A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals (60 face diagonals, 100 space diagonals). It is represented by the

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mo ...

Dimensions

If the edge length of a regular dodecahedron is

a

, the

radius In classical geometry, a radius (plural, : radii) 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 name comes from the latin ''radius'', ...

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 vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcircle' ...

(one that touches the regular dodecahedron at all vertices) is :

r_u = a\frac \left(1 + \sqrt\right) \approx 1.401\,258\,538 \cdot a

and the radius of an inscribed sphere (

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

to each of the regular dodecahedron's faces) is :

r_i = a\frac \sqrt \approx 1.113\,516\,364 \cdot a

while the midradius, which touches the middle of each edge, is :

r_m = a\frac \left(3 +\sqrt\right) \approx 1.309\,016\,994 \cdot a

These quantities may also be expressed as :

r_u = a\, \frac \phi

r_i = a\, \frac

r_m = a\, \frac

where ''ϕ'' is the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

. Note that, given a regular dodecahedron of edge length one, ''r_u'' is the radius of a circumscribing sphere about a

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

of edge length ''ϕ'', 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. ...

of a regular pentagon of edge length ''ϕ''.

Surface area and volume

The

surface area The surface area 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 definition of ...

''A'' and the

volume Volume is a measure of occupied 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: :

V = \frac (15+7\sqrt) a^3 \approx 7.663\,118\,9606a^3

Additionally, the surface area and volume of a regular dodecahedron are related to the golden ratio. A dodecahedron with an edge length of one unit has the properties: :

Two-dimensional symmetry projections

The ''regular dodecahedron '' has two high

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...

s, centered, on vertices and pentagonal faces, correspond to the A₂ and H₂ Coxeter planes. The edge-center projection has two orthogonal lines of reflection. In

perspective projection Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation ...

, viewed on top of a pentagonal face, the regular dodecahedron can be seen as a linear-edged

Schlegel diagram In geometry, a Schlegel diagram is a projection of a polytope from \mathbb^d into \mathbb^ through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in \mathbb^ that, together with the orig ...

, or

stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thro ...

as a spherical polyhedron. These projections are also used in showing the four-dimensional

120-cell In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, he ...

, a regular 4-dimensional polytope, constructed from 120 dodecahedra, projecting it down to 3-dimensions.

Spherical tiling

The regular dodecahedron can also be represented as a spherical tiling.

Cartesian coordinates

The following

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...

define the 20 vertices of a regular dodecahedron centered at the origin and suitably scaled and oriented: :(±1, ±1, ±1) :(0, ±''ϕ'', ±) :(±, 0, ±''ϕ'') :(±''ϕ'', ±, 0) where is the golden ratio (also written ''τ'') ≈ 1.618. The edge length is . The

circumradius In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every pol ...

is .

Facet-defining equations

Similar to the symmetry of the vertex coordinates, the equations of the twelve facets of the regular dodecahedron also display symmetry in their coefficients: :''x'' ± ''ϕy'' = ±''ϕ''² :''y'' ± ''ϕz'' = ±''ϕ''² :''z'' ± ''ϕx'' = ±''ϕ''²

Properties

*The

dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the un ...

of a regular dodecahedron is 2

arctan In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spe ...

(''ϕ'') or approximately ° (where again ''ϕ'' = , the golden ratio). Note that the tangent of the dihedral angle is exactly −2. *If the original regular dodecahedron has edge length 1, its dual

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...

has edge length ''ϕ''. *If the five Platonic solids are built with same volume, the regular dodecahedron has the shortest edges. *It has 43,380 nets. *The map-coloring number of a regular dodecahedron's faces is 4. *The distance between the vertices on the same face not connected by an edge is ''ϕ'' times the edge length. *If two edges share a common vertex, then the midpoints of those edges form a 36-72-72

golden triangle Golden Triangle may refer to: Places Asia * Golden Triangle (Southeast Asia), named for its opium production * Golden Triangle (Yangtze), China, named for its rapid economic development * Golden Triangle (India), comprising the popular tourist ...

with the body center.

