In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on th ...
with twelve flat faces. The most familiar dodecahedron is the
regular dodecahedron
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. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges ...
with regular pentagons as faces, which is a
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 edges c ...
. There are also three
regular star dodecahedra, which are constructed as
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 specific el ...
s of the convex form. All of these have
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 of the ...
, order 120.
Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular:
The
pyritohedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
, a common crystal form in
pyrite
The mineral pyrite (), or iron pyrite, also known as fool's gold, is an iron sulfide with the chemical formula Iron, FeSulfur, S2 (iron (II) disulfide). Pyrite is the most abundant sulfide mineral.
Pyrite's metallic Luster (mineralogy), lust ...
, has
pyritohedral symmetry
150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection a ...
, while the
tetartoid
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
has
tetrahedral symmetry
150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection a ...
.
The
rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.
Properties
The rhombic dodecahedro ...
can be seen as a limiting case of the pyritohedron, and it has
octahedral symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...
. The
elongated dodecahedron
In geometry, the elongated dodecahedron, extended rhombic dodecahedron, rhombo-hexagonal dodecahedron or hexarhombic dodecahedron is a convex dodecahedron with 8 rhombic and 4 hexagonal faces. The hexagons can be made equilateral, or regular de ...
and
trapezo-rhombic dodecahedron
In geometry, the trapezo-rhombic dodecahedron or rhombo-trapezoidal dodecahedron is a convex dodecahedron with 6 rhombic and 6 trapezoidal faces. It has symmetry. A concave form can be constructed with an identical net, seen as excavating trig ...
variations, along with the rhombic dodecahedra, are
space-filling. There are numerous
other dodecahedra.
While the regular dodecahedron shares many features with other Platonic solids, one unique property of it is that one can start at a corner of the surface and draw an infinite number of straight lines across the figure that return to the original point without crossing over any other corner.
Regular dodecahedron
The convex regular dodecahedron is one of the five regular
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 edges c ...
s and can be represented by its
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 more ...
.
The
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. ...
is the regular
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 ...
, having five equilateral triangles around each vertex.
The convex regular dodecahedron also has three
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 specific el ...
s, all of which are regular star dodecahedra. They form three of the four
Kepler–Poinsot polyhedra. They are the
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 regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeti ...
, the
great dodecahedron
In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagon ...
, and the
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 each ve ...
. The small stellated dodecahedron and great dodecahedron are dual to each other; the great stellated dodecahedron is dual to the
great icosahedron
In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeti ...
. All of these regular star dodecahedra have regular pentagonal or
pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle aroun ...
mic faces. The convex regular dodecahedron and great stellated dodecahedron are different realisations of the same
abstract regular polyhedron
In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines.
A geometric polytope is said to be ...
; the small stellated dodecahedron and great dodecahedron are different realisations of another abstract regular polyhedron.
Other pentagonal dodecahedra
In
crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...
, two important dodecahedra can occur as crystal forms in some
symmetry classes of the
cubic crystal system
In crystallography, the cubic (or isometric) crystal system is a crystal system where the Crystal_structure#Unit_cell, unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals.
There ...
that are topologically equivalent to the regular dodecahedron but less symmetrical: the pyritohedron with
pyritohedral symmetry
150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection a ...
, and the
tetartoid
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
with
tetrahedral symmetry
150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection a ...
:
Pyritohedron
A pyritohedron is a dodecahedron with
pyritohedral
image:tetrahedron.jpg, 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that c ...
(T
h) symmetry. Like the
regular dodecahedron
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. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges ...
, it has twelve identical
pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
al faces, with three meeting in each of the 20 vertices (see figure). However, the pentagons are not constrained to be regular, and the underlying atomic arrangement has no true fivefold symmetry axis. Its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes of
rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
are three mutually perpendicular twofold axes and four threefold axes.
Although regular dodecahedra do not exist in crystals, the pyritohedron form occurs in the crystals of the mineral
pyrite
The mineral pyrite (), or iron pyrite, also known as fool's gold, is an iron sulfide with the chemical formula Iron, FeSulfur, S2 (iron (II) disulfide). Pyrite is the most abundant sulfide mineral.
Pyrite's metallic Luster (mineralogy), lust ...
