Tetartoid Dodecahedron

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.