Dodecahedron

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Animation of a net of a regular (pentagonal) 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. The most familiar dodecahedron is the regular dodecahedron, which is a Platonic solid. The pyritohedron is an irregular pentagonal dodecahedron, having the same topology as the regular one but pyritohedral symmetry instead. The rhombic dodecahedron has octahedral symmetry. There are a large number of other dodecahedra.

Pentagonal dodecahedron[edit]

Topologically equivalent dodecahedra
Dodecahedron.jpg
Regular dodecahedron
Ih, order 120
Pyritohedron.png
Pyritohedron
Th, order 24.
Tetartoid.png
Tetartoid
T, order 12.
Main article: Regular dodecahedron

The convex regular dodecahedron is one of the five regular Platonic solids and can be represented by its Schläfli symbol {5, 3}.

The dual polyhedron is the regular icosahedron {3, 5}, having five equilateral triangles around each vertex.

In crystallography, two important dodecahedra can occur as crystal forms in some symmetry classes of the cubic crystal system that are topologically equivalent to the regular dodecahedron but less symmetrical: the pyritohedron with pyritohedral symmetry, and the tetartoid with tetrahedral symmetry:

Pyritohedron[edit]

Pyritohedron
Pyritohedron.png
A pyritohedron has 30 edges, divided into two lengths: 24 and 6 in each group.
Face polygon irregular pentagon
Coxeter diagrams CDel node.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png
CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
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
Pyritohedron flat.png

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 constrained to be regular, and the underlying atomic arrangement has no true fivefold symmetry axes. Its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes of rotational symmetry are three mutually perpendicular twofold axes.

Although regular dodecahedra do not exist in crystals, the pyritohedron form occurs in the crystals of the mineral pyrite, and it may be an inspiration for the discovery of the regular Platonic solid form. Note that the true regular dodecahedron can occur as a shape for quasicrystals with icosahedral symmetry, which includes true fivefold rotation axes.

Crystal pyrite[edit]

Its name comes from one of the two common crystal habits shown by pyrite, the other one being the cube.

Pyriteespagne.jpg
Cubic pyrite
Pyrite cristal.jpg
Pyritohedral
Ho-Mg-ZnQuasicrystal.jpg
Ho-Mg-Zn quasicrystal

Cartesian coordinates[edit]

Pyritohedron animation.gif

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))

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.

A reflected pyritohedron is made by swapping the nonzero coordinates above. The two pyritohedra can be superimposed to give the compound of two dodecahedra as seen in the image here.

Geometric freedom[edit]

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
Great stellated dodecahedron.png
Regular star, great stellated dodecahedron, with pentagons distorted into regular pentagrams
Concave pyritohedral dodecahedron.png
Concave pyritohedral dodecahedron
Pyritohedron cube.png
A cube can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions.
Irregular dodecahedron.png
The geometric proportions of the pyritohedron in the Weaire–Phelan structure
Dodecahedron.png
A regular dodecahedron is an intermediate case with equal edge lengths.
Rhombicdodecahedron.jpg
A rhombic dodecahedron is the limiting case with the 6 crossedges reducing to length zero.

Tetartoid[edit]

Tetartoid
Tetragonal pentagonal dodecahedron
Tetartoid.png
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.

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.[1] The mineral cobaltite can have this symmetry form.[2]

cobaltite
Koboltglans.jpg

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.

Cartesian coordinates[edit]

To generate a tetartoid pentagon under the tetrahedral group, choose numbers 0≤a≤b≤c.[3]

Calculate n = a2 c - b c2, d1 = a2 - a b + b2 + a c - 2 b c , d2 = a2 + a b + b2 - a c - 2 b c . If n × d1 × d2 ≠ 0, then the following are vertices of a tetartoid pentagon:

((a, b, c), (-a, -b, c), (-n, -n, n)/d1, (-c, -a, b), (-n, n, n)/d2)

Variations[edit]

It can be seen as a tetrahedron, with edges divided into 3 segments, along with a center point of each triangular face. In Conway polyhedron notation it can be seen as gT, a gyro tetrahedron.

Tetartoid cubic.pngTetartoid tetrahedral.png
Example tetartoid variations
Tetartoid-010.png Tetartoid-020.png Tetartoid-040.png
Tetartoid-060.png Tetartoid-080.png Tetartoid-095.png

Rhombic dodecahedron[edit]

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.[4] 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.[5] This figure is another spacefiller, and can also occur in non-periodic spacefillings along with the rhombic triacontahedron, the rhombic icosahedron and rhombic hexahedra.[6]

Other dodecahedra[edit]

There are 6,384,634 topologically distinct convex dodecahedra, excluding mirror images, having at least 8 vertices.[7] (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)

Dodecahedral graph[edit]

Regular dodecahedron graph
Hamiltonian path.svg
A Hamiltonian cycle in a dodecahedron.
Vertices 20
Edges 30
Radius 5
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.

Orthogonal projection
Dodecahedron t0 H3.png

See also[edit]

References[edit]

  • 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

External links[edit]