Category:Polyhedra-vf image

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This category lists images of vertex figures for the set of uniform polyhedra. Every uniform polyhedron can be generated by a single vertex figure.

Images are used for two list articles and individual articles for each polyhedron.

• List of uniform polyhedra
• List of Wenninger polyhedron models

[edit] Technical details on construction

The vertex figure is a pyramid, defined by a polyhedron vertex at the pyramid apex and the N edge-neighboring vertices a the pyramid base. The base vertices all exist on a plane and on the circumference of a circle.

The lengths of the pyramid base edges are determined by the distance between alternate points on the polygon faces. (Colored faces shown but intersecting faces are not drawn correctly divided.)

The faces on the vertex can be regular polygons {p} or regular star polygons {p/q}. The sequence of faces can go forwards or backwards around the pyramid.

A general vertex figure can be given as a sequence of n polygons and m is the Winding number around the pyramid.

• (p1/q1,p2/q2,p3/q3,...,pn/qn)/m
  • Where p/q is a polygon face of p sides and q turns.
  • Backwards winding faces are represented by {p/(p-q)}. Example {3/2} is a backwards winding triangle {3/1} and {5/3} is a backwards pentagram {5/2}.
  • The winding number m can be zero or greater and is usually only shown if it is two or greater.

Some uniform polyhedra (with two forward and two identical backwards faces) have a winding number of zero and require an explicit base circle radius for it to be contrained to a unique geometry. Example (4.⁸/₃.⁴/₃.⁸/₅) Image:Great_rhombihexahedron_vertfig.png

NOTE: Star polygon faces have an ambiguous "interior" definition. They can be drawn in two ways: (1) Filled by an even/odd intersection interior rule, or (2) Calculating new vertices on intersecting edges and filling all interior surfaces of the resulting net. These images are drawn the first way, while polyhedral model builders like Wenninger construct edge (and face) intersections and use the second rule. The underlying geometry is the same - it's just a slightly different visualization.

[edit] External links

• http://home.aanet.com.au/robertw/VertexDesc.html Consistent Vertex Descriptions
• http://web.ukonline.co.uk/polyhedra/uniform/uniform.html Uniform Polyhedra (By Vertex Figures)
• http://www.queenhill.demon.co.uk/polyhedra/filling/filling.htm Filling polytopes