| Rectification (geometry) |
Article Index for Rectification |
Information AboutRectification (geometry) |
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- edges reduced to vertices, and vertices expanded into new cells.]] In Euclidean Geometry , rectification is the process of truncating a Polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by the Vertex Figures and the rectified facets of the original polytope. EXAMPLE OF RECTIFICATION AS A FINAL TRUNCATION TO AN EDGE Rectification is the final point of a truncation process. For example on a cube this sequence shows four steps of a continuum of truncations between the regular and rectified form: Higher order rectification can be performed on higher dimensional regular polytopes. The highest order of rectification creates the dual polytope. A rectification truncates edges to points. A birectification truncates faces to points. A trirectification truncates cells to points. EXAMPLE OF BIRECTIFICATION AS A FINAL TRUNCATION TO A FACE This sequence shows a ''birectified cube'' as the final sequence from a cube to the dual where the original faces are truncated down to a single point: : IN POLYGONS The dual of a polygon is the same as its rectified form. IN POLYHEDRONS AND PLANE TILINGS Each Platonic Solid and its Dual have the same rectified polyhedron. (This is not true of polytopes in higher dimensions.) The rectified polyhedron turns out to be expressible as the intersection of the original platonic solid with an appropriated scaled concentric version of its dual. For this reason, its name is a combination of the names of the original and the dual: # The rectified Tetrahedron , whose dual is the tetrahedron, is the ''tetratetrahedron'', better known as the Octahedron . # The rectified Octahedron , whose dual is the Cube , is the Cuboctahedron . # The rectified Icosahedron , whose dual is the Dodecahedron , is the Icosidodecahedron . # A rectified Square Tiling is a Square Tiling . # A rectified Triangular Tiling or Hexagonal Tiling is a Trihexagonal Tiling . Examples IN POLYCHORA AND 3D HONEYCOMB TESSELLATIONS Each Convex Regular Polychoron has a rectified form as a Uniform Polychoron . A regular polychoron {p,q,r} has cells {p,q}. Its rectification will have two cell types, a rectified {p,q} polyhedron left from the original cells and {q,r} polyhedron as new cells formed by each truncated vertex. A rectified {p,q,r} is not the same as a rectified {r,q,p}, however. A further truncation, called Bitruncation , is symmetric between a polychoron and its dual. See Uniform_polychoron#Geometric_derivations . Examples ORDERS OF RECTIFICATION A first order rectification truncates edges down to points. If a polytope is Regular , this form is represented by an extended Schläfli Symbol notation t1{p,q,...}. A second order rectification, or birectification, truncates Faces down to points. If regular it has notation t2{p,q,...}. For Polyhedra , a birectification creates a Dual Polyhedron . Higher order rectifications can be constructed for higher order polytopes. In general an n-rectification truncates ''n-faces'' to points. If an n-polytope is (n-1)-rectified, its Facets are reduced to points and the polytope becomes its Dual . Notations and facets There are different equivalent notations for each order of rectification. These tables show the names by dimension and the two type of Facet s for each. Regular Polygon s Facet s are edges, represented as {2}. Regular Polyhedra and Tiling s Facet s are regular polygons. Regular Polychora and Honeycomb s Facet s are regular or rectified polyhedra. Regular Polyteron s and Tetracomb s Facet s are regular or rectified polychora. SEE ALSO
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