A regular dodecahedron or pentagonal dodecahedronStrictly speaking, a pentagonal dodecahedron need not be composed of regular pentagons. The name "pentagonal dodecahedron" therefore covers a wider class of solids than just the Platonic solid, the regular dodecahedron. is a

dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...

composed of regular

pentagonal In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting ''regular pentagon'' (or ''star pentagon'') is cal ...

faces, three meeting at each vertex. It is an example of

Platonic solids In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edge ...

, described as cosmic stellation by

Plato Plato ( ; Greek language, Greek: , ; born BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...

in his dialogues, and it was used as part of

Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Sola ...

proposed by

Johannes Kepler Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best know ...

. However, the regular dodecahedron, including the other Platonic solids, has already been described by other philosophers since antiquity. The regular dodecahedron is a truncated trapezohedron because it is the result of truncating axial vertices of a

pentagonal trapezohedron In geometry, a pentagonal trapezohedron is the third in the infinite family of trapezohedra, face-transitive polyhedra. Its dual polyhedron is the pentagonal antiprism. As a decahedron it has ten faces which are congruent kites. It can be dec ...

. It is also a

Goldberg polyhedron In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (mathematician), Michael Goldberg (1902–1990 ...

because it is the initial polyhedron to construct new polyhedrons by the process of

chamfer A chamfer ( ) is a transitional edge between two faces of an object. Sometimes defined as a form of bevel, it is often created at a 45° angle between two adjoining right-angled faces. Chamfers are frequently used in machining, carpentry, fur ...

ing. It has a relation with other Platonic solids, one of them is the

regular icosahedron The regular icosahedron (or simply ''icosahedron'') is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with Regular polygon, regular faces to each of its pentagonal faces, or by putting ...

as its

dual polyhedron In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other ...

. Other new polyhedrons can be constructed by using regular dodecahedron. The regular dodecahedron's metric properties and construction are associated with the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...

. The regular dodecahedron is featured in some artistic and narrative works. Some toys and artifacts are also shaped like regular dodecahedra, including the

Roman dodecahedron A Roman dodecahedron or Gallo-Roman dodecahedron is a small hollow object made of copper alloy which has been Casting (metalworking), cast into a regular dodecahedron, regular dodecahedral shape with twelve flat pentagonal faces. Each face has a ...

. Regular dodecahedra can also be found in nature and supramolecules, as well as the shape of the universe. The

skeleton A skeleton is the structural frame that supports the body of most animals. There are several types of skeletons, including the exoskeleton, which is a rigid outer shell that holds up an organism's shape; the endoskeleton, a rigid internal fra ...

of a regular dodecahedron can be represented as the

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

called the dodecahedral graph, a

Platonic graph In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-con ...

. Its property of the

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

, a

path A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desir ...

visits all of its vertices exactly once, can be found in a toy called

icosian game The icosian game is a mathematical game invented in 1856 by Irish mathematician William Rowan Hamilton. It involves finding a Hamiltonian cycle on a dodecahedron, a polygon using edges of the dodecahedron that passes through all its vertex (geo ...

As a Platonic solid

Descriptions

The regular dodecahedron is a polyhedron with twelve pentagonal faces, thirty edges, and twenty vertices. It is one of the

Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...

s, a set of polyhedrons in which the faces are

regular polygons In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

that are

congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...

and the same number of faces meet at a vertex. This set of polyhedrons is named after

. In '' Theaetetus'', a dialogue of Plato, Plato hypothesized that the classical elements were made of the five uniform regular solids. Plato described the regular dodecahedron, obscurely remarked, "...the god used tfor arranging the constellations on the whole heaven". Timaeus, as a personage of Plato's dialogue, associates the other four Platonic solids—

regular tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

regular octahedron In geometry, a regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. An octahedron, more generally, can be any eight-sided polyh ...

, and

—with the four

classical element The classical elements typically refer to Earth (classical element), earth, Water (classical element), water, Air (classical element), air, Fire (classical element), fire, and (later) Aether (classical element), aether which were proposed to ...

s, adding that there is a fifth solid pattern which, though commonly associated with the regular dodecahedron, is never directly mentioned as such; "this God used in the delineation of the universe."

