| + Common dodecahedra |
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| Regular | Small stellated | Great | Great stellated |
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| Pyritohedron | Tetartoid | Rhombic | Triangular |
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| Rhombo-hexagonal | Rhombo-square | Trapezo-rhombic | Rhombo-triangular |
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In geometry, a dodecahedron (; ) or duodecahedron[1908 Chambers's Twentieth Century Dictionary of the English Language, 1913 Webster's Revised Unabridged Dictionary] is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as of the convex form. All of these have icosahedral symmetry, 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, a common crystal form in pyrite, has pyritohedral symmetry, while the tetartoid has tetrahedral symmetry.
The rhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron 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 regular dodecahedron is a convex polyhedron with regular pentagonal faces, three meeting at each vertex. It has 12 faces, 30 edges, and 20 vertices.
It is one of the five regular
, named after
Plato who classified the five solids, resembling the classical elements; the regular dodecahedron is a constellation.
Its
Dual polyhedron is the regular icosahedron.
The regular dodecahedron has three , all of which are regular star dodecahedra. They form three of the four Kepler–Poinsot polyhedra. They are the small stellated dodecahedron, the great dodecahedron, and the great stellated dodecahedron. The small stellated dodecahedron and great dodecahedron are dual to each other; the great stellated dodecahedron is dual to the great icosahedron. All of these regular star dodecahedra have regular pentagonal or 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
A
pyritohedron (or pentagonal dodecahedron) is a dodecahedron with pyritohedral symmetry T
h. Like the regular dodecahedron, it has twelve identical
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 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 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 (such as holmium–magnesium–zinc quasicrystal) with icosahedral symmetry, which includes true fivefold rotation axes.
Crystal pyrite
The name
crystal pyrite comes from one of the two common
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°. 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 − h2)),
(±(1 + h), ±(1 − h2), 0) and
(±(1 − h2), 0, ±(1 + h)).
h is the height of the wedge-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).
Another one is h = = 0.618... for the regular dodecahedron. 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.
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| Honeycomb of alternating convex and concave pyritohedra with heights between ± | Heights between 0 (cube)
and 1 (rhombic dodecahedron) |
Geometric freedom
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.
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Versions with equal absolute values and opposing signs form a honeycomb together. (Compare this animation.)
The ratio shown is that of edge lengths, namely those in a set of 24 (touching cube vertices) to those in a set of 6 (corresponding to cube faces). |
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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
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.[Dutch, Steve. The 48 Special Crystal Forms . Natural and Applied Sciences, University of Wisconsin-Green Bay, U.S.] The mineral cobaltite can have this symmetry form.[ Crystal Habit. Galleries.com. Retrieved on 2016-12-02.]
Abstractions sharing the solid's topology 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 this is a gyro tetrahedron.)
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| A tetartoid can be created by enlarging 12 of the 24 faces of a dyakis dodecahedron.
(The tetartoid shown here is based on one that is itself created by enlarging 24 of the 48 faces of the disdyakis dodecahedron.)
{ | |
The
crystal model on the right shows a tetartoid created by enlarging the blue faces of the dyakis dodecahedral core. Therefore, the edges between the blue faces are covered by the red skeleton edges.
|}
Cartesian coordinates
The following points are vertices of a tetartoid pentagon under tetrahedral symmetry:
- ( a, b, c); (− a, − b, c); (−, −, ); (− c, − a, b); (−, , ),
under the following conditions:
[ The Tetartoid. Demonstrations.wolfram.com. Retrieved on 2016-12-02.]
- ,
- n = a2 c − bc2,
- d1 = a2 − ab + b2 + ac − 2 bc,
- d2 = a2 + ab + b2 − ac − 2 bc,
- .
Geometric freedom
The regular dodecahedron is a tetartoid with more than the required symmetry. The triakis tetrahedron 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.)
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 , 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 Stanko Bilinski in 1960.[Hafner, I. and Zitko, T. Introduction to golden rhombic polyhedra. Faculty of Electrical Engineering, University of Ljubljana, Slovenia.] 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, notably distinct dodecahedra (excluding pentagonal and rhombic forms) include:
File:Decagonal prism.png | Decagonal prism, a prism consisting of ten squares and two decagonal bases. Its symmetry group is D10h symmetry of order 40.
File:Pentagonal antiprism.png | Pentagonal antiprism: an antiprism consisting of ten equilateral triangles and two pentagonal bases. It is D5d symmetry of order 20.
File:Pentagonal cupola.png | Pentagonal cupola: a Johnson solid with five triangles, five squares, one pentagon, and one decagon. It is cyclic symmetry symmetry of order 10.
File:Snub disphenoid.png | Snub disphenoid: both Johnson solid and deltahedron, consisting of twelve equilateral triangles. It is D2d of order 8, the same symmetries as a tetragonal disphenoid. Its dual is elongated gyrobifastigium, an octahedron.
File:Elongated square dipyramid.png | Elongated square dipyramid, a Johnson solid. Obtained by augmenting two opposite faces of a cube by equilateral square pyramids, the resulting polyhedron has eight triangles and four squares. It is D4h symmetry of order 16.
File:Metabidiminished icosahedron.png | Metabidiminished icosahedron: a Johnson solid obtained by removing two pentagonal pyramids from a regular icosahedron, resulting in ten triangles and two pentagons. It is cyclic symmetry symmetry of order 4.
File:Hexagonale bipiramide.png | Hexagonal bipyramid: a bipyramid with twelve isosceles , obtained by attaching two hexagonal pyramids base-to-base. If both affixed pyramids have Regular polygon and apices perpendicular to the center of their base, the bipyramid has D6h symmetry of order 24. Like any other bipyramids, the hexagonal bipyramid is face-transitive and the dual of hexagonal prism.
File:Hexagonal trapezohedron.png | Hexagonal trapezohedron: a trapezohedron with twelve kites. It is face-transitive, the dual of hexagonal antiprism, and has D6d symmetry of order 24.
File:Triakistetrahedron.jpg | Triakis tetrahedron: a Catalan solid and the Kleetope of a regular tetrahedron. Obtained by affixing four triangular pyramids onto each face of the regular tetrahedron, the resulting polyhedron has twelve isosceles triangles. It is face-transitive and Rupert property. It is the dual of truncated tetrahedron, shares the same symmetry as Td symmetry of order 24.
File:Hendecagonal pyramid.svg | Hendecagonal pyramid: the twelve isosceles triangles and one regular hendecagon faces in a pyramid. It has cyclic symmetry symmetry of order 11.
File:Trapezo-rhombic dodecahedron.png | Trapezo-rhombic dodecahedron: a polyhedron with six rhombi, six . It is the dual of triangular orthobicupola, both have D3h symmetry of order 12. It is known for tessellating by translating a copy of itself.
File:Rhombo-hexagonal dodecahedron.png | Rhombo-hexagonal dodecahedron (also known as "elongated dodecahedron"): a polyhedron with eight rhombi and four equilateral . it has D4h symmetry of order 16. It is one of five Federov polyhedra or parallelohedra, generated to create its honeycomb.
Practical usage
Armand Spitz used a dodecahedron as the "globe" equivalent for his Digital Dome planetarium projector,
based upon a suggestion from
Albert Einstein.
External links
