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The rhombic dodecahedron is a space-filling polyhedron, meaning it can be applied to tessellate three-dimensional space: it can be stacked to fill a space, much like hexagons fill a plane. It is a parallelohedron because it can be space-filling a honeycomb in which all of its copies meet face-to-face. [ 7 ]
Each vertex with the acute rhombic face angles has 6 cells containing it. 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.
This is an indexed list of the uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger.. The book was written as a guide book to building polyhedra as physical models.
The rhombicosidodecahedron shares its vertex arrangement with three nonconvex uniform polyhedra: the small stellated truncated dodecahedron, the small dodecicosidodecahedron (having the triangular and pentagonal faces in common), and the small rhombidodecahedron (having the square faces in common).
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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It is possible to fill the plane with polygons which do not meet at their corners, for example using rectangles, as in a brick wall pattern: this is not a proper tiling because corners lie part way along the edge of a neighbouring polygon. Similarly, in a proper honeycomb, there must be no edges or vertices lying part way along the face of a ...
The cube is the only Platonic solid that can fill space, although a tiling that combines tetrahedra and octahedra (the tetrahedral-octahedral honeycomb) is possible. Although the regular tetrahedron cannot fill space, other tetrahedra can, including the Goursat tetrahedra derived from the cube, and the Hill tetrahedra.