, and it may be an inspiration for the discovery of the regular
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 edges c ...
form. The true regular dodecahedron can occur as a shape for
quasicrystal
A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical cr ...
s (such as
holmium–magnesium–zinc quasicrystal
A holmium–magnesium–zinc (Ho–Mg–Zn) quasicrystal is a quasicrystal made of an alloy of the three metals holmium, magnesium and zinc that has the shape of a regular dodecahedron, a Platonic solid with 12 five-sided faces. Unlike the simila ...
) with
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 of the ...
, which includes true fivefold rotation axes.
Crystal pyrite
The name ''crystal pyrite'' comes from one of the two common
crystal habit
In mineralogy, crystal habit is the characteristic external shape of an individual crystal or crystal group. The habit of a crystal is dependent on its crystallographic form and growth conditions, which generally creates irregularities due to l ...
s shown by
pyrite
The mineral pyrite (), or iron pyrite, also known as fool's gold, is an iron sulfide with the chemical formula Iron, FeSulfur, S2 (iron (II) disulfide). Pyrite is the most abundant sulfide mineral.
Pyrite's metallic Luster (mineralogy), lust ...
(the other one being the
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 r ...
). In pyritohedral pyrite, the faces have a
Miller index
Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices.
In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers ''h'', ''k'', and '' ...
of (210), which means that 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 uni ...
is 2·arctan(2) ≈ 126.87° and each pentagonal face has one angle of approximately 121.6° in between two angles of approximately 106.6° and opposite two angles of approximately 102.6°. The following formulas show the measurements for the face of a perfect crystal (which is rarely found in nature).
Cartesian coordinates
The eight vertices of a cube have the coordinates (±1, ±1, ±1).
The coordinates of the 12 additional vertices are
(0, ±(1 + ''h''), ±(1 − ''h''
2)
),
(±(1 + ''h''), ±(1 − ''h''
2), 0
) and
(±(1 − ''h''
2), 0, ±(1 + ''h'')
).
''h'' is the height of the
wedge
A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converti ...
-shaped "roof" above the faces of that cube with edge length 2.
An important case is ''h'' = (a quarter of the cube edge length) for perfect natural pyrite (also the pyritohedron in the
Weaire–Phelan structure
In geometry, the Weaire–Phelan structure is a three-dimensional structure representing an idealised foam of equal-sized bubbles, with two different shapes. In 1993, Denis Weaire and Robert Phelan found that this structure was a better solution ...
).
Another one is ''h'' = = 0.618... for the
regular dodecahedron
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. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges ...
. See section ''
Geometric freedom'' for other cases.
Two pyritohedra with swapped nonzero coordinates are in dual positions to each other like the dodecahedra in the
compound of two dodecahedra
This uniform polyhedron compound is a composition of 2 icosahedra. It has octahedral symmetry ''Oh''. As a holosnub, it is represented by Schläfli symbol β and Coxeter diagram .
The triangles in this compound decompose into two orbits under ac ...
.
Geometric freedom
The pyritohedron has a geometric degree of freedom with
limiting cases of a cubic
convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
at one limit of collinear edges, and a
rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.
Properties
The rhombic dodecahedro ...
as the other limit as 6 edges are degenerated to length zero. The regular dodecahedron represents a special intermediate case where all edges and angles are equal.
It is possible to go past these limiting cases, creating concave or nonconvex pyritohedra. The ''endo-dodecahedron'' is concave and equilateral; it can tessellate space with the convex regular dodecahedron. Continuing from there in that direction, we pass through a degenerate case where twelve vertices coincide in the centre, and on to the regular
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 each ve ...
where all edges and angles are equal again, and the faces have been distorted into regular
pentagrams. On the other side, past the rhombic dodecahedron, we get a nonconvex equilateral dodecahedron with fish-shaped self-intersecting equilateral pentagonal faces.
Tetartoid
A tetartoid (also tetragonal pentagonal dodecahedron, pentagon-tritetrahedron, and tetrahedric pentagon dodecahedron) is a dodecahedron with chiral
tetrahedral symmetry
150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection a ...
(T). Like the
regular dodecahedron
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. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges ...
, it has twelve identical
pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
al faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes.
