Construction of Wythoff

In Geometry, a construction of Wythoff , named in the honor of the Mathematician Willem Abraham Wythoff, is a method to build a Polyèdre uniform or a plane Pavage. It is often refers as a kaleidoscopic construction of Wythoff.

It is based on the idea of the Pavage of a Sphère, with spherical triangles. If three mirrors were placed, i.e. their plans are cut in a single point, then the mirrors surround a spherical triangle on surface of an unspecified sphere centered on this point and by repeating these reflections, one obtains a multitude of copies of the triangle. If the angles of the spherical triangle are selected in an adapted way, the triangles will pave the sphere, one or more time.

If one places a top at a suitable point in the spherical triangle surrounded by the mirrors, it is possible to make sure that the reflections of this point will produce a uniform polyhedron. For a spherical triangle ABC , we have four possibilities which will produce a uniform polyhedron:

  1. a top is placed at the point has . This produced a polyhedron with the Symbole of Wythoff has | B   C , where has equalizes π divided by the angle of the triangle into has , and in a similar way for B and C .
  2. a top is placed on a point of the segment AB i.e. it division the angle in C . This produced a polyhedron with the symbol of Wythoff has   B | C .
  3. a top is placed on the center of ABC . This produced a polyhedron with a symbol of Wythoff has   B   C |.
  4. the top is on a point such as, when it twice undergoes a rotation of the angle at this point around an unspecified corner of the triangle, it is moved same distance for each angle. Only the even reflections of the original top are used. The polyhedron has the symbol of Wythoff | has   B   C .

The process in general also applies for regular polytopes of higher size, including the uniform polychores four-dimensional.

See also: Symbol of Wythoff

References

  • Coxeter Regular Polytopes , Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff' S construction)
  • Coxeter The Beauty off Geometry: Twelve Essays , Dover Publications, 1999, ISBN 0-486-40919-8 (Chapter 3: Wythoff' S Construction for Uniform Polytopes)
  • W.A. Wythoff, has relation between the polytopes off the C600-family , Koninklijke Akademie van Wetenschappen you Amsterdam, Proceedings off the Section off Sciences, 20 (1918) 966-970.

External bonds

  • Poster uniform polyhedrons by using the method of construction of Wythoff
  • Description of constructions of Wythoff
  • " Jenn" , Logiciel which generates sights of polyhedrons (spherical) and polychores starting from the groups of symmetry

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