Regular dodecahedron

Last updated on 22 November 2017

A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular composed of twelve regular pentagonal faces, with three meeting at each vertex, and is represented by the Schläfli symbol {5,3}. 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).[1]

Regular dodecahedron
Dodecahedron.jpg
(Click here for rotating model)
Type Platonic solid
Elements F = 12, E = 30
V = 20 (χ = 2)
Faces by sides 12{5}
Conway notation D
Schläfli symbols {5,3}
Face configuration V3.3.3.3.3
Wythoff symbol 3 | 2 5
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
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)
Dodecahedron vertfig.png
5.5.5
(Vertex figure)
Icosahedron.png
Regular icosahedron
(dual polyhedron)
Dodecahedron flat.svg
Net
Dodecaedro desarrollo.gif
Animation of a net of a regular (pentagonal) dodecahedron being folded

Dimensions

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

OEISA179296

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

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

These quantities may also be expressed as

where ϕ is the golden ratio.

Note that, given a regular 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:

Two-dimensional symmetry projections

The regular 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 Dodecahedron t0 A2.png Dodecahedron t0 e.png Dodecahedron t0 H3.png
Projective
symmetry
[[3]] = [6] [2] [[5]] = [10]

In perspective projection, viewed above a pentagonal face, the regular 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
Regular dodecahedron Dodecahedron t0 H3.png Dodecahedron schlegel diagram.png Dodecahedron stereographic projection.png
Dodecaplex
(120-cell)
120-cell t0 H3.svg Schlegel wireframe 120-cell.png Stereographic polytope 120cell faces.png

Cartesian coordinates

Dodecahedron vertices.png
Vertex coordinates:
     The orange vertices lie at (±1, ±1, ±1) and form a cube (dotted lines).
     The green vertices lie at (0, ±ϕ, ±1/ϕ) and form a rectangle on the yz-plane.
     The blue vertices lie at (±1/ϕ, 0, ±ϕ) and form a rectangle on the xz-plane.
     The pink vertices lie at (±ϕ, ±1/ϕ, 0) and form a rectangle on the xy-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 20 vertices of a regular dodecahedron centered at the origin and suitably scaled and oriented:[2]

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

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.

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:

ϕx ± y = ±ϕ2
ϕy ± z = ±ϕ2
ϕz ± x = ±ϕ2

Properties

  • The dihedral angle of a regular dodecahedron is 2 arctan(ϕ) or approximately 116.5650512° (where again ϕ = 1 + 5/2, the golden ratio). OEISA137218 Note that the tangent of the dihedral angle is exactly −2.
  • If the original regular dodecahedron has edge length 1, its dual icosahedron 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 an equilateral triangle with the body center.

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

A rectified regular 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.

Relation to the regular icosahedron

When a regular 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 + 0.6).

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

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.[3] In fact, five cubes may overlap and interlock inside the regular 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 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 : 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

Relation to the golden rectangle

Golden ratio rectangles of ratio (ϕ + 1) : 1 and ϕ : 1 also fit perfectly within a regular dodecahedron.[4] 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 regular dodecahedron).

In addition, the center of each face of the regular dodecahedron form three intersecting golden rectangles.[5]

Relation to the 6-cube and rhombic triacontahedron

6demicube-even-dodecahedron.png
Projection of 6-demicube into regular dodecahedral envelope

It can be projected to 3D from the 6-dimensional 6-demicube using the same basis vectors that form the hull of the rhombic triacontahedron from the 6-cube. Shown here including the inner 12 vertices, which are not connected by the outer hull edges of 6D norm length 2, 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

Roman dodecahedron.jpg
Roman dodecahedron
Rfel vsesmer front.png
Omnidirectional sound source
Dodecahedron climbing wall.jpg
A climbing wall consisting of three dodecahedral pieces

Regular 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."[6] 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 regular dodecahedron, is never directly mentioned as such; "this God used in the delineation of the universe."[7] Aristotle also postulated that the heavens were made of a fifth element, which he called aithêr (aether in Latin, ether in American English).