Although regular dodecahedra do not exist in crystals, the tetartoid form does. The name tetartoid comes from the Greek root for one-fourth because it has one fourth of full octahedral symmetry, and half of pyritohedral symmetry. The mineral
cobaltite
Cobaltite is a sulfide mineral composed of cobalt, arsenic, and sulfur, Co As S. Its impurities may contain up to 10% iron and variable amounts of nickel.Klein, Cornelus and Cornrlius Hurlbut, 1996, ''Manual of Mineralogy'', 20th ed., Wiley, p.2 ...
can have this symmetry form.
Abstractions sharing the solid's
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
and symmetry can be created from the cube and the tetrahedron. In the cube each face is bisected by a slanted edge. In the tetrahedron each edge is trisected, and each of the new vertices connected to a face center. (In
Conway polyhedron notation
In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.
Conway and Hart extended the idea of using op ...
this is a gyro tetrahedron.)
Cartesian coordinates
The following points are vertices of a tetartoid pentagon under
tetrahedral symmetry
150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection a ...
:
:(''a'', ''b'', ''c''); (−''a'', −''b'', ''c''); (−, −, ); (−''c'', −''a'', ''b''); (−, , ),
under the following conditions:
:,
:''n'' = ''a''
2''c'' − ''bc''
2,
:''d''
1 = ''a''
2 − ''ab'' + ''b''
2 + ''ac'' − 2''bc'',
:''d''
2 = ''a''
2 + ''ab'' + ''b''
2 − ''ac'' − 2''bc'',
:.
Geometric freedom
The
regular dodecahedron
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. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges ...
is a tetartoid with more than the required symmetry. The
triakis tetrahedron
In geometry, a triakis tetrahedron (or kistetrahedron) is a Catalan solid with 12 faces. Each Catalan solid is the dual of an Archimedean solid. The dual of the triakis tetrahedron is the truncated tetrahedron.
The triakis tetrahedron can be see ...
is a degenerate case with 12 zero-length edges. (In terms of the colors used above this means, that the white vertices and green edges are absorbed by the green vertices.)
Dual of triangular gyrobianticupola
A lower symmetry form of the regular dodecahedron can be constructed as the dual of a polyhedron constructed from two triangular
anticupola connected base-to-base, called a ''triangular gyrobianticupola.'' It has D
3d symmetry, order 12. It has 2 sets of 3 identical pentagons on the top and bottom, connected 6 pentagons around the sides which alternate upwards and downwards. This form has a hexagonal cross-section and identical copies can be connected as a partial hexagonal honeycomb, but all vertices will not match.
:
Rhombic dodecahedron
![Rhombicdodecahedron](https://upload.wikimedia.org/wikipedia/commons/b/ba/Rhombicdodecahedron.jpg)
The ''
rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.
Properties
The rhombic dodecahedro ...
'' is a
zonohedron
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in ...
with twelve rhombic faces and octahedral symmetry. It is dual to the
quasiregular cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
(an
Archimedean solid
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
) and occurs in nature as a crystal form. The rhombic dodecahedron packs together to fill space.
The ''rhombic dodecahedron'' can be seen as a degenerate
pyritohedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
where the 6 special edges have been reduced to zero length, reducing the pentagons into rhombic faces.
The rhombic dodecahedron has several
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 specific el ...
s, the
first of which is also a
parallelohedral spacefiller.
Another important rhombic dodecahedron, the
Bilinski dodecahedron
In geometry, the Bilinski dodecahedron is a Convex set, convex polyhedron with twelve Congruence (geometry), congruent golden rhombus faces. It has the same topology but a different geometry than the face-transitive rhombic dodecahedron. It is a ...
, has twelve faces congruent to those 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 Cata ...
, i.e. the diagonals are in the ratio of 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 ( ...
. It is also a
zonohedron
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in ...
and was described by
Bilinski in 1960. This figure is another spacefiller, and can also occur in non-periodic
spacefillings along with the rhombic triacontahedron, the rhombic icosahedron and rhombic hexahedra.
Other dodecahedra
There are 6,384,634 topologically distinct ''convex'' dodecahedra, excluding mirror images—the number of vertices ranges from 8 to 20.
Counting polyhedra
Numericana.com (2001-12-31). Retrieved on 2016-12-02. (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)
Topologically distinct dodecahedra (excluding pentagonal and rhombic forms)
*Uniform polyhedra:
**Decagonal prism
In geometry, the decagonal prism is the eighth in the infinite set of prisms, formed by ten square side faces and two regular decagon caps. With twelve faces, it is one of many nonregular dodecahedra. The decagonal prism has 12 faces, 30 edges, a ...