As a configuration

This configuration matrix represents the dodecahedron. The rows and columns correspond to vertices, edges, and faces. The diagonal numbers say how many of each element occur in the whole dodecahedron. The nondiagonal numbers say how many of the column's element occur in or at the row's element.

\begin\begin20 & 3 & 3 \\ 2 & 30 & 2 \\ 5 & 5 & 12  \end\end

Here is the configuration expanded with ''k''-face elements and ''k''-figures. The diagonal element counts are the ratio of the full Coxeter group H₃, order 120, divided by the order of the subgroup with mirror removal.

Geometric relations

The ''regular dodecahedron'' is the third in an infinite set of

truncated trapezohedra In geometry, an truncated trapezohedron is a polyhedron formed by a trapezohedron with pyramids truncated from its two polar axis vertices. If the polar vertices are completely truncated (diminished), a trapezohedron becomes an antiprism. ...

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

pentagonal trapezohedron In geometry, a pentagonal trapezohedron or deltohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedra to the antiprisms. It has ten faces (i.e., it is a decahedron) which are congruent kites. It can ...

. The

stellation In geometry, stellation is the process of extending a polygon in two dimensions, 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 specif ...

s of the regular dodecahedron make up three of the four Kepler–Poinsot polyhedra. A rectified regular dodecahedron forms an

icosidodecahedron In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 i ...

. The regular dodecahedron has

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 ...

I_h, Coxeter group ,3 order 120, with an abstract group structure of ''A''₅ × ''Z''₂.

Relation to the regular icosahedron

The dodecahedron and icosahedron are dual polyhedra. A regular dodecahedron has 12 faces and 20 vertices, whereas a regular icosahedron has 20 faces and 12 vertices. Both have 30 edges. When a regular dodecahedron is inscribed in a

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...

, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.55%). A regular dodecahedron with edge 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...), which ratio is approximately , or in exact terms: or .

Relation to the nested cube

A cube can be embedded within a regular dodecahedron, affixed to eight of its equidistant vertices, in five different positions. In fact, five cubes may overlap and interlock inside the regular dodecahedron to result in the

compound of five cubes The compound of five cubes is one of the five regular polyhedral compounds. It was first described by Edmund Hess in 1876. It is one of five regular compounds, and dual to the compound of five octahedra. It can be seen as a faceting of a regul ...

. The ratio of the edge of a regular dodecahedron to the edge of a cube embedded inside such a regular dodecahedron is 1 : ''ϕ'', or (''ϕ'' − 1) : 1. The ratio of a regular dodecahedron's volume to the volume of a cube embedded inside such a regular dodecahedron is 1 : , or : 1, or (5 + ) : 4. For example, an embedded cube with a volume of 64 (and edge length of 4), will nest within a regular dodecahedron of volume 64 + 32''ϕ'' (and edge length of 4''ϕ'' − 4). Thus, the difference in volume between the encompassing regular dodecahedron and the enclosed cube is always one half the volume of the cube times ''ϕ''. From these ratios are derived simple formulas for the volume of a regular dodecahedron with edge length ''a'' in terms of the golden mean: :''V'' = (''aϕ'')³ · (5 + ) :''V'' = (14''ϕ'' + 8)''a''³

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).

Relation to the golden rectangle

Golden rectangles of ratio (''ϕ'' + 1) : 1 and ''ϕ'' : 1 also fit perfectly within a regular dodecahedron. In proportion to this golden rectangle, an enclosed cube's edge is ''ϕ'', when the long length of the rectangle is ''ϕ'' + 1 (or ''ϕ''²) and the short length is 1 (the edge shared with the regular dodecahedron). In addition, the center of each face of the regular dodecahedron form three intersecting golden rectangles.

