# Dodecahedron

Regular Dodecahedron

Type Platonic solid
Elements F = 12, E = 30
V = 20 (χ = 2)
Faces by sides 12{5}
Conway notation D
gT
Schläfli symbols {5,3}
Wythoff symbol 3 | 2 5
Coxeter diagram
Symmetry Ih, H3, [5,3], (*532)
Rotation group I, [5,3]+, (532)
References U23, C26, W5
Properties Regular convex
Dihedral angle 116.56505° = arccos(-1/√5)

5.5.5
(Vertex figure)

Icosahedron
(dual polyhedron)

Net
Animation of a net of a dodecahedron being folded

In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces, but usually a regular dodecahedron is meant, which is one of the five Platonic solids. It is composed of twelve regular pentagonal faces, with three meeting at each vertex, and is represented by the Schläfli symbol {5,3}. It has 20 vertices, 30 edges and 160 diagonals (60 face diagonals, 100 space diagonals).[1] Its dual polyhedron is the icosahedron, with Schläfli symbol {3,5}.

The pyritohedron is an irregular pentagonal dodecahedron, having the same topology as the regular one but pyritohedral symmetry. The rhombic dodecahedron has octahedral symmetry. There are a large number of other dodecahedra.

## Regular dodecahedron

### Dimensions

If the edge length of a regular dodecahedron is a, the radius of a circumscribed sphere (one that touches the dodecahedron at all vertices) is

$r_u = a\frac{\sqrt{3}}{4} \left(1 + \sqrt{5}\right) \approx 1.401258538 \cdot a$

and the radius of an inscribed sphere (tangent to each of the dodecahedron's faces) is

$r_i = a\frac{1}{2} \sqrt{\frac{5}{2} +\frac{11}{10}\sqrt{5}} \approx 1.113516364 \cdot a$

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

$r_m = a\frac{1}{4} \left(3 +\sqrt{5}\right) \approx 1.309016994 \cdot a$

These quantities may also be expressed as

$r_u = a\, \frac{\sqrt{3}}{2} \phi$
$r_i = a\, \frac{\phi^2}{2 \sqrt{3-\phi}}$
$r_m = a\, \frac{\phi^2}{2}$

where φ is the golden ratio.

Note that, given a regular pentagonal dodecahedron of edge length one, ru is the radius of a circumscribing sphere about a cube of edge length φ, and ri is the apothem of a regular pentagon of edge length φ.

### Area and volume

The surface area A and the volume V of a regular dodecahedron of edge length a are:

$A = 3\sqrt{25+10\sqrt{5}} a^2 \approx 20.645728807a^2$
$V = \frac{1}{4} (15+7\sqrt{5}) a^3 \approx 7.6631189606a^3$

### Two-dimensional symmetry projections

The dodecahedron has two special orthogonal projections, centered, on vertices and pentagonal faces, correspond to the A2 and H2 Coxeter planes.

Orthogonal projections
Centered by Vertex Edge Face
Image
Projective
symmetry
[[3]] = [6] [2] [[5]] = [10]

In perspective projection, viewed above a pentagonal face, the dodecahedron can be seen as a linear-edged schlegel diagram, or stereographic projection as a spherical polyhedron. These projections are also used in showing the four-dimensional 120-cell, a regular 4-dimensional polytope, constructed from 120 dodecahedra, projecting it down to 3-dimensions.

Projection Orthogonal projection Perspective projection
Schlegel diagram Stereographic projection
Dodecahedron
Dodecaplex
(120-cell)

### Spherical tiling

The dodecahedron can also be represented as a spherical tiling.

### Cartesian coordinates

 Vertex coordinates: The orange vertices lie at (±1, ±1, ±1) and form a cube (dotted lines). The green vertices lie at (0, ±1/ϕ, ±ϕ) and form a rectangle on the y–z plane. The blue vertices lie at (±1/ϕ, ±ϕ, 0) and form a rectangle on the x–y plane. The pink vertices lie at (±ϕ, 0, ±1/ϕ) and form a rectangle on the x–z plane. The distance between adjacent vertices is 2/ϕ, and the distance from the origin to any vertex is √3. ϕ = (1 + √5) / 2 is the golden ratio.

