Archimedean solid in the context of "Rhombicuboctahedron"

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.

A polyhedral map projection is a map projection based on a spherical polyhedron. Typically, the polyhedron is overlaid on the globe, and each face of the polyhedron is transformed to a polygon or other shape in the plane. The best-known polyhedral map projection is Buckminster Fuller's Dymaxion map. When the spherical polyhedron faces are transformed to the faces of an ordinary polyhedron instead of laid flat in a plane, the result is a polyhedral globe.

Often the polyhedron used is a Platonic solid or Archimedean solid. However, other polyhedra can be used: the AuthaGraph projection makes use of a polyhedron with 96 faces, and the myriahedral projection allows for an arbitrary large number of faces.Although interruptions between faces are common, and more common with an increasing number of faces, some maps avoid them: the Lee conformal projection only has interruptions at its border, and the AuthaGraph projection scales its faces so that the map fills a rectangle without internal interruptions. Some projections can be tesselated to fill the plane, the Lee conformal projection among them.

In geometry, a net of a polyhedron is an arrangement of non-overlapping edge-joined polygons in the plane that can be folded (along edges) to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard.

An early instance of polyhedral nets appears in the works of Albrecht Dürer, whose 1525 book A Course in the Art of Measurement with Compass and Ruler (Unterweysung der Messung mit dem Zyrkel und Rychtscheyd ) included nets for the Platonic solids and several of the Archimedean solids. These constructions were first called nets in 1543 by Augustin Hirschvogel.

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.

The Catalan solids are the dual polyhedra of Archimedean solids. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and symmetric vertices. The faces of the Catalan solids correspond by duality to the vertices of Archimedean solids, and vice versa.

In geometry, the elongated square gyrobicupola is a polyhedron constructed by two square cupolas attaching onto the bases of octagonal prism, with one of them rotated. It is a canonical polyhedron. It is not considered to be an Archimedean solid because it lacks a set of global symmetries that map every vertex to every other vertex, unlike the 13 Archimedean solids. However, it was once mistakenly considered a rhombicuboctahedron by many mathematicians. For this reason, it is also known as the pseudo-rhombicuboctahedron, Miller solid, or Miller–Askinuze solid.

In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or tiling as the sequence of faces around a vertex. It has variously been called a vertex description, vertex type, vertex symbol, vertex arrangement, vertex pattern, face-vector, vertex sequence. It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book Mathematical Models. For uniform polyhedra, there is only one vertex type and therefore the vertex configuration fully defines the polyhedron. (Chiral polyhedra exist in mirror-image pairs with the same vertex configuration.)

For example, "3.5.3.5" indicates a vertex belonging to 4 faces, alternating triangles and pentagons. This vertex configuration defines the vertex-transitive icosidodecahedron. The notation is cyclic and therefore is equivalent with different starting points, so 3.5.3.5 is the same as 5.3.5.3. The order is important, so 3.3.5.5 is different from 3.5.3.5 (the first has two triangles followed by two pentagons). Repeated elements can be collected as exponents so this example is also represented as (3.5).

In geometry, the truncated icosahedron is a polyhedron that can be constructed by truncating all of the regular icosahedron's vertices. Intuitively, it may be regarded as footballs (or soccer balls) that are typically patterned with white hexagons and black pentagons. Geodesic dome structures such as those whose architecture Buckminster Fuller pioneered are often based on this structure. It is an example of an Archimedean solid, as well as a Goldberg polyhedron.