Expanded cuboctahedron

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Expanded cuboctahedron
Expanded dual cuboctahedron.png
Schläfli symbol rr = 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.png
Properties convex
Expanded cuboctahedron net.png
Net

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.

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:

R1-R3.gif

Honeycomb[]

The expanded cuboctahedron can fill space along with a cuboctahedron, octahedron, and triangular prism.

HC R3-P3-A3-Pr3.png

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.[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.

Excavated expanded cuboctahedron.png

Related polyhedra[]

Name Cube Cubocta-
hedron 		Rhombi-
cuboctahedron 		Expanded
cuboctahedron
Coxeter[2] C CO = rC rCO = rrC rrCO = rrrC
Conway aC = aO eC eaC
Image Uniform polyhedron-43-t0.svg Uniform polyhedron-43-t1.svg Uniform polyhedron-43-t02.png Expanded dual cuboctahedron.png
Conway O = dC jC oC oaC
Dual Uniform polyhedron-43-t2.svg Dual cuboctahedron.png Deltoidalicositetrahedron.jpg Deltoidal tetracontaoctahedron.png

See also[]

References[]

  1. ^ A Dissection of the Expanded Rhombic Dodecahedron
  2. ^ "Uniform Polyhedron".

External links[]

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