FACTOID # 15: A mere 0.8% of West Virginians were born in a foreign country.

 Home Encyclopedia Statistics States A-Z Flags Maps FAQ About

 WHAT'S NEW RELATED ARTICLES People who viewed "Dodecahedron" also viewed:

SEARCH ALL

Search encyclopedia, statistics and forums:

(* = Graphable)

Encyclopedia > Dodecahedron
Regular Dodecahedron

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

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[1] 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. ...

 Great stellated dodecahedron Small ditrigonal icosidodecahedron Ditrigonal dodecadodecahedron Great ditrigonal icosidodecahedron Five cubes Five tetrahedra Ten tetrahedra

### 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.[1]
• 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

Roman dodecahedron
• 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. ([1] Image of background)

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. ...

## References

1. ^ Klein, Douglas J. (2002). "Resistance-Distance Sum Rules" (PDF). Croatica Chemica Acta 75 (2): 633–649. Retrieved on 2006-09-30.

For the Manfred Mann album, see 2006 (album). ... September 30 is the 273rd day of the year (274th in leap years) in the Gregorian calendar. ...

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 »

Share your thoughts, questions and commentary here