Expanded cuboctahedron

The expanded cuboctahedron is a polyhedron, constructed as an 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.

Expanded cuboctahedron
Schläfli symbolrr = rrr{4,3}
Conway notationedaC = aaaC
Faces50:
8 {3}
6+24 {4}
12 rhombs
Edges96
Vertices48
Symmetry groupOh, [4,3], (*432) order 48
Rotation groupO, [4,3]+, (432), order 24
Dual polyhedronDeltoidal tetracontaoctahedron
Propertiesconvex

Net

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 cuboctahedron
Faces86:
8 {3}
6+24+48 {4}
Edges168
Vertices62
Euler characteristic-20
genus11
Symmetry 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.[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.

Name Cube Cubocta-
hedron		 Rhombi-
cuboctahedron		 Expanded
cuboctahedron
Coxeter[2] C CO = rC rCO = rrC rrCO = rrrC
Conway aC = aO eC eaC
Image
Conway O = dC jC oC oaC
Dual

See also

References
  1. A Dissection of the Expanded Rhombic Dodecahedron
  2. http://mathworld.wolfram.com/UniformPolyhedron.html
  • 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
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