Login
Login

expanded cuboctahedron

{{Short description|Type of polyhedron}}

class="wikitable" bgcolor="#ffffff" cellpadding="5" align="right" style="margin-left:10px" width="240"

!bgcolor=#e7dcc3 colspan=2|Expanded cuboctahedron

colspan=2 align=center|240px
bgcolor=#e7dcc3|Schläfli symbolrr\begin{Bmatrix} 4 \\ 3 \end{Bmatrix} = rrr{4,3}
bgcolor=#e7dcc3|Conway notationeaC = aeC = aaaC
bgcolor=#e7dcc3|Faces50:
8 {3}
6+24 {4}
12 rhombs
bgcolor=#e7dcc3|Edges96
bgcolor=#e7dcc3|Vertices48
bgcolor=#e7dcc3|Symmetry groupOh, [4,3], (*432) order 48
bgcolor=#e7dcc3|Rotation groupO, [4,3]+, (432), order 24
bgcolor=#e7dcc3|Dual polyhedronDeltoidal tetracontaoctahedron
80px
bgcolor=#e7dcc3|Propertiesconvex
colspan=2 align=center|280px
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:

:File:R1-R3.gif

{{Clear}}

Honeycomb

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

320px

Dissection

class="wikitable" bgcolor="#ffffff" cellpadding="5" align="right" style="margin-left:10px" width="240"

!bgcolor=#e7dcc3 colspan=2|Excavated expanded cuboctahedron

bgcolor=#e7dcc3|Faces86:
8 {3}
6+24+48 {4}
bgcolor=#e7dcc3|Edges168
bgcolor=#e7dcc3|Vertices62
bgcolor=#e7dcc3|Euler characteristic|
20
bgcolor=#e7dcc3|genus11
bgcolor=#e7dcc3|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.[http://www.orchidpalms.com/polyhedra/rhombic/RD/XRD-dissection.htm 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.

:320px

Related polyhedra

class=wikitable

!Name

!Cube

!Cubocta-
hedron

!Rhombi-
cuboctahedron

!Expanded
Cuboctahedron

!Expanded
Rhombicuboctahedron

align=center

!Coxeter{{Cite web|url=http://mathworld.wolfram.com/UniformPolyhedron.html|title=Uniform Polyhedron}}

!rowspan=2|C

!CO = rC

! rCO = rrC

! rrCO = rrrC

! rrrCO = rrrrC

align=center

!Conway

!aC = aO

!eC

!eaC

!eeC

align=center

!Image

|100px

|100px

|100px

|100px

|[File:Expanded rhombicuboctahedron.png|

align=center

!Conway

!O = dC

!jC

!oC

!oaC

!oeC

align=center

!Dual

|100px

|100px

|100px

|100px

|

See also

References

{{reflist}}

External links

  • [http://www.georgehart.com/virtual-polyhedra/conway_notation.html George Hart's Conway interpreter]: generates polyhedra in VRML, taking Conway notation as input
  • [http://www.orchidpalms.com/polyhedra/rhombic/RTC/rhomvar.htm Variations on a Rhombic Theme]
  • [http://www.software3d.com/DrillRhom.php www.software3d.com: Prism-Expanded Dissected Cuboctahedron]

Category:Polyhedra