Relation to the 6-cube and rhombic triacontahedron

It can be projected to 3D from the 6-dimensional

6-demicube In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a ''6-cube'' ( hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte ...

using the same basis vectors that form the hull of the

rhombic triacontahedron In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Ca ...

from the 6-cube. Shown here including the inner 12 vertices, which are not connected by the outer hull edges of 6D norm length , form a regular icosahedron. The 3D projection basis vectors 'u'',''v'',''w''used are: :''u'' = (1, ''ϕ'', 0, −1, ''ϕ'', 0) :''v'' = (''ϕ'', 0, 1, ''ϕ'', 0, −1) :''w'' = (0, 1, ''ϕ'', 0, −1, ''ϕ'')

History and uses

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

Iamblichus Iamblichus (; grc-gre, Ἰάμβλιχος ; Aramaic: 𐡉𐡌𐡋𐡊𐡅 ''Yamlīḵū''; ) was a Syrian neoplatonic philosopher of Arabic origin. He determined a direction later taken by neoplatonism. Iamblichus was also the biographer o ...

states that Hippasus, a Pythagorean, perished in the sea, because he boasted that he first divulged "the sphere with the twelve pentagons." In ''

Theaetetus Theaetetus (Θεαίτητος) is a Greek name which could refer to: * Theaetetus (mathematician) (c. 417 BC – 369 BC), Greek geometer * ''Theaetetus'' (dialogue), a dialogue by Plato, named after the geometer * Theaetetus (crater) Theaetetus ...

'', a dialogue of Plato, Plato hypothesized that the classical elements were made of the five uniform regular solids; these later became known as the

platonic solid 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 e ...

s. Of the fifth Platonic solid, the dodecahedron, Plato 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 with the four

classical element Classical elements typically refer to earth, water, air, fire, and (later) aether which were proposed to explain the nature and complexity of all matter in terms of simpler substances. Ancient cultures in Greece, Tibet, and India had simi ...

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 (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...

also postulated that the heavens were made of a fifth element, which he called aithêr (''aether'' in Latin, ''ether'' in American English).

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 (; grc-gre, Εὐκλείδης; 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 ...

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. 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 Mediterranean history after Classical Greece, between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as signified by the Battle of Actium in 3 ...

, small, hollow bronze

Roman dodecahedra A Roman dodecahedron or Gallo-Roman dodecahedron is a small hollow object made of copper alloy which has been cast into a regular dodecahedral shape: twelve flat pentagonal faces, each face having a circular hole of varying diameter in the middl ...

were made and have been found in various Roman ruins in Europe. Their 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 (Les Nabis), Art Nouveau and Symbolism led to the first twentieth-century ...

, 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 mathematically inspired woodcuts, lithographs, and mezzotints. Despite wide popular interest, Escher was for most of his life neglected in t ...

, such as his lithographs ''

Reptiles Reptiles, as most commonly defined are the animals in the class Reptilia ( ), a paraphyletic grouping comprising all sauropsids except birds. Living reptiles comprise turtles, crocodilians, squamates ( lizards and snakes) and rhynchoceph ...

'' (1943) and ''

Gravitation In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...

'' (1952). In

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

's painting ''

The Sacrament of the Last Supper ''The Sacrament of the Last Supper'' is a painting by Salvador Dalí. Completed in 1955, after nine months of work, it remains one of his most popular compositions. Since its arrival at the National Gallery of Art in Washington, D.C. in 1955, it ...

'' (1955), the room is a hollow regular dodecahedron.

Gerard Caris Gerard Caris (born 20 March 1925) is a Dutch sculptor and artist who has pursued a single motif throughout the course of his artistic career, the pentagon. Early life and education He was born in Maastricht, the Netherlands. After attending ...

based his entire artistic oeuvre on the regular dodecahedron and the pentagon, which is presented as a new art movement coined as Pentagonism. Dodecahedron climbing wall

In modern

role-playing games A role-playing game (sometimes spelled roleplaying game, 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, either through literal ac ...

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

polyhedral dice Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing ga ...

Immersive Media Company Immersive Media Company (IMC) was a digital imaging company specializing in spherical immersive video, founded in 1994. The parent company Immersive Ventures was headquartered in Kelowna, British Columbia with Immersive Media Company offices ...

