![]() ![]() ![]() The symmetry of the icosahedron to come up with some fairly What about the other regular polyhedra in three-space? We can exploit Yields the coordinates of the 2 n vertices of the n-dimensionalĬube-dual as the points at unit distance from the origin on the In n-dimensional space, this same construction These six points are the vertices of a regular octahedron with The squares are the six points at unit distance on the positive and Where all vertices have first coordinate -1 is given by the four Out to be more convenient to start with a cube centered at the origin Having all coordinates 0 or 1, but in considering the dual, it turns In the previous paragraph, we studied a cube with vertices Square faces and obtain the coordinates for the vertices of the Of the cube, we can figure out the coordinates of the centers of If we choose coordinates for the vertices Vertices of a regular octahedron can be obtained as the centers of the Three-dimensional coordinate description for the octahedron by takingĪdvantage of the fact that the octahedron is the dual of the cube: the Regular three-dimensional octahedron thought of as the middle slice ofĪ hypercube in four-dimensional space. We have just obtained a set of coordinates for the vertices of a The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality.Show Extra Information Links Hide Extra Information Links Editorial Version Chapter 8 : Coordinate Geometry Coordinates for Regular Polyhedra Images from Kepler's Harmonices Mundi (1619) In general this creates only a topological dual. Kinds of duality The dual of a Platonic solid can be constructed by connecting the face centers. The dual of an isotoxal polyhedron (one in which any two edges are equivalent ) is also isotoxal.ĭuality is closely related to polar reciprocity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron. The dual of an isogonal polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is an isohedral polyhedron (one in which any two faces are equivalent ), and vice versa. For example, the regular polyhedra – the (convex) Platonic solids and (star) Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. Starting with any given polyhedron, the dual of its dual is the original polyhedron.ĭuality preserves the symmetries of a polyhedron. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. 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. Vertices of one correspond to faces of the other, and edges correspond to each other. Polyhedron associated with another by swapping vertices for faces The dual of a cube is an octahedron. ![]()
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