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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]
A polyhedron with only equilateral triangles as faces is called a deltahedron. There are eight convex deltahedra, one of which is a triangular bipyramid with regular polygonal faces. [ 1 ] A convex polyhedron in which all of its faces are regular polygons is the Johnson solid , and every convex deltahedron is a Johnson solid.
Their dual, the Archimedean solids, are vertex-transitive but not face-transitive. Each Catalan solid has constant dihedral angles , meaning the angle between any two adjacent faces is the same. [ 1 ]
Its dual is an unequal n-antiprism, with the top and bottom n-gons of different radii. If the kites are twisted and are of two different shapes, the n -trapezohedron can only have C n (cyclic) symmetry, order n , and is called an unequal twisted trapezohedron .
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.
Simple examples of Goldberg polyhedra include the dodecahedron and truncated icosahedron. Other forms can be described by taking a chess knight move from one pentagon to the next: first take m steps in one direction, then turn 60° to the left and take n steps. Such a polyhedron is denoted GP(m,n).
The triakis icosahedron is a Catalan solid, the dual polyhedron of the truncated dodecahedron. The truncated dodecahedron is an Archimedean solid , with faces that are regular decagons and equilateral triangles , and with all edges having unit length; its vertices lie on a common sphere, the circumsphere of the truncated decahedron.