Pentagonal dodecahedron Dodecahedron

1 pentagonal dodecahedron

1.1 pyritohedron

1.1.1 crystal pyrite
1.1.2 cartesian coordinates
1.1.3 geometric freedom


1.2 tetartoid

1.2.1 cartesian coordinates
1.2.2 variations


1.3 dual of triangular gyrobianticupola

pentagonal dodecahedron

the convex regular dodecahedron 1 of 5 regular platonic solids , can represented schläfli symbol {5, 3}.

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

in crystallography, 2 important dodecahedra can occur crystal forms in symmetry classes of cubic crystal system topologically equivalent regular dodecahedron less symmetrical: pyritohedron pyritohedral symmetry, , tetartoid tetrahedral symmetry:

pyritohedron

a pyritohedron dodecahedron pyritohedral (th) symmetry. regular dodecahedron, has twelve identical pentagonal faces, 3 meeting in each of 20 vertices. however, pentagons not constrained regular, , underlying atomic arrangement has no true fivefold symmetry axes. 30 edges divided 2 sets – containing 24 , 6 edges of same length. axes of rotational symmetry 3 mutually perpendicular twofold axes , 4 threefold axes.

although regular dodecahedra not exist in crystals, pyritohedron form occurs in crystals of mineral pyrite, , may inspiration discovery of regular platonic solid form. note true regular dodecahedron can occur shape quasicrystals icosahedral symmetry, includes true fivefold rotation axes.

crystal pyrite

its name comes 1 of 2 common crystal habits shown pyrite, other 1 being cube.

cartesian coordinates

the coordinates of 8 vertices of original cube are:

(±1, ±1, ±1)

the coordinates of 12 vertices of cross-edges are:

(0, ±(1 + h), ±(1 − h))
(±(1 + h), ±(1 − h), 0)
(±(1 − h), 0, ±(1 + h))

where h height of wedge-shaped roof above faces of cube. when h = 1, 6 cross-edges degenerate points , rhombic dodecahedron formed. when h = 0, cross-edges absorbed in facets of cube, , pyritohedron reduces cube. when h = √5 − 1/2, inverse of golden ratio, result regular dodecahedron.

pyritohedra in dual positions

a reflected pyritohedron made swapping nonzero coordinates above. 2 pyritohedra can superimposed give compound of 2 dodecahedra seen in image here.

geometric freedom

the pyritohedron has geometric degree of freedom limiting cases of cubic convex hull @ 1 limit of colinear edges, , rhombic dodecahedron other limit 6 edges degenerated length zero. regular dodecahedron represents special intermediate case edges , angles equal.

tetartoid


tetartoid

a tetartoid (also tetragonal pentagonal dodecahedron, pentagon-tritetrahedron, , tetrahedric pentagon dodecahedron) dodecahedron chiral tetrahedral symmetry (t). regular dodecahedron, has twelve identical pentagonal faces, 3 meeting in each of 20 vertices. however, pentagons not regular , figure has no fivefold symmetry axes.

although regular dodecahedra not exist in crystals, tetartoid form does. name tetartoid comes greek root one-fourth because has 1 fourth of full octahedral symmetry, , half of pyritohedral symmetry. mineral cobaltite can have symmetry form.

its topology can cube square faces bisected 2 rectangles pyritohedron, , bisection lines slanted retaining 3-fold rotation @ 8 corners.

cartesian coordinates

the following points vertices of 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 following conditions:

0 ≤ ≤ b ≤ c,
n = ac − bc,
d1 = − ab + b + ac − 2bc,
d2 = + ab + b − ac − 2bc,
nd1d2 ≠ 0.

variations

it can seen tetrahedron, edges divided 3 segments, along center point of each triangular face. in conway polyhedron notation can seen gt, gyro tetrahedron.

dual of triangular gyrobianticupola

a lower symmetry form of regular dodecahedron can constructed dual of polyhedra constructed 2 triangular anticupola connected base-to-base, called triangular gyrobianticupola. has d3d symmetry, order 12. has 2 sets of 3 identical pentagons on top , bottom, connected 6 pentagons around sides alternate upwards , downwards. form has hexagonal cross-section , identical copies can connected partial hexagonal honeycomb, vertices not match.

^ dutch, steve. 48 special crystal forms. natural , applied sciences, university of wisconsin-green bay, u.s.
^ crystal habit. galleries.com. retrieved on 2016-12-02.
^ tetartoid. demonstrations.wolfram.com. retrieved on 2016-12-02.