Dodecahedron

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. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

Common dodecahedra
Ih, order 120
Regular- Small stellated- Great- Great stellated-
Th, order 24 T, order 12 Oh, order 48 Johnson (J84)
Pyritohedron Tetartoid Rhombic- Triangular-
D4h, order 16 D3h, order 12
Rhombo-hexagonal- Rhombo-square- Trapezo-rhombic- Rhombo-triangular-

The pyritohedron, a common crystal form in pyrite, is an irregular pentagonal dodecahedron, having the same topology (in terms of its vertices as a graph) as the regular one but pyritohedral symmetry while the tetartoid has tetrahedral symmetry. The rhombic dodecahedron, seen as a limiting case of the pyritohedron, has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling. There are numerous other dodecahedra.

Regular dodecahedra
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.

Four kinds of regular dodecahedra

Convex regular dodecahedron
Small stellated dodecahedron
Great dodecahedron
Great stellated dodecahedron

The convex regular dodecahedron also has three stellations, all of which are regular star dodecahedra. They form three of the four Kepler–Poinsot polyhedra. They are the small stellated dodecahedron {5/2, 5}, the great dodecahedron {5, 5/2}, and the great stellated dodecahedron {5/2, 3}. The small stellated dodecahedron and great dodecahedron are dual to each other; the great stellated dodecahedron is dual to the great icosahedron {3, 5/2}. All of these regular star dodecahedra have regular pentagonal or pentagrammic faces. The convex regular dodecahedron and great stellated dodecahedron are different realisations of the same abstract regular polyhedron; the small stellated dodecahedron and great dodecahedron are different realisations of another abstract regular polyhedron.

Other pentagonal dodecahedra

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
Pyritohedron

A pyritohedron has 30 edges: 6 corresponding to cube faces, and 24 touching cube vertices.
Face polygonirregular pentagon
Coxeter diagrams
Faces12
Edges30 (6 + 24)
Vertices20 (8 + 12)
Symmetry groupTh, [4,3+], (3*2), order 24
Rotation groupT, [3,3]+, (332), order 12
Dual polyhedronPseudoicosahedron
Propertiesface transitive
Net (for perfect natural pyrite)

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 (see figure).[1] 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 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, and it may be an inspiration for the discovery of the regular Platonic solid form. The true regular dodecahedron can occur as a shape for quasicrystals (such as holmium–magnesium–zinc quasicrystal) with icosahedral symmetry, which includes true fivefold rotation axes.

Crystal pyrite

Its name comes from one of the two common crystal habits shown by pyrite, the other one being the cube. In pyritohedral pyrite, the faces have a Miller index of (210), which means that the dihedral angle 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°. In a perfect crystal, the measurements of an ideal face would be:

These ideal proportions are rarely found in nature.


Cubic pyrite
Pyritohedral pyrite...
...with corner angles

Cartesian coordinates

If the eight vertices of a cube have coordinates of:

(±1, ±1, ±1)

Then a pyritohedron has 12 additional vertices:

(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 the pyritohedron reduces to a rhombic dodecahedron. When h = 0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h = −1 + √5/2, the multiplicative inverse of the golden ratio, the result is a regular dodecahedron. When h = −1 − √5/2, the conjugate of this value, the result is a regular great stellated dodecahedron. For natural pyrite, h = 1/2.

Orthogonal projections of a pyritohedron with a wedge height h = 1/2, or 1/4 the cube edge length. This is the same as natural pyrite. These proportions are also found in the Weaire–Phelan structure.
At left, h = 1/2. At right, h = 1/φ (a regular dodecahedron).
Pyritohedra in dual positions

A reflected pyritohedron is made by swapping the nonzero coordinates above. The two pyritohedra can be superimposed to give the compound of two dodecahedra. The image to the left shows the case where the pyritohedra are convex regular dodecahedra.

Geometric freedom
Animation of convex/concave pyritohedral honeycomb, between h=±√5 − 1/2

The pyritohedron has a geometric degree of freedom with limiting cases of a cubic convex hull at one limit of collinear 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.

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 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.

Special cases of the pyritohedron
1 : 1 0 : 1 1 : 1 2 : 1 1 : 1 0 : 1 1 : 1
h = −√5 + 1/2 h = −1 h = −√5 + 1/2 h = 0 h = √5 − 1/2 h = 1 h = √5 + 1/2

Regular star, great stellated dodecahedron, with regular pentagram faces
Degenerate, 12 vertices in the center
The concave equilateral dodecahedron, called an endo-dodecahedron.
A cube can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions.
A regular dodecahedron is an intermediate case with equal edge lengths.
A rhombic dodecahedron is a degenerate case with the 6 crossedges reduced to length zero.
Self-intersecting equilateral dodecahedron

Tetartoid
Tetartoid
Tetragonal pentagonal dodecahedron
Face polygonirregular pentagon
Conway notationgT
Faces12
Edges30 (6+12+12)
Vertices20 (4+4+12)
Symmetry groupT, [3,3]+, (332), order 12
Propertiesconvex, 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 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.[2] The mineral cobaltite can have this symmetry form.[3]

cobaltite

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

The following points are vertices of a tetartoid pentagon under tetrahedral symmetry:

(a, b, c); (−a, −b, c); (−n/d1, −n/d1, n/d1); (−c, −a, b); (−n/d2, n/d2, n/d2),

under the following conditions:[4]

0 ≤ a ≤ b ≤ c,
n = a2c − bc2,
d1 = a2 − ab + b2 + ac − 2bc,
d2 = a2 + ab + b2 − ac − 2bc,
nd1d2 ≠ 0.

Variations

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.

Example tetartoid variations

Dual of triangular gyrobianticupola

A lower symmetry form of the regular dodecahedron can be constructed as the dual of a polyhedra constructed from two triangular anticupola connected base-to-base, called a triangular gyrobianticupola. It has D3d 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
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. 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 parallelohedral spacefiller.

Another important rhombic dodecahedron, the Bilinski 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

There are 6,384,634 topologically distinct convex dodecahedra, excluding mirror images—the number of vertices ranges from 8 to 20.[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)

Practical usage

Armand Spitz used a dodecahedron as the "globe" equivalent for his Digital Dome planetarium projector.[8] based upon a suggestion from Albert Einstein.

gollark: Mon*ads* are just monoids in the category of endofunctors.
gollark: (It is JOKE®)
gollark: .
gollark: Which are just endofunctors in the category of monads
gollark: It's for monoids.

See also

References
  1. Crystal Habit. Galleries.com. Retrieved on 2016-12-02.
  2. Dutch, Steve. The 48 Special Crystal Forms Archived 2013-09-18 at the Wayback Machine. Natural and Applied Sciences, University of Wisconsin-Green Bay, U.S.
  3. Crystal Habit. Galleries.com. Retrieved on 2016-12-02.
  4. The Tetartoid. Demonstrations.wolfram.com. Retrieved on 2016-12-02.
  5. Hafner, I. and Zitko, T. Introduction to golden rhombic polyhedra. Faculty of Electrical Engineering, University of Ljubljana, Slovenia.
  6. Lord, E. A.; Ranganathan, S.; Kulkarni, U. D. (2000). "Tilings, coverings, clusters and quasicrystals". Curr. Sci. 78: 64–72.
  7. Counting polyhedra. Numericana.com (2001-12-31). Retrieved on 2016-12-02.
  8. Ley, Willy (February 1965). "Forerunners of the Planetarium". For Your Information. Galaxy Science Fiction. pp. 87–98.
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 Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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