Expanded cuboctahedron

Expanded cuboctahedron
Schläfli symbol	rr$\begin{Bmatrix} 4 \\ 3 \end{Bmatrix}$ = rrr{4,3}
Conway notation	edaC = aaaC
Faces	50: 8 {3} 6+24 {4} 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 80px
Properties	convex
280px Net

The expanded cuboctahedron is a polyhedron, constructed as a expanded 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

Expansion

The expansion operation from the rhombic dodecahedron can be seen in this animation:

Dissection

Excavated expanded cuboctahedron
Faces	86: 8 {3} 6+24+48 {4}
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. 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.

320px

Related polyhedra

Name	Cube	Cubocta- hedron	Rhombi- cuboctahedron	Expanded cuboctahedron
Coxeter	C	CO = rC	rCO = rrC	rrCO = rrrC
Conway		aC = aO	eC	eaC
Image
Conway	O = dC	jC	oC	oaC
Dual				100px

See also

• Rhombicuboctahedron (expanded cube)
• Expanded icosidodecahedron
• Truncated rhombicuboctahedron

References

• Coxeter Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (pp. 145–154 Chapter 8: Truncation)
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5

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