Expanded cuboctahedronSchläfli symbolrr = rrr{4,3}Conway notationeaC = aeC = aaaCFaces50:8 {3}6+24 {4}12 rhombsEdges96Vertices48Symmetry groupOh, [4,3], (*432) order 48Rotation groupO, [4,3]+, (432), order 24Dual polyhedronDeltoidal tetracontaoctahedronPropertiesconvexNet The expanded cuboctahedron is a polyhedron constructed by expansion of the cuboctahedron. It has 50 faces: 8 triangles, 30 squares, and 12 rhombs. The 48 vertices exist at two sets of 24, with a slightly different distance from its center.
It can also be constructed as a rectified rhombicuboctahedron.
Other names Expanded rhombic dodecahedron Rectified rhombicuboctahedron Rectified small rhombicuboctahedron Rhombirhombicuboctahedron Expanded expanded tetrahedron
Expansion The expansion operation from the rhombic dodecahedron can be seen in this animation:
Honeycomb The expanded cuboctahedron can fill space along with a cuboctahedron, octahedron, and triangular prism.
Dissection Excavated expanded cuboctahedronFaces86:8 {3}6+24+48 {4}Edges168Vertices62Euler characteristic-20genus11Symmetry groupOh, [4,3], (*432) order 48 This polyhedron can be dissected into a central rhombic dodecahedron surrounded by: 12 rhombic prisms, 8 tetrahedra, 6 square pyramids, and 24 triangular prisms.
If the central rhombic dodecahedron and the 12 rhombic prisms are removed, you can create a toroidal polyhedron with all regular polygon faces.A Dissection of the Expanded Rhombic Dodecahedron This toroid has 86 faces (8 triangles and 78 squares), 168 edges, and 62 vertices. 14 of the 62 vertices are on the interior, defining the removed central rhombic dodecahedron. With Euler characteristic χ = f + v - e = -20, its genus, g = (2-χ)/2 is 11.
Related polyhedra
NameCubeCubocta-hedronRhombi-cuboctahedronExpandedCuboctahedronExpandedRhombicuboctahedronCoxeterCCO = rC rCO = rrC rrCO = rrrC rrrCO = rrrrCConwayaC = aOeCeaCeeCImageConwayO = dCjCoCoaCoeCDual
See also Rhombicuboctahedron (expanded cube) Expanded icosidodecahedron Truncated rhombicuboctahedron
References
Coxeter Regular Polytopes, Third edition, (1973), Dover edition, (pp. 145–154 Chapter 8: Truncation) John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008,
External links George Hart's Conway interpreter: generates polyhedra in VRML, taking Conway notation as input Variations on a Rhombic Theme www.software3d.com: Prism-Expanded Dissected Cuboctahedron