– 10 squares, 2 decagons, D10h symmetry, order 40.
**Pentagonal antiprism
In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for ...
– 10 equilateral triangles, 2 pentagons, D5d symmetry, order 20
*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 isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...
s (regular faced):
**Pentagonal cupola
In geometry, the pentagonal cupola is one of the Johnson solids (). It can be obtained as a slice of the rhombicosidodecahedron. The pentagonal cupola consists of 5 equilateral triangles, 5 squares, 1 pentagon, and 1 decagon.
Formulae
The fol ...
– 5 triangles, 5 squares, 1 pentagon, 1 decagon, C5v symmetry, order 10
**Snub disphenoid
In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron is a convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some vert ...
– 12 triangles, D2d, order 8
**Elongated square dipyramid
In geometry, the elongated square bipyramid (or elongated octahedron) is one of the Johnson solids (). As the name suggests, it can be constructed by elongating an octahedron by inserting a cube between its congruent halves.
It has been named ...
– 8 triangles and 4 squares, D4h symmetry, order 16
**Metabidiminished icosahedron
In geometry, the metabidiminished icosahedron is one of the Johnson solids (). The name refers to one way of constructing it, by removing two pentagonal pyramids () from a regular icosahedron, replacing two sets of five triangular faces of the ...
– 10 triangles and 2 pentagons, C2v symmetry, order 4
*Congruent irregular faced: (face-transitive
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...
)
**Hexagonal bipyramid
A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramid (geometry), pyramids joined at their bases. The resulting solid has 12 triangular face (geometry), faces, 8 vertex (geometry), vertices and 18 edges. The 12 faces are identic ...
– 12 isosceles triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
s, dual of hexagonal prism
In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices..
Since it has 8 faces, it is an octahedron. However, the term ''octahedron'' is primarily used to ...
, D6h symmetry, order 24
**Hexagonal trapezohedron
In geometry, a hexagonal trapezohedron or deltohedron is the fourth in an infinite series of trapezohedra which are dual polyhedra to the antiprisms. It has twelve faces which are congruence (geometry), congruent kite (geometry), kites. It can be ...
– 12 kites
A kite is a tethered heavier than air flight, heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. ...
, dual of hexagonal antiprism
In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.
Antiprisms are similar to prisms except the bases are twisted relative to each oth ...
, D6d symmetry, order 24
**Triakis tetrahedron
In geometry, a triakis tetrahedron (or kistetrahedron) is a Catalan solid with 12 faces. Each Catalan solid is the dual of an Archimedean solid. The dual of the triakis tetrahedron is the truncated tetrahedron.
The triakis tetrahedron can be see ...
– 12 isosceles triangles, dual of truncated tetrahedron
In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedro ...
, Td symmetry, order 24
*Other less regular faced:
**Hendecagonal pyramid
A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...
– 11 isosceles triangles and 1 regular hendecagon
In geometry, a hendecagon (also undecagon or endecagon) or 11-gon is an eleven-sided polygon. (The name ''hendecagon'', from Greek ''hendeka'' "eleven" and ''–gon'' "corner", is often preferred to the hybrid ''undecagon'', whose first part is f ...
, C11v, order 11
**Trapezo-rhombic dodecahedron
In geometry, the trapezo-rhombic dodecahedron or rhombo-trapezoidal dodecahedron is a convex dodecahedron with 6 rhombic and 6 trapezoidal faces. It has symmetry. A concave form can be constructed with an identical net, seen as excavating trig ...
– 6 rhombi, 6 trapezoid
A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium ().
A trapezoid is necessarily a Convex polygon, convex quadri ...
s – dual of triangular orthobicupola
In geometry, the triangular orthobicupola is one of the Johnson solids (). As the name suggests, it can be constructed by attaching two triangular cupolas () along their bases. It has an equal number of squares and triangles at each vertex; howev ...
, D3h symmetry, order 12
**Rhombo-hexagonal dodecahedron
In geometry, the elongated dodecahedron, extended rhombic dodecahedron, rhombo-hexagonal dodecahedron or hexarhombic dodecahedron is a convex dodecahedron with 8 rhombic and 4 hexagonal faces. The hexagons can be made equilateral, or regular de ...
or ''elongated Dodecahedron'' – 8 rhombi and 4 equilateral hexagon
In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°.