Regular 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 regular dodecahedron. Gerard Caris based his entire artistic oeuvre on the regular dodecahedron and the pentagon, which is presented as a new art movement coined as Pentagonism.

In modern role-playing games, the regular 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.[8]

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 regular dodecahedron.

The Megaminx 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 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.

Dodecahedron is the name of an avant-garde black metal band from Netherlands.[9]

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 and colleagues in 2003,[10][11] and an optimal orientation on the sky for the model was estimated in 2008.[12]

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."

Space filling with cube and bilunabirotunda

Regular dodecahedra fill space with cubes and bilunabirotundae (Johnson solid 91), in the ratio of 1 to 1 to 3.[13][14] 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.

J91.jpg
Block model
Honeycomb of regular dodecahedra-cubes-J91.png Dodecahedron lattice.png
Lattice of dodecahedra
Bilunabirotunda augmented cube.png
6 bilunabirotundae around a cube

Related polyhedra and tilings

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

*n32 symmetry mutation of regular tilings: {n,3}
Spherical Euclidean Compact hyperb. Paraco. Noncompact hyperbolic
Spherical trigonal hosohedron.png Uniform tiling 332-t0-1-.png Uniform tiling 432-t0.png Uniform tiling 532-t0.png Uniform polyhedron-63-t0.png H2 tiling 237-1.png H2 tiling 238-1.png H2 tiling 23i-1.png H2 tiling 23j12-1.png H2 tiling 23j9-1.png H2 tiling 23j6-1.png H2 tiling 23j3-1.png
{2,3} {3,3} {4,3} {5,3} {6,3} {7,3} {8,3} {∞,3} {12i,3} {9i,3} {6i,3} {3i,3}

The regular 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)
Uniform polyhedron-53-t0.png Uniform polyhedron-53-t01.png Uniform polyhedron-53-t1.png Uniform polyhedron-53-t12.png Uniform polyhedron-53-t2.png Uniform polyhedron-53-t02.png Uniform polyhedron-53-t012.png Uniform polyhedron-53-s012.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
{5,3} t{5,3} r{5,3} t{3,5} {3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
Icosahedron.svg Triakisicosahedron.jpg Rhombictriacontahedron.svg Pentakisdodecahedron.jpg Dodecahedron.svg Deltoidalhexecontahedron.jpg Disdyakistriacontahedron.jpg Pentagonalhexecontahedronccw.jpg
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}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png =
CDel nodes 10ru.pngCDel split2.pngCDel node.png or CDel nodes 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel nodes 10ru.pngCDel split2.pngCDel node 1.png or CDel nodes 01rd.pngCDel split2.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h0.png =
CDel node h.pngCDel split1.pngCDel nodes hh.png
Uniform polyhedron-43-t0.svg Uniform polyhedron-43-t01.svg Uniform polyhedron-43-t1.svg
Uniform polyhedron-33-t02.png
Uniform polyhedron-43-t12.svg
Uniform polyhedron-33-t012.png
Uniform polyhedron-43-t2.svg
Uniform polyhedron-33-t1.png
Uniform polyhedron-43-t02.png
Rhombicuboctahedron uniform edge coloring.png
Uniform polyhedron-43-t012.png Uniform polyhedron-43-s012.png Uniform polyhedron-33-t0.pngUniform polyhedron-33-t2.png Uniform polyhedron-33-t01.pngUniform polyhedron-33-t12.png Uniform polyhedron-43-h01.svg
Uniform polyhedron-33-s012.png
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
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 4.pngCDel node.png
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Octahedron.svg Triakisoctahedron.jpg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Hexahedron.svg Deltoidalicositetrahedron.jpg Disdyakisdodecahedron.jpg Pentagonalicositetrahedronccw.jpg Tetrahedron.svg Triakistetrahedron.jpg Dodecahedron.svg

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.

