Topologically equivalent forms Rhombic dodecahedron

1 topologically equivalent forms

1.1 parallelohedron

1.1.1 dihedral rhombic dodecahedron
1.1.2 bilinski dodecahedron


1.2 deltoidal dodecahedron

topologically equivalent forms

parallelohedron

the rhombic dodecahedron parallelohedron, space-filling polyhedron, dodecahedrille, being dual tetroctahedrille or half cubic honeycomb, , described 2 coxeter diagrams: , .

dihedral rhombic dodecahedron

other symmetry constructions of rhombic dodecahedron space-filling, , parallelotopes similar variations of space-filling truncated octahedra.

for example, 4 square faces, , 60-degree rhombic faces, , d4h dihedral symmetry, order 16. seen cuboctahedron square pyramids augmented on top , bottom.

bilinski dodecahedron

in 1960 stanko bilinski discovered second rhombic dodecahedron 12 congruent rhombus faces, bilinski dodecahedron. has same topology different geometry. rhombic faces in form have golden ratio.

deltoidal dodecahedron

example net (3/4,3/2)

another topologically equivalent variation, called deltoidal dodecahedron or trapezoidal dodecahedron, isohedral tetrahedral symmetry order 24, distorting rhombic faces kites (deltoids). has 8 vertices adjusted in or out in alternate sets of 4, limiting case tetrahedral envelope. variations can parametrized (a,b), b determined planar faces. (1,1) rhombic solution. (a) approaches 1/2, (b) approaches infinity.

(±2, 0, 0), (0, ±2, 0), (0, 0, ±2)
(a, a, a), (−a, −a, a), (−a, a, −a), (a, −a, −a)
(−b, −b, −b), (−b, b, b), (b, −b, b), (b, b, −b)