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1.1.6 Families of variable degree. ... but a diagram showing how the elements meet. Tessellations ... Truncated rhombic dodecahedron;
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. As a Catalan solid, it is the dual polyhedron of the cuboctahedron. As a parallelohedron, the rhombic dodecahedron can be used to tesselate its copies in space creating a rhombic dodecahedral honeycomb.
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.
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 space-filler, dual to cuboctahedron Rhombic Triacontahedron 7.5 radius = phi/sqrt(2) Icosahedron
rrt{2,3} rrs{2,6} rrCO rrID Conway notation eP3 eA4 eaO = eaC eaI = eaD Polyhedra to be expanded Triangular prism or triangular bipyramid: Square antiprism or tetragonal trapezohedron: Cuboctahedron or rhombic dodecahedron: Icosidodecahedron or rhombic triacontahedron: Image Animation
The illustration here shows the vertex figure (red) of the cuboctahedron being used to derive the corresponding face (blue) of the rhombic dodecahedron.. For a uniform polyhedron, each face of the dual polyhedron may be derived from the original polyhedron's corresponding vertex figure by using the Dorman Luke construction. [2]
Rhombic triacontahedron V(3.5) 2 Johannes Kepler coined the category semiregular in his book Harmonices Mundi (1619), including the 13 Archimedean solids , two infinite families ( prisms and antiprisms on regular bases), and two edge-transitive Catalan solids , the rhombic dodecahedron and rhombic triacontahedron .
The regular hyperbolic honeycombs thus include two with four or five dodecahedra meeting at each edge; their dihedral angles thus are π/2 and 2π/5, both of which are less than that of a Euclidean dodecahedron. Apart from this effect, the hyperbolic honeycombs obey the same topological constraints as Euclidean honeycombs and polychora.