Aristotle Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...

also postulated that the heavens were made of a fifth element, which he called aithêr (''aether'' in Latin, ''ether'' in American English). Following its attribution with nature by Plato,

in his ''

Harmonices Mundi ''Harmonice Mundi'' (Latin: ''The Harmony of the World'', 1619) is a book by Johannes Kepler. In the work, written entirely in Latin, Kepler discusses harmony and congruence in geometrical forms and physical phenomena. The final section of t ...

'' sketched each of the Platonic solids, one of them is a regular dodecahedron. In his '' Mysterium Cosmographicum'', Kepler also proposed the

by using the Platonic solids setting into another one and separating them with six spheres resembling the six planets. The ordered solids started from the innermost to the outermost: regular octahedron, regular icosahedron, regular dodecahedron, regular tetrahedron, and cube. Many antiquity philosophers described the regular dodecahedron, including the rest of the Platonic solids. Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist.

Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...

completely mathematically described the Platonic solids in the ''Elements'', the last book (Book XIII) of which is devoted to their properties. Propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid, Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. In Proposition 18 he argues that there are no further convex regular polyhedra.

Iamblichus Iamblichus ( ; ; ; ) was a Neoplatonist philosopher who determined a direction later taken by Neoplatonism. Iamblichus was also the biographer of the Greek mystic, philosopher, and mathematician Pythagoras. In addition to his philosophical co ...

states that

Hippasus Hippasus of Metapontum (; , ''Híppasos''; c. 530 – c. 450 BC) was a Greek philosopher and early follower of Pythagoras. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irra ...

, a Pythagorean, perished in the sea, because he boasted that he first divulged "the sphere with the twelve pentagons". The regular dodecahedron, as the family of Platonic solids, is a

regular polyhedron A regular polyhedron is a polyhedron whose symmetry group acts transitive group action, transitively on its Flag (geometry), flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In ...

. It is isogonal, isohedral, and

isotoxal In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given tw ...

: any two vertices, two faces, and two edges of a regular dodecahedron can be transformed by rotations and reflections under its symmetry orbit respectively, which preserves the appearance.

Relation to the regular icosahedron

The

of a dodecahedron is the

. One property of the dual polyhedron generally is that the original polyhedron and its dual share the same three-dimensional symmetry group. In the case of the regular dodecahedron, it has the same symmetry as the regular icosahedron, the

icosahedral symmetry In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual polyhedr ...

\mathrm_\mathrm

. The regular dodecahedron has ten three-fold axes passing through pairs of opposite vertices, six five-fold axes passing through the opposite faces centers, and fifteen two-fold axes passing through the opposite sides midpoints. When a regular dodecahedron is inscribed in a

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.55%). The resulting of both spheres' volumes initially began from the problem by ancient Greeks, determining which of two shapes has a larger volume: an icosahedron inscribed in a sphere, or a dodecahedron inscribed in the same sphere. The problem was solved by

Hero of Alexandria Hero of Alexandria (; , , also known as Heron of Alexandria ; probably 1st or 2nd century AD) was a Greek mathematician and engineer who was active in Alexandria in Egypt during the Roman era. He has been described as the greatest experimental ...

Pappus of Alexandria Pappus of Alexandria (; ; AD) was a Greek mathematics, Greek mathematician of late antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem in projective geometry. Almost nothing is known a ...

, and

Fibonacci Leonardo Bonacci ( – ), commonly known as Fibonacci, was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''Fibonacci ...

, among others.

Apollonius of Perga Apollonius of Perga ( ; ) was an ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the earlier contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention o ...

discovered the curious result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas. Both volumes have formulas involving the golden ratio but are taken to different powers.

Golden rectangle In geometry, a golden rectangle is a rectangle with side lengths in golden ratio \tfrac :1, or with approximately equal to or Golden rectangles exhibit a special form of self-similarity: if a square is added to the long side, or removed from ...

may also related to both regular icosahedron and regular dodecahedron. The regular icosahedron can be constructed by intersecting three golden rectangles perpendicularly, arranged in two-by-two orthogonal, and connecting each of the golden rectangle's vertices with a segment line. There are 12 regular icosahedron vertices, considered as the center of 12 regular dodecahedron faces.

