Dodecahedron

Pyritohedron
A pyritohedron has 30 edges: 6 corresponding to cube faces, and 24 touching cube vertices.
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 (for perfect natural pyrite)

A pyritohedron is a dodecahedron with pyritohedral (T_h) 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.

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:

${\displaystyle {\text{Height}}={\frac {\sqrt {5}}{2}}\cdot {\text{Long side}}}$

${\displaystyle {\text{Width}}={\frac {4}{3}}\cdot {\text{Long side}}}$

${\displaystyle {\text{Short sides}}={\sqrt {\frac {7}{12}}}\cdot {\text{Long side}}}$

These ideal proportions are rarely found in nature.

Cubic pyrite

Pyritohedral pyrite...

...with corner angles

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.

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 Tetragonal pentagonal dodecahedron
Face polygon	irregular pentagon
Conway 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 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.

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.

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

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 D_3d 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.