The following Cartesian coordinates define the vertices of a dodecahedron centered at the origin and suitably scaled and oriented:[2]

(±1, ±1, ±1)
(0, ±1/ϕ, ±ϕ)
(±1/ϕ, ±ϕ, 0)
ϕ, 0, ±1/ϕ)

where ϕ = (1 + √5) / 2 is the golden ratio (also written τ) ≈ 1.618. The edge length is 2/ϕ = √5 – 1. The containing sphere has a radius of √3.

### Properties

• The dihedral angle of a dodecahedron is 2 arctan(φ) or approximately 116.5650512 degrees (where again φ = (1 + √5) / 2, the golden ratio).
• If the original dodecahedron has edge length 1, its dual icosahedron has edge length φ.
• If the five Platonic solids are built with same volume, the 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.

### Space filling with cube and bilunabirotunda

Regular dodecahedra fill the space with cubes and bilunabirotundae, Johnson solid 91, in the ratio of 1 to 1 to 3.[3][4] The dodecahedra alone make a lattice of edge-to-edge pyritohedra. The bilunabirotundae fill the rhombic gaps. Each cube meets six bilunabirotundae in three orientations.

 Block model Lattice of dodecahedra 6 bilunabirotundae around a cube

### Geometric relations

The regular dodecahedron is the third in an infinite set of truncated trapezohedra which can be constructed by truncating the two axial vertices of a pentagonal trapezohedron.

The stellations of the dodecahedron make up three of the four Kepler–Poinsot polyhedra.

A rectified dodecahedron forms an icosidodecahedron.

The regular dodecahedron has icosahedral symmetry Ih, Coxeter group [5,3], order 120, with an abstract group structure of A5 × Z2.

#### Icosahedron vis-à-vis dodecahedron

When a dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.54%).

A 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 3.51246117975, or in exact terms: (3/5)(3φ +1) or (1.8φ + .6).

A regular dodecahedron has 12 faces and 20 vertices, whereas a regular icosahedron has 20 faces and 12 vertices. Both have 30 edges.

#### Nested cube

A cube can be embedded within a regular dodecahedron, affixed to eight of its equidistant vertices, in five different positions.[5] In fact, five cubes may overlap and interlock inside the dodecahedron to result in the compound of five cubes.

The ratio of the edge of a regular dodecahedron to the edge of a cube embedded inside such a 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 dodecahedron is 1 : 2/(2 + φ), or (1 + φ/2) : 1, or (5+√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 + 32φ (and edge length of 4φ − 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 = ()3 · (1/4)(5 + √5)
V = (1/4)(14φ + 8)a3

#### The dodecahedron's golden frame

Golden ratio rectangles of ratio φ + 1 : 1 and φ to 1 also fit perfectly within a regular dodecahedron.[6] In proportion to this golden rectangle, an enclosed cube's edge is φ, when the long length of the rectangle is φ + 1 (or φ2) and the short length is 1 (the edge shared with the dodecahedron).

In addition, the center of each face of the dodecahedron form three intersecting golden rectangles.[7]

### Related polyhedra and tilings

The regular dodecahedron is topologically related to a series of tilings by vertex figure n3.

Spherical Euclidean Compact hyperb. Paraco. Noncompact hyperbolic
{2,3} {3,3} {4,3} {5,3} {6,3} {7,3} {8,3} {∞,3} {12i,3} {9i,3} {6i,3} {3i,3}

The dodecahedron can be transformed by a truncation sequence into its dual, the icosahedron:

Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
{5,3} t{5,3} r{5,3} 2t{5,3}=t{3,5} 2r{5,3}={3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
V5.5.5 V3.10.10 V3.5.3.5 V5.6.6 V3.3.3.3.3 V3.4.5.4 V4.6.10 V3.3.3.3.5
Uniform octahedral polyhedra
Symmetry: [4,3], (*432) [4,3]+
(432)
[1+,4,3] = [3,3]
(*332)
[3+,4]
(3*2)
{4,3} t{4,3} r{4,3}
r{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3} sr{4,3} h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{3,4}
s{31,1}

=

=

=
=
or
=
or
=

Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35

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 (n32) rotational symmetry.