Relation to the regular tetrahedron

As two opposing tetrahedra can be inscribed in a cube, and five cubes can be inscribed in a dodecahedron, ten tetrahedra in five cubes can be inscribed in a dodecahedron: two opposing sets of five, with each set covering all 20 vertices and each vertex in two tetrahedra (one from each set, but not the opposing pair). As stated by ,

Configuration matrix

The configuration matrix is a

matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...

in which the rows and columns correspond to the elements of a polyhedron as in the vertices, edges, and faces. The

diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek � ...

of a matrix denotes the number of each element that appears in a polyhedron, whereas the non-diagonal of a matrix denotes the number of the column's elements that occur in or at the row's element. The regular dodecahedron can be represented in the following matrix:

\begin
 20 & 3 & 3 \\
 2 & 30 & 2 \\
 5 & 5 & 12
\end

Relation to the golden ratio

The

is the ratio between two numbers equal to the ratio of their sum to the larger of the two quantities. It is one of two

root In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...

s of a polynomial, expressed as

\phi = (1 + \sqrt)/2 \approx 1.618

. The golden ratio can be applied to the regular dodecahedron's metric properties, as well as to construct the regular dodecahedron. The

surface area The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...

A

and the

volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...

V

of a regular dodecahedron of edge length

a

are:

A = 3\sqrta^2, \qquad V = \fraca^3.

The following

Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

define the twenty vertices of a regular dodecahedron centered at the origin and suitably scaled and oriented:

\begin
 (\pm 1, \pm 1, \pm 1), &\qquad (0, \pm \phi, \pm 1/\phi), \\
 (\pm 1/\phi, 0, \pm \phi), &\qquad (\pm \phi, \pm 1/\phi, 0).
\end

If the edge length of a regular dodecahedron is

a

, the

radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

of a

circumscribed sphere In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's Vertex (geometry), vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the te ...

r_c

(one that touches the regular dodecahedron at all vertices), the radius of an inscribed sphere

r_i

(

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

to each of the regular dodecahedron's faces), and the midradius

r_m

(one that touches the middle of each edge) are:

\begin
 r_c &= \frac a \approx 1.401a, \\
 r_i &= \sqrt a \approx 1.114a, \\
 r_m &= \fraca \approx 1.309a.
\end

Given a regular dodecahedron of edge length one,

r_c

is the radius of a circumscribing sphere about a

of edge length

\phi

, and

r_i

is the

apothem The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. T ...

of a regular pentagon of edge length

\phi

. The dihedral angle of a regular dodecahedron between every two adjacent pentagonal faces is

2 \arctan (\phi)

, approximately 116.565°.

Other related geometric objects

The regular dodecahedron can be interpreted as a truncated trapezohedron. It is the set of polyhedrons that can be constructed by truncating the two axial vertices of a

trapezohedron In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron Remarks: the faces of a deltohedron are deltoids; a (non-twisted) kite or deltoid can be Dissection (geometry), dissected into two isosceles triangle ...

. Here, the regular dodecahedron is constructed by truncating the pentagonal trapezohedron. The regular dodecahedron can be interpreted as the

. It is a set of polyhedrons containing hexagonal and pentagonal faces. Other than two Platonic solids—tetrahedron and cube—the regular dodecahedron is the initial of Goldberg polyhedron construction, and the next polyhedron is resulted by truncating all of its edges, a process called

. This process can be continuously repeated, resulting in more new Goldberg's polyhedrons. These polyhedrons are classified as the first class of a Goldberg polyhedron. The

stellation In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific ...

s of the regular dodecahedron make up three of the four Kepler–Poinsot polyhedra. The first stellation of a regular dodecahedron is constructed by attaching its layer with pentagonal pyramids, forming a

small stellated dodecahedron In geometry, the small stellated dodecahedron is a Kepler–Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex List of regular polytopes#Non-convex 2, regular polyhedra. It is composed of 12 pentag ...

. The second stellation is by attaching the small stellated dodecahedron with wedges, forming a

great dodecahedron In geometry, the great dodecahedron is one of four Kepler–Poinsot polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), intersecting each other making a pentagrammic path, with five pentagons meeting at each vert ...

. The third stellation is by attaching the great dodecahedron with the sharp triangular pyramids, forming a

great stellated dodecahedron In geometry, the great stellated dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at eac ...

Appearances

In arts and popular cultures

Regular dodecahedra have been used as dice and probably also as divinatory devices. During the

Hellenistic era In classical antiquity, the Hellenistic period covers the time in Greek history after Classical Greece, between the death of Alexander the Great in 323 BC and the death of Cleopatra VII in 30 BC, which was followed by the ascendancy of the Roma ...

