cuboctahedron
A cuboctahedron, rectified cube, or rectified octahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e., an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral. Its dual polyhedron is the rhombic dodecahedron.
Key facts
- Polyhedron.name
- Cuboctahedron
- Polyhedron.image
- Cuboctahedron (green).png
- Polyhedron.type
- Archimedean solid
- Polyhedron.faces
- 14
- Polyhedron.edges
- 24
- Polyhedron.vertices
- 12
- Polyhedron.vertex_config
- 3.4.3.4
- Polyhedron.schläfli
- r{4,3}
- Polyhedron.conway
- aC
- Polyhedron.symmetry
- Octahedral \mathrm{O}_\mathrm{h}
- Polyhedron.dual
- Rhombic dodecahedron
- Polyhedron.angle
- approximately 125°
- Polyhedron.properties
Article
13 sectionsContents
- Construction
- Properties
- Measurement and other metric properties
- Symmetry and classification
- Radial equilateral symmetry
- Configuration matrix
- Graph
- Related polyhedra and honeycomb
- Appearance
- References
- Footnotes
- Works cited
- External links
{{infobox polyhedron | name = Cuboctahedron | image = Cuboctahedron (green).png | type = Archimedean solid | faces = 14 | edges = 24 | vertices = 12 | vertex_config = 3.4.3.4 | coxeter = | schläfli = r{4,3} | conway = aC | symmetry = Octahedral \mathrm{O}_\mathrm{h} | dual = Rhombic dodecahedron | angle = approximately 125° | properties = convex, vector equilibrium, Rupert property | vertex_figure = Polyhedron 6-8 vertfig.svg | net = Polyhedron 6-8 net.svg }} A cuboctahedron, rectified cube, or rectified octahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e., an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral. Its dual polyhedron is the rhombic dodecahedron.
== Construction == The cuboctahedron can be constructed in many ways: Its construction can be started by attaching two regular triangular cupolas base-to-base. This is similar to one of the Johnson solids, triangular orthobicupola. The difference is that the triangular orthobicupola is constructed with one of the cupolas twisted so that similar polygonal faces are adjacent, whereas the cuboctahedron is not. As a result, the cuboctahedron may also called the triangular gyrobicupola. Its construction can be started from a cube or a regular octahedron, marking the midpoints of their edges, and cutting off all the vertices at those points. This process is known as rectification, making the cuboctahedron being named the rectified cube and rectified octahedron. An alternative construction is by cutting off all vertices (truncation) of a regular tetrahedron and beveling the edges. This process is termed cantellation, lending the cuboctahedron an alternate name of cantellated tetrahedron. From all of these constructions, the cuboctahedron has 14 faces: 8 equilateral triangles and 6 squares. It also has 24 edges and 12 vertices.