow object made of bronze or stone, with a dodecahedral shape: twelve flat pentagonal faces, each having a circular hole in the middle which connects to the hollowed-out center. ... Maurits Cornelis Escher (June 17, 1898 â€“ March 27, 1972), usually referred to as M. C. Escher, was a Dutch graphic artist known for his often mathematically inspired woodcuts, lithographs and mezzotints which feature impossible constructions, explorations of infinity, architecture, and tessellations. ... Reptiles is a lithograph print by the Dutch artist M. C. Escher which was first printed in March, 1943. ... Gravitation (also known as Gravity) is a lithograph print by the Dutch artist M. C. Escher which was first printed in June, 1952. ... Salvador Felipe Jacinto DalÃ­ DomÃ¨nech Marquis of Pubol (May 11, 1904 â€“ January 23, 1989), popularly known as Salvador DalÃ­, was a Spanish artist and one of the most important painters of the 20th century. ... A pictorial representation of a graph In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. ... Hunt the Wumpus was an early computer game. ... The Phantom Tollbooth (1961) is a childrens book and a modern fairy tale full of wordplay and adventure. ... Doctor Who is a long-running British science fiction television programme (and a 1996 television movie) produced by the BBC. The programme shows the adventures of a mysterious time-traveller known as the Doctor, who explores time and space in his TARDIS time ship with his companions, solving problems and... Meglos is a serial in the British science fiction television series Doctor Who, which was first broadcast in four weekly parts from September 27 to October 18, 1980. ... Insert non-formatted text here Carl Edward Sagan (November 9, 1934 â€“ December 20, 1996) was an American astronomer and astrobiologist and a highly successful popularizer of astronomy, astrophysics, and other natural sciences. ... Look up Contact in Wiktionary, the free dictionary. ... Blood Feud is the last episode of the second season of The Simpsons. ... Simpsons redirects here. ... Lisa Marie Simpson is a character in the animated television series The Simpsons, voiced by Yeardley Smith. ... Margaret Maggie Simpson is a fictional character featured in the animated television series The Simpsons. ... Aphex Twin (born Richard David James on August 18, 1971 in Limerick, Ireland) is an electronic music artist, credited with pushing forward the genres of techno, ambient, acid, and drum and bass. ... Periodicity is the quality of occurring at regular intervals (e. ... WMAP image of the CMB anisotropy,Cosmic microwave background radiation(June 2003) The cosmic microwave background radiation (CMB) is a form of electromagnetic radiation that fills the whole of the universe. ... The Paris Observatory (in French, Observatoire de Paris or Observatoire de Paris-Meudon) is the foremost astronomical observatory of France, and one of the largest astronomical centers in the world. ... The shape of the Universe is an informal name for a subject of investigation within physical cosmology. ... In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n â‰¥ 1. ... Paper Mario, known in Japan as Mario Story ), is a role-playing video game developed by Intelligent Systems and published by Nintendo for the Nintendo 64. ...

## See also

Spinning dodecahedron, first 17 frames accidentally rendered with flashiness=1, while last 43 are rendered with flashiness=0. ... The truncated dodecahedron is an Archimedean solid. ... A pentakis dodecahedron is a catalan solid which looks a bit like an overinflated dodecahedron. ... An hamiltonian path over a graph. ... In geometry, the 120-cell (or hecatonicosachoron) is the convex regular 4-polytope with SchlÃ¤fli symbol {5,3,3}. It is sometimes thought of as the 4-dimensional analog of the dodecahedron. ... In mathematics, a convex regular 4-polytope (or polychoron) is 4-dimensional polytope which is both a regular and convex. ... Results from FactBites:

 Dodecahedron - LoveToKnow 1911 (167 words) The "small stellated dodecahedron," the "great dodecahedron" and the "great stellated dodecahedron" are Kepler-Poinsot solids; and the "truncated" and "snub dodecahedra" are Archimedean solids (see Polyhedron). In crystallography, the regular or ordinary dodecahedron is an impossible form since the faces cut the axes in irrational ratios; the "pentagonal dodecahedron" of crystallographers has irregular pentagons for faces, while the geometrical solid, on the other hand, has regular ones. The "rhombic dodecahedron," one of the geometrical semiregular solids, is an important crystal form.
More results at FactBites »

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