n32 symmetry mutations of snub tilings: 3.3.3.3.n
Symmetry
n32
Spherical Euclidean Compact hyperbolic Paracomp.
232 332 432 532 632 732 832 ∞32
Snub
figures
Spherical trigonal antiprism.png Spherical snub tetrahedron.png Spherical snub cube.png Spherical snub dodecahedron.png Uniform tiling 63-snub.png Uniform tiling 73-snub.png Uniform tiling 83-snub.png Uniform tiling i32-snub.png
Config. 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.∞
Gryro
figures
Uniform tiling 432-t0.png Uniform tiling 532-t0.png Spherical pentagonal icositetrahedron.png Spherical pentagonal hexecontahedron.png Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg Ord7 3 floret penta til.png Order-3-infinite floret pentagonal tiling.png
Config. 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.∞

Vertex arrangement

The regular 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 regular 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 regular dodecahedron.

Great stellated dodecahedron.png
Great stellated dodecahedron
Small ditrigonal icosidodecahedron.png
Small ditrigonal icosidodecahedron
Ditrigonal dodecadodecahedron.png
Ditrigonal dodecadodecahedron
Great ditrigonal icosidodecahedron.png
Great ditrigonal icosidodecahedron
Compound of five cubes.png
Compound of five cubes
Compound of five tetrahedra.png
Compound of five tetrahedra
Compound of ten tetrahedra.png
Compound of ten tetrahedra

Stellations

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

0 1 2 3
Stellation Dodecahedron.png
Regular dodecahedron
Small stellated dodecahedron.png
Small stellated dodecahedron
Great dodecahedron.png
Great dodecahedron
Great stellated dodecahedron.png
Great stellated dodecahedron
Facet diagram Zeroth stellation of dodecahedron facets.svg First stellation of dodecahedron facets.svg Second stellation of dodecahedron facets.svg Third stellation of dodecahedron facets.svg

Dodecahedral graph

Regular dodecahedron graph
Hamiltonian path.svg
A Hamiltonian cycle in a dodecahedron.
Vertices 20
Edges 30
Radius 5
Diameter 5
Girth 5
Automorphisms 120 (A5 × Z2)[15]
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.[16]

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.

Orthogonal projection
Dodecahedron t0 H3.png

See also

References

  1. ^ Sutton, Daud (2002), Platonic & Archimedean Solids, Wooden Books, Bloomsbury Publishing USA, p. 55, ISBN 9780802713865.
  2. ^ Weisstein, Eric W. "Icosahedral group". MathWorld.
  3. ^ http://mathworld.wolfram.com/images/eps-gif/DodecahedronCube_700.gif
  4. ^ http://davidf.faricy.net/polyhedra/images/dodecarect.gif
  5. ^ http://www.toshen.com/images/dodecahedronwithgoldrectang.gif
  6. ^ Florian Cajori, A History of Mathematics (1893)
  7. ^ Plato, Timaeus, Jowett translation [line 1317–8]; the Greek word translated as delineation is diazographein, painting in semblance of life.
  8. ^ Dodecahedral Crystal Habit
  9. ^ "Dodecahedron on Metal Archives".
  10. ^ Dumé, Belle (Oct 8, 2003). "Is The Universe A Dodecahedron?". PhysicsWorld. Archived from the original on 2012-04-25.
  11. ^ 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/0310253Freely accessible. Bibcode:2003Natur.425..593L. doi:10.1038/nature01944. PMID 14534579.
  12. ^ 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:0801.0006Freely accessible. Bibcode:2008A&A...482..747L. doi:10.1051/0004-6361:20078777.
  13. ^ http://demonstrations.wolfram.com/DodecahedronAndBilunabirotunda/
  14. ^ http://www.lcv.ne.jp/~hhase/memo/m09_08b.html
  15. ^ Frucht, Roberto (1936–1937), "Die gruppe des Petersen'schen Graphen und der Kantensysteme der regulären Polyeder", Comment. Math. Helv., 9: 217–223, doi:10.5169/seals-10183
  16. ^ Weisstein, Eric W. "Dodecahedral Graph". MathWorld.

External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds

This page is based on a Wikipedia article written by authors (here).
Text is available under the CC BY-SA 3.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.