Expanded cuboctahedron explained

bgcolor=#e7dcc3 colspan=2	Expanded cuboctahedron
Schläfli symbol	rr\begin{Bmatrix} 4 \ 3 \end{Bmatrix} = rrr
Conway notation	edaC = aaaC
Faces	50: 8 6+24 12 rhombs
Edges	96
Vertices	48
Symmetry group	Oh, [4,3], (*432) order 48
Rotation group	O, [4,3]^+, (432), order 24
Dual polyhedron	Deltoidal tetracontaoctahedron
Properties	convex
Net

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

bgcolor=#e7dcc3 colspan=2	Excavated expanded cuboctahedron
Faces	86: 8 6+24+48
Edges	168
Vertices	62
Euler characteristic	-20
genus	11
Symmetry group	Oh, [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.^[1] 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.

Notes and References

1. http://www.orchidpalms.com/polyhedra/rhombic/RD/XRD-dissection.htm A Dissection of the Expanded Rhombic Dodecahedron
2. Web site: Uniform Polyhedron.