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The dual of a cube is an octahedron.Vertices of one correspond to faces of the other, and edges correspond to each other. In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. [1]
The illustration here shows the vertex figure (red) of the cuboctahedron being used to derive the corresponding face (blue) of the rhombic dodecahedron.. For a uniform polyhedron, each face of the dual polyhedron may be derived from the original polyhedron's corresponding vertex figure by using the Dorman Luke construction. [2]
For every convex polyhedron, there exists a dual polyhedron having faces in place of the original's vertices and vice versa, and; the same number of edges. The dual of a convex polyhedron can be obtained by the process of polar reciprocation. [34] Dual polyhedra exist in pairs, and the dual of a dual is just the original polyhedron again.
A Goldberg polyhedron is a dual polyhedron of a geodesic polyhedron. A consequence of Euler's polyhedron formula is that a Goldberg polyhedron always has exactly 12 pentagonal faces. Icosahedral symmetry ensures that the pentagons are always regular and that there are always 12 of them.
Their dual tilings are called Laves tilings in honor of crystallographer Fritz Laves. [ 1 ] [ 2 ] They're also called Shubnikov–Laves tilings after Aleksei Shubnikov . [ 3 ] John Conway called the uniform duals Catalan tilings , in parallel to the Catalan solid polyhedra.
In geometry, an n-gonal trapezohedron, n-trapezohedron, n-antidipyramid, n-antibipyramid, or n-deltohedron [3], [4] is the dual polyhedron of an n-gonal antiprism.The 2n faces of an n-trapezohedron are congruent and symmetrically staggered; they are called twisted kites.
Since the quadrilaterals are chiral and non-regular, the dyakis dodecahedron is a non-uniform polyhedron, a type of polyhedron that is not vertex-transitive and does not have regular polygon faces. It is an isohedron, [4] meaning that it is face transitive. The dual polyhedron of a dyakis dodecahedron is the cantic snub octahedron.
The great dodecahemicosacron is the dual of the great dodecahemicosahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the small dodecahemicosacron. Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at ...