Dimensional family of snub polyhedra and tilings: 3.3.3.3.n
Symmetry
n32
[n,3]+
Spherical Euclidean Compact hyperbolic Paracompact
232
[2,3]+
D3
332
[3,3]+
T
432
[4,3]+
O
532
[5,3]+
I
632
[6,3]+
P6
732
[7,3]+
832
[8,3]+...
∞32
[∞,3]+
Snub
figure

3.3.3.3.2

3.3.3.3.3

3.3.3.3.4

3.3.3.3.5

3.3.3.3.6

3.3.3.3.7

3.3.3.3.8

3.3.3.3.∞
Coxeter
Schläfli

sr{2,3}

sr{3,3}

sr{4,3}

sr{5,3}

sr{6,3}

sr{7,3}

sr{8,3}

sr{∞,3}
Snub
dual
figure

V3.3.3.3.2

V3.3.3.3.3

V3.3.3.3.4

V3.3.3.3.5

V3.3.3.3.6

V3.3.3.3.7
V3.3.3.3.8
V3.3.3.3.∞
Coxeter

#### Vertex arrangement

The dodecahedron shares its vertex arrangement with four nonconvex uniform polyhedra and three uniform polyhedron compounds.

Five cubes fit within, with their edges as diagonals of the dodecahedron's faces, and together these make up the regular polyhedral compound of five cubes. Since two tetrahedra can fit on alternate cube vertices, five and ten tetrahedra can also fit in a dodecahedron.

#### Stellations

The 3 stellations of the dodecahedron are all regular (nonconvex) polyhedra: (Kepler–Poinsot polyhedra)

0 1 2 3
Stellation
Dodecahedron

Small stellated dodecahedron

Great dodecahedron

Great stellated dodecahedron
Facet diagram

## Dodecahedral graph

Regular dodecahedron graph
A Hamiltonian cycle in a dodecahedron.
Vertices 20
Edges 30
Diameter 5
Girth 5
Automorphisms 120 (S5)
Chromatic number 3
Properties Hamiltonian, regular, symmetric, distance-regular, distance-transitive, 3-vertex-connected, planar graph

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

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 has order 120. The vertices can be colored with 3 colors, as can the edges, and the diameter is 5.[8]

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

## Pyritohedron

Pyritohedron

A pyritohedron has 30 edges, divided into two lengths: 24 and 6 in each group.
Face polygon irregular pentagon
Coxeter diagrams
Faces 12
Edges 30 (6+24)
Vertices 20 (8+12)
Symmetry group Th, [4,3+], (3*2), order 24
Rotation group T, [3,3]+, (332), order 12
Dual polyhedron Pseudoicosahedron
Properties face transitive
Net

A pyritohedron is a dodecahedron with pyritohedral (Th) symmetry. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not necessarily regular, so the structure normally has no fivefold symmetry axes. Its 30 edges are divided into two sets – containing 24 and 6 edges of the same length.

Although regular dodecahedra do not exist in crystals, the distorted, pyritohedron form occurs in the crystal pyrite, and it may be an inspiration for the discovery of the regular Platonic solid form.

### Crystal pyrite

Its name comes from one of the two common crystal forms of pyrite, the other one being cubical.

 Cubic pyrite Pyritohedral Ho-Mg-Zn quasicrystal

### Cartesian coordinates

The coordinates of the eight vertices of the original cube are:

(±1, ±1, ±1)

The coordinates of the 12 vertices of the cross-edges are:

(0, ±(1 + h), ±(1 − h2))
(±(1 + h), ±(1 − h2), 0)
(±(1 − h2), 0, ±(1 + h))

A dual position pyritohedron is positioned by swapping the nonzero coordinates above.

where h is the height of the wedge-shaped "roof" above the faces of the cube. When h = 1, the six cross-edges degenerate to points and a rhombic dodecahedron is formed. When h = 0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h = (√5 − 1)/2, the inverse of the golden ratio, the result is a regular dodecahedron.