, small hollow bronze Roman dodecahedra were made and have been found in various Roman ruins in Europe. Its purpose is not certain. In

20th-century art Twentieth-century art—and what it became as modern art—began with modernism in the late nineteenth century. Overview Nineteenth-century movements of Post-Impressionism (), Art Nouveau and Symbolism led to the first twentieth-century art mov ...

, regular dodecahedra appear in the work of

M. C. Escher Maurits Cornelis Escher (; ; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithography, lithographs, and mezzotints, many of which were Mathematics and art, inspired by mathematics. Despite wide popular int ...

, such as his lithographs ''

Reptiles Reptiles, as commonly defined, are a group of tetrapods with an ectothermic metabolism and Amniotic egg, amniotic development. Living traditional reptiles comprise four Order (biology), orders: Testudines, Crocodilia, Squamata, and Rhynchocepha ...

'', and

Salvador Dalí Salvador Domingo Felipe Jacinto Dalí i Domènech, Marquess of Dalí of Púbol (11 May 190423 January 1989), known as Salvador Dalí ( ; ; ), was a Spanish Surrealism, surrealist artist renowned for his technical skill, precise draftsmanship, ...

's painting '' The Sacrament of the Last Supper'' in which the room is a hollow regular dodecahedron). Gerard Caris based his entire artistic oeuvre on the regular dodecahedron and the pentagon, presented as a new art movement coined Pentagonism. In modern

role-playing games A role-playing game (sometimes spelled roleplaying game, or abbreviated as RPG) is a game in which players assume the roles of characters in a fictional setting. Players take responsibility for acting out these roles within a narrative, eith ...

, the regular dodecahedron is often used as a twelve-sided die, one of the more common polyhedral dice. The

Megaminx The Megaminx or Mégaminx (, ) is a dodecahedron-shaped puzzle similar to the Rubik's Cube. It has a total of 50 movable pieces to rearrange, compared to the 20 movable pieces of the Rubik's Cube. History The Megaminx, or Magic Dodecahedro ...

is a twisted puzzle similar to the Rubik's Cube but the shape is pentagonal faces dodecahedral. In the children's novel ''

The Phantom Tollbooth ''The Phantom Tollbooth'' is a children's fantasy adventure novel written by Norton Juster, with illustrations by Jules Feiffer, first published in 1961 in literature, 1961. The story follows a bored young boy named Milo who unexpectedly recei ...

'', the regular dodecahedron appears as a character in the land of Mathematics. Each face of the regular dodecahedron describes the various

facial expression Facial expression is the motion and positioning of the muscles beneath the skin of the face. These movements convey the emotional state of an individual to observers and are a form of nonverbal communication. They are a primary means of conveying ...

s, swiveling to the front as required to match his mood. In

Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...

's 1954 short story "The Mathematician's Nightmare: The Vision of Professor Squarepunt", the number 5 said: "I am the number of fingers on a hand. I make pentagons and pentagrams. And but for me dodecahedra could not exist; and, as everyone knows, the universe is a dodecahedron. So, but for me, there could be no universe."

In nature

The fossil

coccolithophore Coccolithophores, or coccolithophorids, are single-celled organisms which are part of the phytoplankton, the autotrophic (self-feeding) component of the plankton community. They form a group of about 200 species, and belong either to the kingdom ...

'' Braarudosphaera bigelowii'' (see figure), a unicellular coastal

phytoplanktonic Phytoplankton () are the autotrophic (self-feeding) components of the plankton community and a key part of ocean and freshwater ecosystems. The name comes from the Greek words (), meaning 'plant', and (), meaning 'wanderer' or 'drifter'. Phyt ...

alga Algae ( , ; : alga ) is an informal term for any organisms of a large and diverse group of photosynthetic organisms that are not plants, and includes species from multiple distinct clades. Such organisms range from unicellular microalgae, suc ...

, has a calcium carbonate shell with a regular dodecahedral structure about 10 micrometers across. The hydrocarbon dodecahedrane, some

quasicrystal A quasiperiodicity, quasiperiodic crystal, or quasicrystal, is a structure that is Order and disorder (physics), ordered but not Bravais lattice, periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks trans ...

s and cages have dodecahedral shape (see figure). Some regular crystals such as

garnet Garnets () are a group of silicate minerals that have been used since the Bronze Age as gemstones and abrasives. Garnet minerals, while sharing similar physical and crystallographic properties, exhibit a wide range of chemical compositions, de ...

and

diamond Diamond is a Allotropes of carbon, solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Diamond is tasteless, odourless, strong, brittle solid, colourless in pure form, a poor conductor of e ...

are also said to exhibit "dodecahedral"

habit A habit (or wont, as a humorous and formal term) is a routine of behavior that is repeated regularly and tends to occur subconsciously. A 1903 paper in the '' American Journal of Psychology'' defined a "habit, from the standpoint of psychology, ...

