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.
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 is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides ; it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform before they refer to it as a “Johnson solid”.
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at each one. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.
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, a cupola is a solid formed by joining two polygons, one with twice as many edges as the other, by an alternating band of isosceles triangles and rectangles. If the triangles are equilateral and the rectangles are squares, while the base and its opposite face are regular polygons, the triangular, square, and pentagonal cupolae all count among the Johnson solids, and can be formed by taking sections of the cuboctahedron, rhombicuboctahedron, and rhombicosidodecahedron, respectively.
In geometry, the square orthobicupola is one of the Johnson solids (J28). As the name suggests, it can be constructed by joining two square cupolae (J4) along their octagonal bases, matching like faces. A 45-degree rotation of one cupola before the joining yields a square gyrobicupola (J29).
In geometry, the square gyrobicupola is one of the Johnson solids (J29). Like the square orthobicupola (J28), it can be obtained by joining two square cupolae (J4) along their bases. The difference is that in this solid, the two halves are rotated 45 degrees with respect to one another.
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 or cantellated triangular prism is one of the Johnson solids. As the name suggests, it can be constructed by elongating a triangular orthobicupola by inserting a hexagonal prism between its two halves. The resulting solid is superficially similar to the rhombicuboctahedron, with the difference that it has threefold rotational symmetry about its axis instead of fourfold symmetry.
In geometry, the elongated triangular gyrobicupola is one of the Johnson solids (J36). As the name suggests, it can be constructed by elongating a "triangular gyrobicupola," or cuboctahedron, by inserting a hexagonal prism between its two halves, which are congruent triangular cupolae (J3). Rotating one of the cupolae through 60 degrees before the elongation yields the triangular orthobicupola (J35).
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 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, moving faces apart and outward, but also maintains the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original edge.
In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty' and from Ancient Greek ἕδρα (hédra) ' seat'. The plural can be either "icosahedra" or "icosahedrons".
In geometry, the elongated gyrobifastigium or gabled rhombohedron is a space-filling octahedron with 4 rectangles and 4 right-angled pentagonal faces.