### Geometric freedom

The pyritohedron has a geometric degree of freedom with limiting cases of a cubic convex hull at one limit of colinear edges, and a rhombic dodecahedron 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.

Special cases of the pyritohedron
1 : 1 1 : 1 2 : 1 1.3092... : 1 1 : 1 0 : 1
h = - (√5 + 1)/2 h = 0 h = (√5 − 1)/2 h = 1

Regular star, great stellated dodecahedron, with pentagons distorted into regular pentagrams

Concave pyritohedral dodecahedron

A cube can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions.

The geometric proportions of the pyritohedron in the Weaire–Phelan structure

A regular dodecahedron is an intermediate case with equal edge lengths.

A rhombic dodecahedron is the limiting case with the 6 crossedges reducing to length zero.

## Tetartoid

Tetartoid
Tetragonal pentagonal dodecahedron
Face polygon irregular pentagon
Conway polyhedron notation gT
Faces 12
Edges 30 (6+12+12)
Vertices 20 (4+4+12)
Symmetry group T, [3,3]+, (332), order 12
Properties convex, face transitive
Tetartoid

A tetartoid (also tetragonal pentagonal dodecahedron, pentagon-tritetrahedron, and tetrahedric pentagon dodecahedron) is a dodecahedron with chiral tetrahedral symmetry, (T). Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes. In Conway polyhedron notation it can be seen as gT, a gyro tetrahedron.

Although regular dodecahedra do not exist in crystals, the tetartoid form occurs in the crystals. The name tetartoid comes from the Greek root for one-fourth because it has one forth of full octahedral symmetry, and half that of pyritohedral symmetry.[9] The mineral cobaltite can have this symmetry form.[10]

Its topology can be as a cube with square faces bisected into 2 rectangles like the pyritohedron, and then the bisection lines are slanted retaining 3-fold rotation at the 8 corners.

It can be seen as a tetrahedron, with edges divided into 3 segments, along with a center point of each triangular face.

## Rhombic dodecahedra

Rhombic dodecahedron

The rhombic dodecahedron is a zonohedron with twelve rhombic faces and octahedral symmetry. It is dual to the quasiregular cuboctahedron (an Archimedean solid) and occurs in nature as a crystal form.[11] The rhombic dodecahedron packs together to fill space.

The rhombic dodecahedron can be seen as a degenerate pyritohedron where the 6 special edges have been reduced to zero length, reducing the pentagons into rhombic faces.

The rhombic dodecahedron has several stellations, the first of which is also a spacefiller.

Another important rhombic dodecahedron has twelve faces congruent to those of the rhombic triacontahedron, i.e. the diagonals are in the ratio of the golden ratio. It is also a zonohedron and was described by Bilinski in 1960.[12] This figure is another spacefiller, and can also occur in non-periodic spacefillings along with the rhombic triacontahedron, the rhombic icosahedron and rhombic hexahedra.[13]

## Other dodecahedra

 The truncated pentagonal trapezohedron has D5d dihedral symmetry, order 20. The pyritohedron has Th pyritohedral symmetry, order 24. The tetartoid has T tetrahedral symmetry, order 12.

There are 6,384,634 topologically distinct convex dodecahedra, excluding mirror images, having at least 8 vertices.[15] (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 include:

## History and uses

Roman dodecahedron
Ho-Mg-Zn quasicrystal
Omnidirectional sound source
A climbing wall consisting of three dodecahedral pieces

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

Iamblichus states that Hippasus, a Pythagorean, perished in the sea, because he boasted that he first divulged "the sphere with the twelve pentagons." [16] In Theaetetus, a dialogue of Plato, Plato was able to prove that there are just five uniform regular solids; they later became known as the platonic solids. Timaeus (c. 360 B.C.), as a personage of Plato's dialogue, associates the other four platonic solids with the four classical elements, adding that there is a fifth solid pattern which, though commonly associated with the dodecahedron, is never directly mentioned as such; "this God used in the delineation of the universe."[17] Aristotle also postulated that the heavens were made of a fifth element, which he called aithêr (aether in Latin, ether in American English).