, but this statement actually refers to the

rhombic dodecahedron In geometry, the rhombic dodecahedron is a Polyhedron#Convex_polyhedra, convex polyhedron with 12 congruence (geometry), congruent rhombus, rhombic face (geometry), faces. It has 24 edge (geometry), edges, and 14 vertex (geometry), vertices of 2 ...

shape.Dodecahedral Crystal Habit
Various models have been proposed for the global geometry of the universe. These proposals include the Poincaré dodecahedral space, a positively curved space consisting of a regular dodecahedron whose opposite faces correspond (with a small twist). This was proposed by

Jean-Pierre Luminet Jean-Pierre Luminet (born 3 June 1951) is a French astrophysicist, specializing in black holes and cosmology. He is an emeritus research director at the CNRS ( Centre national de la recherche scientifique). Luminet is a member of the Laboratoir ...

and colleagues in 2003, and an optimal orientation on the sky for the model was estimated in 2008.

Dodecahedral graph

According to

Steinitz's theorem In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedron, convex polyhedra: they are exactly the vertex connect ...

, the

can be represented as the

of a polyhedron; roughly speaking, a framework of a polyhedron. Such a graph has two properties. It is planar, meaning the edges of a graph are connected to every vertex without crossing other edges. It is also three-connected graph, meaning that, whenever a graph with more than three vertices, and two of the vertices are removed, the edges remain connected. The skeleton of a regular dodecahedron can be represented as a graph, and it is called the dodecahedral graph, a

. This graph can also be constructed as the

generalized Petersen graph In graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. They include the Petersen graph and generalize one of the ways o ...

G(10,2)

, where the vertices of a decagon are connected to those of two pentagons, one pentagon connected to odd vertices of the decagon and the other pentagon connected to the even vertices. Geometrically, this can be visualized as the ten-vertex equatorial belt of the dodecahedron connected to the two 5-vertex polar regions, one on each side. The high degree of symmetry of the polygon is replicated in the properties of this graph, which are distance-transitive, distance-regular, and

symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

. The

automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...

has order a hundred and twenty. The vertices can be

colored ''Colored'' (or ''coloured'') is a racial descriptor historically used in the United States during the Jim Crow era to refer to an African American. In many places, it may be considered a slur. Dictionary definitions The word ''colored'' wa ...

with 3 colors, as can the edges, and the

diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...

is five. The dodecahedral graph is

, meaning a

visits all of its vertices exactly once. The name of this property is named after

William Rowan Hamilton Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathema ...

, who invented a

mathematical game A mathematical game is a game whose rules, strategies, and outcomes are defined by clear mathematics, mathematical parameters. Often, such games have simple rules and match procedures, such as tic-tac-toe and dots and boxes. Generally, mathemati ...

known as the

. The game's object was to find a

Hamiltonian cycle In the mathematics, mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path (graph theory), path in an undirected or directed graph that visits each vertex (graph theory), vertex exactly once. A Hamiltonian cycle (or ...

along the edges of a dodecahedron.

Notes

References

External links

* *
Editable printable net of a dodecahedron with interactive 3D viewThe Uniform PolyhedraOrigami Polyhedra
– Models made with Modular Origami
Dodecahedron
– 3-d model that works in your browser

The Encyclopedia of Polyhedra **

VRML VRML (Virtual Reality Modeling Language, pronounced ''vermal'' or by its initials, originally—before 1995—known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graph ...

br>Regular dodecahedron
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Stella: Polyhedron Navigator
Software used to create some of the images on this page.
How to make a dodecahedron from a Styrofoam cube
{{Authority control Goldberg polyhedra Planar graphs Platonic solids 12 (number)

As a Platonic solid

Descriptions

Relation to the regular icosahedron

Relation to the regular tetrahedron

Configuration matrix

Relation to the golden ratio

Other related geometric objects

Appearances

In arts and popular cultures

In nature

Dodecahedral graph

Notes

See also

References

External links