Regular hexa ...
s, D4h symmetry, order 16
** Truncated pentagonal trapezohedron, D5d, order 20, topologically equivalent to regular dodecahedron
Practical usage
Armand Spitz
Armand Neustadter Spitz (July 7, 1904 – April 14, 1971) was an American planetarium designer.
Biography
Armand Spitz, the son of Louis Spitz and Rose (Neustadter), was born in Philadelphia, Pennsylvania and was educated at the University of ...
used a dodecahedron as the "globe" equivalent for his Digital Dome planetarium projector. based upon a suggestion from Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
.
See also
* 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, heca ...
– a regular polychoron (4D polytope) whose surface consists of 120 dodecahedral cells
* – a dodecahedron shaped 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 kingdo ...
(a unicellular
A unicellular organism, also known as a single-celled organism, is an organism that consists of a single cell, unlike a multicellular organism that consists of multiple cells. Organisms fall into two general categories: prokaryotic organisms and ...
phytoplankton
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'.
Ph ...
algae
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 ...
)
* Pentakis dodecahedron
In geometry, a pentakis dodecahedron or kisdodecahedron is the polyhedron created by attaching a pentagonal pyramid to each face of a regular dodecahedron; that is, it is the Kleetope of the dodecahedron. It is a Catalan solid, meaning that i ...
* Roman dodecahedron
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 midd ...
* Snub dodecahedron
In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.
The snub dodecahedron has 92 faces (the most ...
* Truncated dodecahedron
In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.
Geometric relations
This polyhedron can be formed from a regular dodecahedron by truncat ...
References
External links
*''Plato's Fourth Solid and the "Pyritohedron"'', by Paul Stephenson, 1993, The Mathematical Gazette, Vol. 77, No. 479 (Jul., 1993), pp. 220–22
Stellation of Pyritohedron
VRML models and animations of Pyritohedron and its 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 specific el ...
s
*
Editable printable net of a dodecahedron with interactive 3D view
The Uniform Polyhedra
Origami Polyhedra
– Models made with Modular Origami
The Encyclopedia of Polyhedra
*[http://www.bodurov.com/VectorVisualizer/?vectors=-0.94/-2.885/-3.975/-1.52/-4.67/-0.94v-3.035/0/-3.975/-4.91/0/-0.94v3.975/-2.885/-0.94/1.52/-4.67/0.94v1.52/-4.67/0.94/-1.52/-4.67/-0.94v0.94/-2.885/3.975/1.52/-4.67/0.94v-3.975/-2.885/0.94/-1.52/-4.67/-0.94v-3.975/-2.885/0.94/-4.91/0/-0.94v-3.975/2.885/0.94/-4.91/0/-0.94v-3.975/2.885/0.94/-1.52/4.67/-0.94v-2.455/1.785/3.975/-3.975/2.885/0.94v-2.455/-1.785/3.975/-3.975/-2.885/0.94v-1.52/4.67/-0.94/-0.94/2.885/-3.975v4.91/0/0.94/3.975/-2.885/-0.94v3.975/2.885/-0.94/2.455/1.785/-3.975v2.455/-1.785/-3.975/3.975/-2.885/-0.94v1.52/4.67/0.94/-1.52/4.67/-0.94v3.035/0/3.975/0.94/2.885/3.975v0.94/2.885/3.975/-2.455/1.785/3.975v-2.455/1.785/3.975/-2.455/-1.785/3.975v-2.455/-1.785/3.975/0.94/-2.885/3.975v0.94/-2.885/3.975/3.035/0/3.975v2.455/1.785/-3.975/-0.94/2.885/-3.975v-0.94/2.885/-3.975/-3.035/0/-3.975v-3.035/0/-3.975/-0.94/-2.885/-3.975v-0.94/-2.885/-3.975/2.455/-1.785/-3.975v2.455/-1.785/-3.975/2.455/1.785/-3.97v3.035/0/3.975/4.91/0/0.94v4.91/0/0.94/3.975/2.885/-0.94v3.975/2.885/-0.94/1.52/4.67/0.94v1.52/4.67/0.94/0.94/2.885/3.975 Dodecahedron 3D Visualization]
Stella: Polyhedron Navigator
Software used to create some of the images on this page.
How to make a dodecahedron from a Styrofoam cube
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Individual graphs
Planar graphs
Platonic solids
12 (number)