Dodecahedra have been used as dice and probably also as divinatory devices. During the hellenistic era, small, hollow bronze Roman dodecahedra were made and have been found in various Roman ruins in Europe. Their purpose is not certain.

In 20th-century art, dodecahedra appear in the work of M.C. Escher, such as his lithographs Reptiles (1943) and Gravitation (1952). In Salvador Dalí's painting The Sacrament of the Last Supper (1955), the room is a hollow dodecahedron. Gerard Caris based his entire artistic oeuvre on the dodecahedron and the pentagon, which is presented as a new art movement coined as Pentagonism.

In modern role-playing games, the dodecahedron is often used as a twelve-sided die, one of the more common polyhedral dice. Some quasicrystals have dodecahedral shape (see figure). Some regular crystals such as garnet and diamond are also said to exhibit "dodecahedral" habit, but this statement actually refers to the rhombic dodecahedron shape.[11]

Immersive Media, a camera manufacturing company, has 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 dodecahedron.

The popular puzzle game Megaminx is in the shape of a dodecahedron.

In the children's novel The Phantom Tollbooth, the 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.

Dodecahedron is the name of an avant-garde black metal band from Netherlands.[18]

### 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 dodecahedron whose opposite faces correspond (with a small twist). This was proposed by Jean-Pierre Luminet and colleagues in 2003[19][20] and an optimal orientation on the sky for the model was estimated in 2008.[21]

In Bertrand Russell'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."

## References

1. ^ Sutton, Daud (2002), Platonic & Archimedean Solids, Wooden Books, Bloomsbury Publishing USA, p. 55, ISBN 9780802713865.
2. ^
3. ^ http://demonstrations.wolfram.com/DodecahedronAndBilunabirotunda/
4. ^ http://www.lcv.ne.jp/~hhase/memo/m09_08b.html
5. ^ http://mathworld.wolfram.com/images/eps-gif/DodecahedronCube_700.gif
6. ^ http://davidf.faricy.net/polyhedra/images/dodecarect.gif
7. ^ http://www.toshen.com/images/dodecahedronwithgoldrectang.gif
8. ^
9. ^ https://www.uwgb.edu/dutchs/symmetry/xlforms.htm
10. ^ http://www.galleries.com/minerals/property/crystal.htm#dodecahe
11. ^ a b Dodecahedral Crystal Habit
12. ^ Hafner, I. and Zitko, T.; Introduction to golden rhombic polyhedra
13. ^ Lord, K.; Tilings, coverings, clusters and quasicrystals
14. ^ Polyhedra are "topologically identical" if they have the same intrinsic arrangement of faces and vertices, such that one can be distorted into the other simply by changing the lengths of edges or the angles between edges or faces.
15. ^ Counting polyhedra
16. ^
17. ^ Plato, Timaeus, Jowett translation [line 1317-8]; the Greek word translated as delineation is diazographein, painting in semblance of life.
18. ^
19. ^ "Is the universe a dodecahedron?", article at PhysicsWeb.
20. ^ Luminet, Jean-Pierre; Jeff Weeks; Alain Riazuelo; Roland Lehoucq; Jean-Phillipe Uzan (2003-10-09). "Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background". Nature 425 (6958): 593–5. arXiv:astro-ph/0310253. Bibcode:2003Natur.425..593L. doi:10.1038/nature01944. PMID 14534579.
21. ^ Roukema, Boudewijn; Zbigniew Buliński; Agnieszka Szaniewska; Nicolas E. Gaudin (2008). "A test of the Poincaré dodecahedral space topology hypothesis with the WMAP CMB data". Astronomy and Astrophysics 482 (3): 747. arXiv:0801.0006. Bibcode:2008A&A...482..747L. doi:10.1051/0004-6361:20078777.
• Plato's Fourth Solid and the "Pyritohedron", by Paul Stephenson, 1993, The Mathematical Gazette, Vol. 77, No. 479 (Jul., 1993), pp. 220–226 [1]
• THE GREEK ELEMENTS