A cuboctahedron 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.
In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.
In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a strictly convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two solids with such a property: the first solids are the pyramids, cupolas. and a rotunda; some of the solids may be constructed by attaching with those previous solids, whereas others may not. These solids are named after mathematicians Norman Johnson and Victor Zalgaller.
In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.
In geometry, the square cupola the cupola with octagonal base. In the case of edges are equal in length, it is the Johnson solid, a convex polyhedron with faces are regular. It can be used to construct many polyhedrons, particularly in other Johnson solids.
In geometry, the square orthobicupola is one of the Johnson solids. As the name suggests, it can be constructed by joining two square cupolae along their octagonal bases, matching like faces. A 45-degree rotation of one cupola before the joining yields a square gyrobicupola.
In geometry, the gyroelongated square bicupola is the Johnson solid constructed by attaching two square cupolae on each base of octagonal antiprism. It has the property of chirality.
In geometry, the gyroelongated triangular cupola is one of the Johnson solids (J22). It can be constructed by attaching a hexagonal antiprism to the base of a triangular cupola (J3). This is called "gyroelongation", which means that an antiprism is joined to the base of a solid, or between the bases of more than one solid.
In geometry, the triangular orthobicupola is one of the Johnson solids. As the name suggests, it can be constructed by attaching two triangular cupolas along their bases. It has an equal number of squares and triangles at each vertex; however, it is not vertex-transitive. It is also called an anticuboctahedron, twisted cuboctahedron or disheptahedron. It is also a canonical polyhedron.
In geometry, the elongated triangular orthobicupola is a polyhedron constructed by attaching two regular triangular cupola into the base of a regular hexagonal prism. It is an example of Johnson solid.
In geometry, the elongated triangular gyrobicupola is a polyhedron constructed by attaching two regular triangular cupolas to the base of a regular hexagonal prism, with one of them rotated in . It is an example of Johnson solid.
In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.
A pseudo-uniform polyhedron is a polyhedron which has regular polygons as faces and has the same vertex configuration at all vertices but is not vertex-transitive: it is not true that for any two vertices, there exists a symmetry of the polyhedron mapping the first isometrically onto the second. Thus, although all the vertices of a pseudo-uniform polyhedron appear the same, it is not isogonal. They are called pseudo-uniform polyhedra due to their resemblance to some true uniform polyhedra.
In geometry, an octadecahedron is a polyhedron with 18 faces. No octadecahedron is regular; hence, the name does not commonly refer to one specific polyhedron.
In geometry, the elongated cupolae are an infinite set of polyhedra, constructed by adjoining an n-gonal cupola to an 2n-gonal prism.
In geometry, the gyroelongated cupolae are an infinite set of polyhedra, constructed by adjoining an n-gonal cupola to an 2n-gonal antiprism.
In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion: it moves the faces apart (outward), and adds a new face between each two adjacent faces; but contrary to expansion, it maintains the original vertices. For a polyhedron, this operation adds a new hexagonal face in place of each original edge.
In geometry, the elongated gyrobifastigium or gabled rhombohedron is a space-filling octahedron with 4 rectangles and 4 right-angled pentagonal faces.