, a former Canadian

digital imaging Digital imaging or digital image acquisition is the creation of a digital representation of the visual characteristics of an object, such as a physical scene or the interior structure of an object. The term is often assumed to imply or include t ...

company, made the Dodeca 2360 camera, the world's first 360° full-motion camera which captures high-resolution video from every direction simultaneously at more than 100 million pixels per second or 30 frames per second. It is based on regular dodecahedron. 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 Dodecahedron, ...

twisty puzzle, alongside its larger and smaller order analogues, is in the shape of a regular dodecahedron. 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. The story follows a bored young boy named Milo who unexpectedly receives a magic tollboo ...

'', the regular dodecahedron appears as a character in the land of Mathematics. Each of his faces wears a different expression – ''e.g.'' happy, angry, sad – which he swivels to the front as required to match his mood.

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 king ...

Braarudosphaera bigelowii ''Braarudosphaera bigelowii'' is a coastal coccolithophore in the fossil record going back 100 million years. The family Braarudosphaeraceae are single-celled coastal phytoplanktonic algae with calcareous Calcareous () is an adjective meanin ...

'' (see figure), a unicellular coastal phytoplanktonic

alga Algae (; singular alga ) is an informal term for a large and diverse group of photosynthetic eukaryotic organisms. It is a polyphyletic grouping that includes species from multiple distinct clades. Included organisms range from unicellular mic ...

, has a calcium carbonate shell with a regular dodecahedral structure about 10 micrometers across. Some quasicrystals 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. All species of garnets possess similar physical properties and crystal forms, but differ in chemical composition. The different ...

and

diamond Diamond is a solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Another solid form of carbon known as graphite is the chemically stable form of carbon at room temperature and pressure, b ...

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.

, but this statement actually refers to the rhombic dodecahedron shape.Dodecahedral Crystal Habit

Shape of the universe

Various models have been proposed for the global geometry of the universe. In addition to the primitive geometries, 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 Laboratoire ...

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

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

'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."

Space filling with cube and bilunabirotunda

Regular dodecahedra fill space with

s and bilunabirotundas (

Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnso ...

91), in the ratio of 1 to 1 to 3. The dodecahedra alone make a lattice of edge-to-edge pyritohedra. The bilunabirotundas fill the rhombic gaps. Each cube meets six bilunabirotundas in three orientations.

Related polyhedra and tilings

The regular dodecahedron is topologically related to a series of tilings by

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...

''n''³. The regular dodecahedron can be transformed by a

truncation In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathb ...

sequence into its dual, the icosahedron: The regular dodecahedron is a member of a sequence of otherwise non-uniform polyhedra and tilings, composed of pentagons with face configurations (V3.3.3.3.''n''). (For ''n'' > 6, the sequence consists of tilings of the hyperbolic plane.) These face-transitive figures have (''n''32) rotational

symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

Vertex arrangement

The regular dodecahedron shares its

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. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...

with four nonconvex uniform polyhedra and three uniform polyhedron compounds. Five

s fit within, with their edges as diagonals of the regular dodecahedron's faces, and together these make up the regular

polyhedral compound In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be connec ...

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

Stellations

The 3

s of the regular dodecahedron are all regular (

nonconvex A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

) polyhedra: ( Kepler–Poinsot polyhedra)

Dodecahedral graph

The

skeleton A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...

of the dodecahedron (the vertices and edges) form a graph. It is one of 5 Platonic graphs, each a skeleton of its

Platonic solid 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 e ...

. This graph can also be constructed as the generalized Petersen graph ''G''(10,2). The high degree of symmetry of the polygon is replicated in the properties of this graph, which is distance-transitive, distance-regular, and symmetric. 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 120. 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, though it has taken on a special meaning in Sout ...

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 center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...

is 5. The dodecahedral graph is Hamiltonian – there is a cycle containing all the vertices. Indeed, this name derives from a mathematical game invented in 1857 by

William Rowan Hamilton Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...

, the icosian game. The game's object was to find a Hamiltonian cycle along the edges of a dodecahedron.

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 ** VRMLbr>Regular dodecahedron
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Stella: Polyhedron Navigator
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How to make a dodecahedron from a Styrofoam cube
{{Polytopes Goldberg polyhedra Planar graphs Platonic solids 12 (number)