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The rhombic dodecahedron can be seen as a degenerate limiting case of a pyritohedron, with permutation of coordinates (±1, ±1, ±1) and (0, 1 + h, 1 − h 2) with parameter h = 1. These coordinates illustrate that a rhombic dodecahedron can be seen as a cube with six square pyramids attached to each face, allowing them to fit together into a ...
Set of Catalan solids The rhombic dodecahedron's construction, the dual polyhedron of a cuboctahedron, by Dorman Luke construction. The Catalan solids are the dual polyhedron of Archimedean solids, a set of thirteen polyhedrons with highly symmetric forms semiregular polyhedrons in which two or more polygonal of their faces are met at a vertex. [1]
The vertices with the obtuse rhombic face angles have 4 cells. The vertices with the acute rhombic face angles have 6 cells. The rhombic dodecahedron can be twisted on one of its hexagonal cross-sections to form a trapezo-rhombic dodecahedron, which is the cell of a somewhat similar tessellation, the Voronoi diagram of hexagonal close-packing.
Any parallelepiped tessellates Euclidean 3-space, as do the five parallelohedra including the cube, hexagonal prism, truncated octahedron, and rhombic dodecahedron. Other space-filling polyhedra include the plesiohedra and stereohedra , polyhedra whose tilings have symmetries taking every tile to every other tile, including the gyrobifastigium ...
Rhombic hexahedron (Dual of tetratetrahedron) — V(3.3.3.3) arccos (0) = π / 2 90° Rhombic dodecahedron (Dual of cuboctahedron) — V(3.4.3.4) arccos (- 1 / 2 ) = 2 π / 3 120° Rhombic triacontahedron (Dual of icosidodecahedron) — V(3.5.3.5) arccos (- √ 5 +1 / 4 ) = 4 π / 5 144° Medial rhombic ...
space-filler, 2As + 1B Tetrahedron 1 self dual, unit volume Coupler 1 space filling oblate octa Cuboctahedron 2.5 edges 1/2, vol. = 1/8 of 20 Duo-Tet Cube 3 24 MITEs Octahedron 4 dual of cube, spacefills w/ tet Rhombic Triacontahedron 5 radius = ~0.9994, vol. = 120 Ts Rhombic Triacontahedron 5+ radius = 1, vol. = 120 Es Rhombic Dodecahedron 6
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These are just the rules for dualising four-dimensional 4-polytopes, except that the usual finite method of reciprocation about a concentric hypersphere can run into problems. The more regular honeycombs dualise neatly: The cubic honeycomb is self-dual. That of octahedra and tetrahedra is dual to that of rhombic dodecahedra.